id sid tid token lemma pos ejpam-87 1 1 european european PROPN ejpam-87 1 2 journal journal PROPN ejpam-87 1 3 of of ADP ejpam-87 1 4 pure pure ADJ ejpam-87 1 5 and and CCONJ ejpam-87 1 6 applied apply VERB ejpam-87 1 7 mathematics mathematic NOUN ejpam-87 1 8 vol vol NOUN ejpam-87 1 9 . . PROPN ejpam-87 2 1 1 1 NUM ejpam-87 2 2 , , PUNCT ejpam-87 2 3 no no INTJ ejpam-87 2 4 . . NOUN ejpam-87 2 5 1 1 NUM ejpam-87 2 6 , , PUNCT ejpam-87 2 7 2008 2008 NUM ejpam-87 2 8 , , PUNCT ejpam-87 2 9 ( ( PUNCT ejpam-87 2 10 60 60 NUM ejpam-87 2 11 - - SYM ejpam-87 2 12 81 81 NUM ejpam-87 2 13 ) ) PUNCT ejpam-87 2 14 issn issn PROPN ejpam-87 2 15 1307 1307 NUM ejpam-87 2 16 - - SYM ejpam-87 2 17 5543 5543 NUM ejpam-87 2 18 – – PUNCT ejpam-87 2 19 www.ejpam.com www.ejpam.com X ejpam-87 2 20 honorary honorary PROPN ejpam-87 2 21 invited invite VERB ejpam-87 2 22 paper paper NOUN ejpam-87 2 23 sharp sharp ADJ ejpam-87 2 24 bounds bound NOUN ejpam-87 2 25 for for ADP ejpam-87 2 26 the the DET ejpam-87 2 27 probability probability NOUN ejpam-87 2 28 of of ADP ejpam-87 2 29 the the DET ejpam-87 2 30 union union NOUN ejpam-87 2 31 of of ADP ejpam-87 2 32 events event NOUN ejpam-87 2 33 under under ADP ejpam-87 2 34 unimodality unimodality NOUN ejpam-87 2 35 condition condition NOUN ejpam-87 2 36 andrás andrás PROPN ejpam-87 2 37 prékopa∗ prékopa∗ PROPN ejpam-87 2 38 , , PUNCT ejpam-87 2 39 mine mine NOUN ejpam-87 2 40 subasi subasi NOUN ejpam-87 2 41 , , PUNCT ejpam-87 2 42 ersoy ersoy NOUN ejpam-87 2 43 subasi subasi NOUN ejpam-87 2 44 rutcor rutcor NOUN ejpam-87 2 45 , , PUNCT ejpam-87 2 46 rutgers rutgers PROPN ejpam-87 2 47 center center PROPN ejpam-87 2 48 for for ADP ejpam-87 2 49 operations operation NOUN ejpam-87 2 50 research research NOUN ejpam-87 2 51 , , PUNCT ejpam-87 2 52 640 640 NUM ejpam-87 2 53 bartholomew bartholomew NOUN ejpam-87 2 54 road road NOUN ejpam-87 2 55 piscataway piscataway PROPN ejpam-87 2 56 , , PUNCT ejpam-87 2 57 nj nj PROPN ejpam-87 2 58 08854 08854 NUM ejpam-87 2 59 - - SYM ejpam-87 2 60 8003 8003 NUM ejpam-87 2 61 , , PUNCT ejpam-87 2 62 usa usa PROPN ejpam-87 2 63 . . PROPN ejpam-87 2 64 abstract abstract PROPN ejpam-87 2 65 . . PUNCT ejpam-87 3 1 linear linear ADJ ejpam-87 3 2 programming programming NOUN ejpam-87 3 3 problem problem NOUN ejpam-87 3 4 is be AUX ejpam-87 3 5 formulated formulate VERB ejpam-87 3 6 for for ADP ejpam-87 3 7 bounding bound VERB ejpam-87 3 8 the the DET ejpam-87 3 9 probability probability NOUN ejpam-87 3 10 of of ADP ejpam-87 3 11 the the DET ejpam-87 3 12 union union NOUN ejpam-87 3 13 of of ADP ejpam-87 3 14 events event NOUN ejpam-87 3 15 , , PUNCT ejpam-87 3 16 where where SCONJ ejpam-87 3 17 the the DET ejpam-87 3 18 probability probability NOUN ejpam-87 3 19 distribution distribution NOUN ejpam-87 3 20 of of ADP ejpam-87 3 21 the the DET ejpam-87 3 22 occurrences occurrence NOUN ejpam-87 3 23 is be AUX ejpam-87 3 24 supposed suppose VERB ejpam-87 3 25 to to PART ejpam-87 3 26 be be AUX ejpam-87 3 27 unimodal unimodal ADJ ejpam-87 3 28 with with ADP ejpam-87 3 29 known know VERB ejpam-87 3 30 mode mode NOUN ejpam-87 3 31 and and CCONJ ejpam-87 3 32 some some PRON ejpam-87 3 33 of of ADP ejpam-87 3 34 the the DET ejpam-87 3 35 binomial binomial ADJ ejpam-87 3 36 moments moment NOUN ejpam-87 3 37 of of ADP ejpam-87 3 38 the the DET ejpam-87 3 39 events event NOUN ejpam-87 3 40 are be AUX ejpam-87 3 41 also also ADV ejpam-87 3 42 known know VERB ejpam-87 3 43 . . PUNCT ejpam-87 4 1 using use VERB ejpam-87 4 2 a a DET ejpam-87 4 3 theorem theorem NOUN ejpam-87 4 4 on on ADP ejpam-87 4 5 combinatorial combinatorial ADJ ejpam-87 4 6 determinants determinant NOUN ejpam-87 4 7 the the DET ejpam-87 4 8 dual dual ADJ ejpam-87 4 9 feasible feasible ADJ ejpam-87 4 10 bases basis NOUN ejpam-87 4 11 of of ADP ejpam-87 4 12 a a DET ejpam-87 4 13 relaxed relaxed ADJ ejpam-87 4 14 problem problem NOUN ejpam-87 4 15 are be AUX ejpam-87 4 16 fully fully ADV ejpam-87 4 17 described describe VERB ejpam-87 4 18 . . PUNCT ejpam-87 5 1 the the DET ejpam-87 5 2 bounds bound NOUN ejpam-87 5 3 for for ADP ejpam-87 5 4 the the DET ejpam-87 5 5 probability probability NOUN ejpam-87 5 6 of of ADP ejpam-87 5 7 the the DET ejpam-87 5 8 union union NOUN ejpam-87 5 9 are be AUX ejpam-87 5 10 presented present VERB ejpam-87 5 11 in in ADP ejpam-87 5 12 the the DET ejpam-87 5 13 form form NOUN ejpam-87 5 14 of of ADP ejpam-87 5 15 formulas formula NOUN ejpam-87 5 16 as as ADV ejpam-87 5 17 well well ADV ejpam-87 5 18 as as ADP ejpam-87 5 19 the the DET ejpam-87 5 20 results result NOUN ejpam-87 5 21 of of ADP ejpam-87 5 22 customized customize VERB ejpam-87 5 23 algorithmic algorithmic ADJ ejpam-87 5 24 solution solution NOUN ejpam-87 5 25 of of ADP ejpam-87 5 26 the the DET ejpam-87 5 27 lp lp NOUN ejpam-87 5 28 ’s ’ VERB ejpam-87 5 29 involved involve VERB ejpam-87 5 30 . . PUNCT ejpam-87 6 1 ams am NOUN ejpam-87 6 2 subject subject ADJ ejpam-87 6 3 classifications classification NOUN ejpam-87 6 4 : : PUNCT ejpam-87 6 5 90c05 90c05 NUM ejpam-87 6 6 , , PUNCT ejpam-87 6 7 90b25 90b25 NOUN ejpam-87 6 8 , , PUNCT ejpam-87 6 9 60e15 60e15 NUM ejpam-87 6 10 . . PUNCT ejpam-87 7 1 key key ADJ ejpam-87 7 2 words word NOUN ejpam-87 7 3 : : PUNCT ejpam-87 7 4 binomial binomial ADJ ejpam-87 7 5 moment moment NOUN ejpam-87 7 6 problem problem NOUN ejpam-87 7 7 , , PUNCT ejpam-87 7 8 linear linear ADJ ejpam-87 7 9 programming programming NOUN ejpam-87 7 10 , , PUNCT ejpam-87 7 11 bounding bound VERB ejpam-87 7 12 probabilities probability NOUN ejpam-87 7 13 , , PUNCT ejpam-87 7 14 discrete discrete ADJ ejpam-87 7 15 unimodality unimodality NOUN ejpam-87 7 16 , , PUNCT ejpam-87 7 17 reliability reliability NOUN ejpam-87 7 18 . . PUNCT ejpam-87 8 1 1 1 X ejpam-87 8 2 . . X ejpam-87 8 3 introduction introduction NOUN ejpam-87 8 4 let let VERB ejpam-87 8 5 a1 a1 PROPN ejpam-87 8 6 , , PUNCT ejpam-87 8 7 ... ... PUNCT ejpam-87 8 8 , , PUNCT ejpam-87 8 9 an an PRON ejpam-87 8 10 be be AUX ejpam-87 8 11 arbitrary arbitrary ADJ ejpam-87 8 12 events event NOUN ejpam-87 8 13 in in ADP ejpam-87 8 14 an an DET ejpam-87 8 15 arbitrary arbitrary ADJ ejpam-87 8 16 probability probability NOUN ejpam-87 8 17 space space NOUN ejpam-87 8 18 . . PUNCT ejpam-87 9 1 the the DET ejpam-87 9 2 kth kth PROPN ejpam-87 9 3 binomial binomial ADJ ejpam-87 9 4 moment moment NOUN ejpam-87 9 5 of of ADP ejpam-87 9 6 them they PRON ejpam-87 9 7 is be AUX ejpam-87 9 8 designated designate VERB ejpam-87 9 9 by by ADP ejpam-87 9 10 sk sk NOUN ejpam-87 9 11 and and CCONJ ejpam-87 9 12 is be AUX ejpam-87 9 13 defined define VERB ejpam-87 9 14 by by ADP ejpam-87 9 15 the the DET ejpam-87 9 16 equation equation NOUN ejpam-87 9 17 : : PUNCT ejpam-87 9 18 sk sk PROPN ejpam-87 9 19 = = PUNCT ejpam-87 9 20 ∑ ∑ NOUN ejpam-87 9 21 1≤i1< 1≤i1< PROPN ejpam-87 9 22 ... ... PUNCT ejpam-87 9 23 > X ejpam-87 132 7 0 0 PUNCT ejpam-87 133 1 and and CCONJ ejpam-87 133 2 if if SCONJ ejpam-87 133 3 0 0 NUM ejpam-87 133 4 ∈ ∈ PROPN ejpam-87 133 5 ib ib NOUN ejpam-87 133 6 , , PUNCT ejpam-87 133 7 then then ADV ejpam-87 133 8 the the DET ejpam-87 133 9 determinant determinant NOUN ejpam-87 133 10 that that PRON ejpam-87 133 11 comes come VERB ejpam-87 133 12 out out ADP ejpam-87 133 13 of of ADP ejpam-87 133 14 ∣∣∣ ∣∣∣ NOUN ejpam-87 134 1 cp cp INTJ ejpam-87 134 2 ct ct PROPN ejpam-87 134 3 b b PROPN ejpam-87 134 4 ap ap PROPN ejpam-87 134 5 b b PROPN ejpam-87 134 6 ∣∣∣ ∣∣∣ ADJ ejpam-87 134 7 , , PUNCT ejpam-87 134 8 if if SCONJ ejpam-87 134 9 we we PRON ejpam-87 134 10 put put VERB ejpam-87 134 11 ( ( PUNCT ejpam-87 134 12 cp cp INTJ ejpam-87 134 13 ap ap PROPN ejpam-87 134 14 ) ) PUNCT ejpam-87 134 15 in in ADP ejpam-87 134 16 its its PRON ejpam-87 134 17 right right ADJ ejpam-87 134 18 place place NOUN ejpam-87 134 19 ( ( PUNCT ejpam-87 134 20 the the DET ejpam-87 134 21 column column NOUN ejpam-87 134 22 subscripts subscript NOUN ejpam-87 134 23 are be AUX ejpam-87 134 24 in in ADP ejpam-87 134 25 increasing increase VERB ejpam-87 134 26 order order NOUN ejpam-87 134 27 ) ) PUNCT ejpam-87 134 28 , , PUNCT ejpam-87 134 29 is be AUX ejpam-87 134 30 also also ADV ejpam-87 134 31 positive positive ADJ ejpam-87 134 32 , , PUNCT ejpam-87 134 33 where where SCONJ ejpam-87 134 34 p p NOUN ejpam-87 134 35 is be AUX ejpam-87 134 36 a a DET ejpam-87 134 37 nonbasic nonbasic ADJ ejpam-87 134 38 subscript subscript NOUN ejpam-87 134 39 . . PUNCT ejpam-87 135 1 proof proof NOUN ejpam-87 135 2 of of ADP ejpam-87 135 3 the the DET ejpam-87 135 4 lemma lemma PROPN ejpam-87 135 5 . . PUNCT ejpam-87 136 1 since since SCONJ ejpam-87 136 2 b b PROPN ejpam-87 136 3 is be AUX ejpam-87 136 4 a a DET ejpam-87 136 5 basis basis NOUN ejpam-87 136 6 , , PUNCT ejpam-87 136 7 it it PRON ejpam-87 136 8 follows follow VERB ejpam-87 136 9 that that SCONJ ejpam-87 136 10 |b| |b| PROPN ejpam-87 136 11 6= 6= ADP ejpam-87 136 12 0 0 NUM ejpam-87 136 13 . . PUNCT ejpam-87 137 1 we we PRON ejpam-87 137 2 prove prove VERB ejpam-87 137 3 that that SCONJ ejpam-87 137 4 this this DET ejpam-87 137 5 value value NOUN ejpam-87 137 6 is be AUX ejpam-87 137 7 positive positive ADJ ejpam-87 137 8 . . PUNCT ejpam-87 138 1 the the DET ejpam-87 138 2 entries entry NOUN ejpam-87 138 3 in in ADP ejpam-87 138 4 the the DET ejpam-87 138 5 first first ADJ ejpam-87 138 6 row row NOUN ejpam-87 138 7 can can AUX ejpam-87 138 8 be be AUX ejpam-87 138 9 written write VERB ejpam-87 138 10 up up ADP ejpam-87 138 11 as as ADP ejpam-87 138 12 sum sum NOUN ejpam-87 138 13 of of ADP ejpam-87 138 14 1 1 NUM ejpam-87 138 15 ’s ’s NOUN ejpam-87 138 16 so so SCONJ ejpam-87 138 17 that that SCONJ ejpam-87 138 18 the the DET ejpam-87 138 19 number number NOUN ejpam-87 138 20 of of ADP ejpam-87 138 21 terms term NOUN ejpam-87 138 22 in in ADP ejpam-87 138 23 any any DET ejpam-87 138 24 position position NOUN ejpam-87 138 25 in in ADP ejpam-87 138 26 that that DET ejpam-87 138 27 row row NOUN ejpam-87 138 28 is be AUX ejpam-87 138 29 equal equal ADJ ejpam-87 138 30 to to ADP ejpam-87 138 31 the the DET ejpam-87 138 32 number number NOUN ejpam-87 138 33 of of ADP ejpam-87 138 34 terms term NOUN ejpam-87 138 35 in in ADP ejpam-87 138 36 any any DET ejpam-87 138 37 entry entry NOUN ejpam-87 138 38 in in ADP ejpam-87 138 39 its its PRON ejpam-87 138 40 column column NOUN ejpam-87 138 41 . . PUNCT ejpam-87 139 1 then then ADV ejpam-87 139 2 we we PRON ejpam-87 139 3 apply apply VERB ejpam-87 139 4 a a DET ejpam-87 139 5 column column NOUN ejpam-87 139 6 subtraction subtraction NOUN ejpam-87 139 7 procedure procedure NOUN ejpam-87 139 8 , , PUNCT ejpam-87 139 9 further far ADV ejpam-87 139 10 , , PUNCT ejpam-87 139 11 split split VERB ejpam-87 139 12 the the DET ejpam-87 139 13 obtained obtain VERB ejpam-87 139 14 determinant determinant ADJ ejpam-87 139 15 into into ADP ejpam-87 139 16 a a DET ejpam-87 139 17 sum sum NOUN ejpam-87 139 18 of of ADP ejpam-87 139 19 determinants determinant NOUN ejpam-87 139 20 . . PUNCT ejpam-87 140 1 any any DET ejpam-87 140 2 determinant determinant ADJ ejpam-87 140 3 in in ADP ejpam-87 140 4 the the DET ejpam-87 140 5 obtained obtain VERB ejpam-87 140 6 sum sum NOUN ejpam-87 140 7 is be AUX ejpam-87 140 8 either either DET ejpam-87 140 9 zero zero NUM ejpam-87 140 10 , , PUNCT ejpam-87 140 11 or or CCONJ ejpam-87 140 12 positive positive ADJ ejpam-87 140 13 , , PUNCT ejpam-87 140 14 by by ADP ejpam-87 140 15 prékopa prékopa NOUN ejpam-87 140 16 , , PUNCT ejpam-87 140 17 m. m. NOUN ejpam-87 140 18 subasi subasi PROPN ejpam-87 140 19 , , PUNCT ejpam-87 140 20 e. e. PROPN ejpam-87 140 21 subasi subasi PROPN ejpam-87 140 22 / / SYM ejpam-87 140 23 eur eur PROPN ejpam-87 140 24 . . PUNCT ejpam-87 141 1 j. j. PROPN ejpam-87 141 2 pure pure PROPN ejpam-87 141 3 appl appl PROPN ejpam-87 141 4 . . PROPN ejpam-87 141 5 math math PROPN ejpam-87 141 6 , , PUNCT ejpam-87 141 7 1 1 NUM ejpam-87 141 8 ( ( PUNCT ejpam-87 141 9 2008 2008 NUM ejpam-87 141 10 ) ) PUNCT ejpam-87 141 11 , , PUNCT ejpam-87 141 12 ( ( PUNCT ejpam-87 141 13 60 60 NUM ejpam-87 141 14 - - SYM ejpam-87 141 15 81 81 NUM ejpam-87 141 16 ) ) PUNCT ejpam-87 141 17 66 66 NUM ejpam-87 141 18 theorem theorem VERB ejpam-87 141 19 1 1 NUM ejpam-87 141 20 , , PUNCT ejpam-87 141 21 because because SCONJ ejpam-87 141 22 they they PRON ejpam-87 141 23 are be AUX ejpam-87 141 24 minors minor NOUN ejpam-87 141 25 , , PUNCT ejpam-87 141 26 crossed cross VERB ejpam-87 141 27 out out ADP ejpam-87 141 28 of of ADP ejpam-87 141 29 the the DET ejpam-87 141 30 matrix matrix NOUN ejpam-87 141 31 p p NOUN ejpam-87 141 32 . . PUNCT ejpam-87 142 1 at at ADV ejpam-87 142 2 least least ADV ejpam-87 142 3 one one NUM ejpam-87 142 4 term term NOUN ejpam-87 142 5 must must AUX ejpam-87 142 6 be be AUX ejpam-87 142 7 positive positive ADJ ejpam-87 142 8 because because SCONJ ejpam-87 142 9 |b| |b| PROPN ejpam-87 142 10 6= 6= ADP ejpam-87 142 11 0 0 NUM ejpam-87 142 12 . . PUNCT ejpam-87 143 1 it it PRON ejpam-87 143 2 follows follow VERB ejpam-87 143 3 that that SCONJ ejpam-87 143 4 |b| |b| PROPN ejpam-87 143 5 > > X ejpam-87 143 6 0 0 X ejpam-87 143 7 . . PUNCT ejpam-87 144 1 now now ADV ejpam-87 144 2 we we PRON ejpam-87 144 3 prove prove VERB ejpam-87 144 4 the the DET ejpam-87 144 5 second second ADJ ejpam-87 144 6 assertion assertion NOUN ejpam-87 144 7 . . PUNCT ejpam-87 145 1 if if SCONJ ejpam-87 145 2 ( ( PUNCT ejpam-87 145 3 cp cp INTJ ejpam-87 145 4 ap ap PROPN ejpam-87 145 5 ) ) PUNCT ejpam-87 145 6 is be AUX ejpam-87 145 7 put put VERB ejpam-87 145 8 in in ADP ejpam-87 145 9 its its PRON ejpam-87 145 10 right right ADJ ejpam-87 145 11 place place NOUN ejpam-87 145 12 , , PUNCT ejpam-87 145 13 then then ADV ejpam-87 145 14 the the DET ejpam-87 145 15 first first ADJ ejpam-87 145 16 column column NOUN ejpam-87 145 17 of of ADP ejpam-87 145 18 ( ( PUNCT ejpam-87 145 19 ct ct PROPN ejpam-87 145 20 a a PRON ejpam-87 145 21 ) ) PUNCT ejpam-87 145 22 will will AUX ejpam-87 145 23 be be AUX ejpam-87 145 24 the the DET ejpam-87 145 25 first first ADJ ejpam-87 145 26 column column NOUN ejpam-87 145 27 of of ADP ejpam-87 145 28 the the DET ejpam-87 145 29 new new ADJ ejpam-87 145 30 determinant determinant ADJ ejpam-87 145 31 . . PUNCT ejpam-87 146 1 if if SCONJ ejpam-87 146 2 we we PRON ejpam-87 146 3 subtract subtract VERB ejpam-87 146 4 the the DET ejpam-87 146 5 first first ADJ ejpam-87 146 6 row row NOUN ejpam-87 146 7 from from ADP ejpam-87 146 8 the the DET ejpam-87 146 9 second second ADJ ejpam-87 146 10 row row NOUN ejpam-87 146 11 in in ADP ejpam-87 146 12 the the DET ejpam-87 146 13 determinant determinant NOUN ejpam-87 146 14 , , PUNCT ejpam-87 146 15 then then ADV ejpam-87 146 16 the the DET ejpam-87 146 17 first first ADJ ejpam-87 146 18 entry entry NOUN ejpam-87 146 19 in in ADP ejpam-87 146 20 the the DET ejpam-87 146 21 second second ADJ ejpam-87 146 22 row row NOUN ejpam-87 146 23 becomes become VERB ejpam-87 146 24 −1 −1 NOUN ejpam-87 146 25 and and CCONJ ejpam-87 146 26 the the DET ejpam-87 146 27 others other NOUN ejpam-87 146 28 0 0 PUNCT ejpam-87 146 29 . . PUNCT ejpam-87 147 1 if if SCONJ ejpam-87 147 2 we we PRON ejpam-87 147 3 develop develop VERB ejpam-87 147 4 the the DET ejpam-87 147 5 determinant determinant NOUN ejpam-87 147 6 according accord VERB ejpam-87 147 7 to to ADP ejpam-87 147 8 the the DET ejpam-87 147 9 second second ADJ ejpam-87 147 10 row row NOUN ejpam-87 147 11 , , PUNCT ejpam-87 147 12 then then ADV ejpam-87 147 13 , , PUNCT ejpam-87 147 14 due due ADP ejpam-87 147 15 to to ADP ejpam-87 147 16 the the DET ejpam-87 147 17 special special ADJ ejpam-87 147 18 structure structure NOUN ejpam-87 147 19 of of ADP ejpam-87 147 20 the the DET ejpam-87 147 21 determinant determinant NOUN ejpam-87 147 22 , , PUNCT ejpam-87 147 23 we we PRON ejpam-87 147 24 obtain obtain VERB ejpam-87 147 25 a a DET ejpam-87 147 26 minor minor NOUN ejpam-87 147 27 of of ADP ejpam-87 147 28 order order NOUN ejpam-87 147 29 m m VERB ejpam-87 147 30 + + ADJ ejpam-87 147 31 1 1 NUM ejpam-87 147 32 crossed cross VERB ejpam-87 147 33 out out ADP ejpam-87 147 34 of of ADP ejpam-87 147 35 the the DET ejpam-87 147 36 matrix matrix NOUN ejpam-87 147 37 a. a. NOUN ejpam-87 147 38 if if SCONJ ejpam-87 147 39 ib ib X ejpam-87 147 40 = = PUNCT ejpam-87 147 41 { { PUNCT ejpam-87 147 42 0 0 PROPN ejpam-87 147 43 , , PUNCT ejpam-87 147 44 i1 i1 PROPN ejpam-87 147 45 , , PUNCT ejpam-87 147 46 ... ... PUNCT ejpam-87 147 47 , , PUNCT ejpam-87 147 48 i i PRON ejpam-87 147 49 m m VERB ejpam-87 147 50 } } PUNCT ejpam-87 147 51 , , PUNCT ejpam-87 147 52 where where SCONJ ejpam-87 147 53 1 1 NUM ejpam-87 147 54 ≤ ≤ NUM ejpam-87 147 55 i1 i1 PROPN ejpam-87 147 56 < < X ejpam-87 147 57 ... ... PUNCT ejpam-87 147 58 < < X ejpam-87 147 59 i i X ejpam-87 147 60 m m PROPN ejpam-87 147 61 , , PUNCT ejpam-87 147 62 then then ADV ejpam-87 147 63 the the DET ejpam-87 147 64 subscript subscript NOUN ejpam-87 147 65 set set NOUN ejpam-87 147 66 of of ADP ejpam-87 147 67 the the DET ejpam-87 147 68 columns column NOUN ejpam-87 147 69 of of ADP ejpam-87 147 70 the the DET ejpam-87 147 71 minors minor NOUN ejpam-87 147 72 is be AUX ejpam-87 147 73 { { PUNCT ejpam-87 147 74 i1 i1 PROPN ejpam-87 147 75 , , PUNCT ejpam-87 147 76 ... ... PUNCT ejpam-87 147 77 , , PUNCT ejpam-87 147 78 p p X ejpam-87 147 79 , , PUNCT ejpam-87 147 80 ... ... PUNCT ejpam-87 147 81 , , PUNCT ejpam-87 147 82 i i PRON ejpam-87 147 83 m m VERB ejpam-87 147 84 } } PUNCT ejpam-87 147 85 , , PUNCT ejpam-87 147 86 where where SCONJ ejpam-87 147 87 i1 i1 PROPN ejpam-87 147 88 < < X ejpam-87 147 89 ... ... PUNCT ejpam-87 148 1 < < X ejpam-87 148 2 p p X ejpam-87 148 3 < < X ejpam-87 148 4 ... ... PUNCT ejpam-87 148 5 < < X ejpam-87 148 6 i i PRON ejpam-87 148 7 m. m. NOUN ejpam-87 148 8 thus thus ADV ejpam-87 148 9 , , PUNCT ejpam-87 148 10 0 0 NUM ejpam-87 148 11 is be AUX ejpam-87 148 12 removed remove VERB ejpam-87 148 13 from from ADP ejpam-87 148 14 ib ib NOUN ejpam-87 148 15 and and CCONJ ejpam-87 148 16 p p NOUN ejpam-87 148 17 is be AUX ejpam-87 148 18 included include VERB ejpam-87 148 19 . . PUNCT ejpam-87 149 1 it it PRON ejpam-87 149 2 is be AUX ejpam-87 149 3 not not PART ejpam-87 149 4 difficult difficult ADJ ejpam-87 149 5 to to PART ejpam-87 149 6 see see VERB ejpam-87 149 7 that that SCONJ ejpam-87 149 8 the the DET ejpam-87 149 9 positivity positivity NOUN ejpam-87 149 10 of of ADP ejpam-87 149 11 |b| |b| PROPN ejpam-87 149 12 implies imply VERB ejpam-87 149 13 the the DET ejpam-87 149 14 positivity positivity NOUN ejpam-87 149 15 of of ADP ejpam-87 149 16 the the DET ejpam-87 149 17 minor minor ADJ ejpam-87 149 18 . . PUNCT ejpam-87 150 1 we we PRON ejpam-87 150 2 have have AUX ejpam-87 150 3 proved prove VERB ejpam-87 150 4 the the DET ejpam-87 150 5 lemma lemma PROPN ejpam-87 150 6 . . PUNCT ejpam-87 151 1 returning return VERB ejpam-87 151 2 to to ADP ejpam-87 151 3 the the DET ejpam-87 151 4 proof proof NOUN ejpam-87 151 5 of of ADP ejpam-87 151 6 the the DET ejpam-87 151 7 theorem theorem NOUN ejpam-87 151 8 2 2 NUM ejpam-87 151 9 , , PUNCT ejpam-87 151 10 consider consider VERB ejpam-87 151 11 first first ADV ejpam-87 151 12 the the DET ejpam-87 151 13 case case NOUN ejpam-87 151 14 0 0 NUM ejpam-87 151 15 /∈ /∈ SYM ejpam-87 152 1 ib ib INTJ ejpam-87 152 2 . . PUNCT ejpam-87 153 1 then then ADV ejpam-87 153 2 |b| |b| PROPN ejpam-87 153 3 > > X ejpam-87 153 4 0 0 PUNCT ejpam-87 153 5 and and CCONJ ejpam-87 153 6 ∣∣∣∣ ∣∣∣∣ NOUN ejpam-87 154 1 cp cp INTJ ejpam-87 154 2 ct ct PROPN ejpam-87 154 3 b b PROPN ejpam-87 154 4 ap ap PROPN ejpam-87 154 5 b b PROPN ejpam-87 154 6 ∣∣∣∣ ∣∣∣∣ PROPN ejpam-87 154 7 = = SYM ejpam-87 154 8 { { PUNCT ejpam-87 154 9 0 0 NUM ejpam-87 154 10 if if SCONJ ejpam-87 154 11 p p PROPN ejpam-87 154 12 6= 6= ADP ejpam-87 154 13 0 0 PUNCT ejpam-87 154 14 < < X ejpam-87 154 15 0 0 PUNCT ejpam-87 155 1 if if SCONJ ejpam-87 155 2 p p NOUN ejpam-87 155 3 = = NOUN ejpam-87 155 4 0 0 NUM ejpam-87 155 5 . . PUNCT ejpam-87 156 1 hence hence ADV ejpam-87 156 2 , , PUNCT ejpam-87 156 3 b b PROPN ejpam-87 156 4 is be AUX ejpam-87 156 5 a a DET ejpam-87 156 6 dual dual ADJ ejpam-87 156 7 feasible feasible ADJ ejpam-87 156 8 basis basis NOUN ejpam-87 156 9 in in ADP ejpam-87 156 10 the the DET ejpam-87 156 11 maximization maximization NOUN ejpam-87 156 12 problem problem NOUN ejpam-87 156 13 . . PUNCT ejpam-87 157 1 if if SCONJ ejpam-87 157 2 , , PUNCT ejpam-87 157 3 on on ADP ejpam-87 157 4 the the DET ejpam-87 157 5 other other ADJ ejpam-87 157 6 hand hand NOUN ejpam-87 157 7 , , PUNCT ejpam-87 157 8 0 0 NUM ejpam-87 157 9 ∈ ∈ NOUN ejpam-87 157 10 ib ib NOUN ejpam-87 157 11 , , PUNCT ejpam-87 157 12 then then ADV ejpam-87 157 13 still still ADV ejpam-87 157 14 |b| |b| X ejpam-87 157 15 > > X ejpam-87 157 16 0 0 PUNCT ejpam-87 157 17 and and CCONJ ejpam-87 157 18 by by ADP ejpam-87 157 19 the the DET ejpam-87 157 20 lemma lemma PROPN ejpam-87 157 21 , , PUNCT ejpam-87 157 22 the the DET ejpam-87 157 23 determinant∣∣∣ determinant∣∣∣ NOUN ejpam-87 157 24 cp cp INTJ ejpam-87 157 25 ct ct PROPN ejpam-87 157 26 b b PROPN ejpam-87 157 27 ap ap PROPN ejpam-87 157 28 b b PROPN ejpam-87 157 29 ∣∣∣ ∣∣∣ NOUN ejpam-87 157 30 is be AUX ejpam-87 157 31 equal equal ADJ ejpam-87 157 32 to to ADP ejpam-87 157 33 the the PRON ejpam-87 157 34 ( ( PUNCT ejpam-87 157 35 m m PROPN ejpam-87 157 36 + + ADJ ejpam-87 157 37 1 1 NUM ejpam-87 157 38 ) ) PUNCT ejpam-87 157 39 × × NOUN ejpam-87 157 40 ( ( PUNCT ejpam-87 157 41 m m VERB ejpam-87 157 42 + + ADJ ejpam-87 157 43 1 1 NUM ejpam-87 157 44 ) ) PUNCT ejpam-87 157 45 minor minor ADJ ejpam-87 157 46 taken take VERB ejpam-87 157 47 from from ADP ejpam-87 157 48 a a PRON ejpam-87 157 49 , , PUNCT ejpam-87 157 50 corresponding correspond VERB ejpam-87 157 51 to to ADP ejpam-87 157 52 the the DET ejpam-87 157 53 columns column NOUN ejpam-87 157 54 { { PUNCT ejpam-87 157 55 i1 i1 PROPN ejpam-87 157 56 , , PUNCT ejpam-87 157 57 ... ... PUNCT ejpam-87 157 58 , , PUNCT ejpam-87 157 59 p p X ejpam-87 157 60 , , PUNCT ejpam-87 157 61 ... ... PUNCT ejpam-87 157 62 , , PUNCT ejpam-87 157 63 i i PRON ejpam-87 157 64 m m VERB ejpam-87 157 65 } } PUNCT ejpam-87 157 66 , , PUNCT ejpam-87 157 67 multiplied multiply VERB ejpam-87 157 68 by by ADP ejpam-87 157 69 ( ( PUNCT ejpam-87 157 70 −1)h(p −1)h(p PROPN ejpam-87 157 71 ) ) PUNCT ejpam-87 157 72 , , PUNCT ejpam-87 157 73 where where SCONJ ejpam-87 157 74 h(p h(p NOUN ejpam-87 157 75 ) ) PUNCT ejpam-87 157 76 is be AUX ejpam-87 157 77 the the DET ejpam-87 157 78 number number NOUN ejpam-87 157 79 of of ADP ejpam-87 157 80 subscripts subscript NOUN ejpam-87 157 81 in in ADP ejpam-87 157 82 b b NOUN ejpam-87 157 83 that that PRON ejpam-87 157 84 are be AUX ejpam-87 157 85 smaller small ADJ ejpam-87 157 86 than than ADP ejpam-87 157 87 p. p. NOUN ejpam-87 157 88 the the DET ejpam-87 157 89 minor minor ADJ ejpam-87 157 90 is be AUX ejpam-87 157 91 positive positive ADJ ejpam-87 157 92 by by ADP ejpam-87 157 93 the the DET ejpam-87 157 94 lemma lemma PROPN ejpam-87 157 95 . . PUNCT ejpam-87 158 1 we we PRON ejpam-87 158 2 want want VERB ejpam-87 158 3 to to PART ejpam-87 158 4 ensure ensure VERB ejpam-87 158 5 the the DET ejpam-87 158 6 positivity positivity NOUN ejpam-87 158 7 of of ADP ejpam-87 158 8 ∣∣∣ ∣∣∣ NOUN ejpam-87 159 1 cp cp INTJ ejpam-87 159 2 ct ct PROPN ejpam-87 159 3 b b PROPN ejpam-87 159 4 ap ap PROPN ejpam-87 159 5 b b PROPN ejpam-87 159 6 ∣∣∣ ∣∣∣ NOUN ejpam-87 159 7 for for ADP ejpam-87 159 8 any any DET ejpam-87 159 9 nonbasic nonbasic ADJ ejpam-87 159 10 p. p. NOUN ejpam-87 159 11 now now ADV ejpam-87 159 12 , , PUNCT ejpam-87 159 13 if if SCONJ ejpam-87 159 14 it it PRON ejpam-87 159 15 is be AUX ejpam-87 159 16 a a DET ejpam-87 159 17 minimization minimization NOUN ejpam-87 159 18 problem problem NOUN ejpam-87 159 19 , , PUNCT ejpam-87 159 20 then then ADV ejpam-87 159 21 h(p h(p NOUN ejpam-87 159 22 ) ) PUNCT ejpam-87 159 23 must must AUX ejpam-87 159 24 be be AUX ejpam-87 159 25 even even ADV ejpam-87 159 26 for for ADP ejpam-87 159 27 any any DET ejpam-87 159 28 nonbasic nonbasic NOUN ejpam-87 159 29 p p NOUN ejpam-87 159 30 which which PRON ejpam-87 159 31 implies imply VERB ejpam-87 159 32 that that SCONJ ejpam-87 159 33 { { PUNCT ejpam-87 159 34 i1 i1 PROPN ejpam-87 159 35 , , PUNCT ejpam-87 159 36 ... ... PUNCT ejpam-87 159 37 , , PUNCT ejpam-87 159 38 i i PRON ejpam-87 159 39 m m VERB ejpam-87 159 40 } } PUNCT ejpam-87 159 41 = = PRON ejpam-87 159 42 { { PUNCT ejpam-87 159 43 i i NOUN ejpam-87 159 44 , , PUNCT ejpam-87 159 45 i i PRON ejpam-87 159 46 + + NOUN ejpam-87 159 47 1 1 NUM ejpam-87 159 48 , , PUNCT ejpam-87 159 49 ... ... PUNCT ejpam-87 159 50 , , PUNCT ejpam-87 159 51 j j PROPN ejpam-87 159 52 , , PUNCT ejpam-87 159 53 j j PROPN ejpam-87 160 1 + + CCONJ ejpam-87 160 2 1 1 X ejpam-87 160 3 } } PUNCT ejpam-87 160 4 if if SCONJ ejpam-87 160 5 m m NOUN ejpam-87 160 6 is be AUX ejpam-87 160 7 even even ADV ejpam-87 160 8 ( ( PUNCT ejpam-87 160 9 m m VERB ejpam-87 160 10 + + ADJ ejpam-87 160 11 1 1 NUM ejpam-87 160 12 is be AUX ejpam-87 160 13 odd odd ADJ ejpam-87 160 14 ) ) PUNCT ejpam-87 160 15 and and CCONJ ejpam-87 160 16 { { PUNCT ejpam-87 160 17 i1 i1 PROPN ejpam-87 160 18 , , PUNCT ejpam-87 160 19 ... ... PUNCT ejpam-87 160 20 , , PUNCT ejpam-87 160 21 i i PRON ejpam-87 160 22 m m VERB ejpam-87 160 23 } } PUNCT ejpam-87 160 24 = = PRON ejpam-87 160 25 { { PUNCT ejpam-87 160 26 i i NOUN ejpam-87 160 27 , , PUNCT ejpam-87 160 28 i i PRON ejpam-87 160 29 + + NOUN ejpam-87 160 30 1 1 NUM ejpam-87 160 31 , , PUNCT ejpam-87 160 32 ... ... PUNCT ejpam-87 160 33 , , PUNCT ejpam-87 160 34 j j PROPN ejpam-87 160 35 , , PUNCT ejpam-87 160 36 j j PROPN ejpam-87 161 1 + + CCONJ ejpam-87 161 2 1 1 NUM ejpam-87 161 3 , , PUNCT ejpam-87 161 4 n n CCONJ ejpam-87 161 5 } } PUNCT ejpam-87 161 6 if if SCONJ ejpam-87 161 7 m m PROPN ejpam-87 161 8 is be AUX ejpam-87 161 9 odd odd ADJ ejpam-87 161 10 ( ( PUNCT ejpam-87 161 11 m m VERB ejpam-87 161 12 + + ADJ ejpam-87 161 13 1 1 NUM ejpam-87 161 14 is be AUX ejpam-87 161 15 even even ADV ejpam-87 161 16 ) ) PUNCT ejpam-87 161 17 . . PUNCT ejpam-87 162 1 if if SCONJ ejpam-87 162 2 it it PRON ejpam-87 162 3 is be AUX ejpam-87 162 4 a a DET ejpam-87 162 5 maximization maximization NOUN ejpam-87 162 6 problem problem NOUN ejpam-87 162 7 , , PUNCT ejpam-87 162 8 then then ADV ejpam-87 162 9 h(p h(p NOUN ejpam-87 162 10 ) ) PUNCT ejpam-87 162 11 must must AUX ejpam-87 162 12 be be AUX ejpam-87 162 13 odd odd ADJ ejpam-87 162 14 for for ADP ejpam-87 162 15 any any DET ejpam-87 162 16 nonbasic nonbasic NOUN ejpam-87 162 17 p p NOUN ejpam-87 162 18 which which PRON ejpam-87 162 19 implies imply VERB ejpam-87 162 20 that that SCONJ ejpam-87 162 21 { { PUNCT ejpam-87 162 22 i1 i1 PROPN ejpam-87 162 23 , , PUNCT ejpam-87 162 24 ... ... PUNCT ejpam-87 162 25 , , PUNCT ejpam-87 162 26 i i PRON ejpam-87 162 27 m m VERB ejpam-87 162 28 } } PUNCT ejpam-87 162 29 = = PUNCT ejpam-87 162 30 { { PUNCT ejpam-87 162 31 1 1 NUM ejpam-87 162 32 , , PUNCT ejpam-87 162 33 i i PRON ejpam-87 162 34 , , PUNCT ejpam-87 162 35 i i PRON ejpam-87 162 36 + + NOUN ejpam-87 162 37 1 1 NUM ejpam-87 162 38 , , PUNCT ejpam-87 162 39 ... ... PUNCT ejpam-87 162 40 , , PUNCT ejpam-87 162 41 j j PROPN ejpam-87 162 42 , , PUNCT ejpam-87 162 43 j j PROPN ejpam-87 162 44 + + CCONJ ejpam-87 162 45 1 1 NUM ejpam-87 162 46 , , PUNCT ejpam-87 162 47 n n CCONJ ejpam-87 162 48 } } PUNCT ejpam-87 162 49 if if SCONJ ejpam-87 162 50 m m NOUN ejpam-87 162 51 is be AUX ejpam-87 162 52 even even ADV ejpam-87 162 53 ( ( PUNCT ejpam-87 162 54 m m VERB ejpam-87 162 55 + + ADJ ejpam-87 162 56 1 1 NUM ejpam-87 162 57 is be AUX ejpam-87 162 58 odd odd ADJ ejpam-87 162 59 ) ) PUNCT ejpam-87 162 60 and and CCONJ ejpam-87 162 61 { { PUNCT ejpam-87 162 62 i1 i1 PROPN ejpam-87 162 63 , , PUNCT ejpam-87 162 64 ... ... PUNCT ejpam-87 162 65 , , PUNCT ejpam-87 162 66 i i PRON ejpam-87 162 67 m m VERB ejpam-87 162 68 } } PUNCT ejpam-87 162 69 = = PUNCT ejpam-87 162 70 { { PUNCT ejpam-87 162 71 1 1 NUM ejpam-87 162 72 , , PUNCT ejpam-87 162 73 i i PRON ejpam-87 162 74 , , PUNCT ejpam-87 162 75 i i PRON ejpam-87 162 76 + + NOUN ejpam-87 162 77 1 1 NUM ejpam-87 162 78 , , PUNCT ejpam-87 162 79 ... ... PUNCT ejpam-87 162 80 , , PUNCT ejpam-87 162 81 j j PROPN ejpam-87 162 82 , , PUNCT ejpam-87 162 83 j j PROPN ejpam-87 163 1 + + CCONJ ejpam-87 163 2 1 1 X ejpam-87 163 3 } } PUNCT ejpam-87 163 4 if if SCONJ ejpam-87 163 5 m m PROPN ejpam-87 163 6 is be AUX ejpam-87 163 7 odd odd ADJ ejpam-87 163 8 ( ( PUNCT ejpam-87 163 9 m m VERB ejpam-87 163 10 + + ADJ ejpam-87 163 11 1 1 NUM ejpam-87 163 12 is be AUX ejpam-87 163 13 even even ADV ejpam-87 163 14 ) ) PUNCT ejpam-87 163 15 . . PUNCT ejpam-87 164 1 this this PRON ejpam-87 164 2 proves prove VERB ejpam-87 164 3 the the DET ejpam-87 164 4 theorem theorem PROPN ejpam-87 164 5 . . PROPN ejpam-87 164 6 remark remark PROPN ejpam-87 164 7 . . PUNCT ejpam-87 165 1 if if SCONJ ejpam-87 165 2 kj kj PROPN ejpam-87 165 3 = = PROPN ejpam-87 165 4 j j PROPN ejpam-87 165 5 , , PUNCT ejpam-87 165 6 j j PROPN ejpam-87 165 7 = = SYM ejpam-87 165 8 1 1 NUM ejpam-87 165 9 , , PUNCT ejpam-87 165 10 ... ... PUNCT ejpam-87 165 11 , , PUNCT ejpam-87 165 12 m m PROPN ejpam-87 165 13 , , PUNCT ejpam-87 165 14 then then ADV ejpam-87 165 15 all all DET ejpam-87 165 16 ( ( PUNCT ejpam-87 165 17 m+1)×(m+1 m+1)×(m+1 PROPN ejpam-87 165 18 ) ) PUNCT ejpam-87 165 19 submatrices submatrice NOUN ejpam-87 165 20 of of ADP ejpam-87 165 21 a a PRON ejpam-87 165 22 are be AUX ejpam-87 165 23 nonsingular nonsingular ADJ ejpam-87 165 24 . . PUNCT ejpam-87 166 1 this this PRON ejpam-87 166 2 is be AUX ejpam-87 166 3 , , PUNCT ejpam-87 166 4 however however ADV ejpam-87 166 5 , , PUNCT ejpam-87 166 6 not not PART ejpam-87 166 7 necessarily necessarily ADV ejpam-87 166 8 the the DET ejpam-87 166 9 case case NOUN ejpam-87 166 10 if if SCONJ ejpam-87 166 11 { { PUNCT ejpam-87 166 12 k1 k1 NOUN ejpam-87 166 13 , , PUNCT ejpam-87 166 14 ... ... PUNCT ejpam-87 166 15 , , PUNCT ejpam-87 166 16 km km PROPN ejpam-87 166 17 } } PUNCT ejpam-87 166 18 6= 6= NUM ejpam-87 166 19 { { PUNCT ejpam-87 166 20 1 1 NUM ejpam-87 166 21 , , PUNCT ejpam-87 166 22 ... ... PUNCT ejpam-87 166 23 , , PUNCT ejpam-87 166 24 m m VERB ejpam-87 166 25 } } PUNCT ejpam-87 166 26 . . PUNCT ejpam-87 167 1 thus thus ADV ejpam-87 167 2 , , PUNCT ejpam-87 167 3 when when SCONJ ejpam-87 167 4 picking pick VERB ejpam-87 167 5 a a DET ejpam-87 167 6 dual dual ADJ ejpam-87 167 7 feasible feasible ADJ ejpam-87 167 8 basis basis NOUN ejpam-87 167 9 satisfying satisfy VERB ejpam-87 167 10 the the DET ejpam-87 167 11 structure structure NOUN ejpam-87 167 12 in in ADP ejpam-87 167 13 theorem theorem NOUN ejpam-87 167 14 2 2 NUM ejpam-87 167 15 , , PUNCT ejpam-87 167 16 we we PRON ejpam-87 167 17 have have VERB ejpam-87 167 18 to to PART ejpam-87 167 19 check check VERB ejpam-87 167 20 on on ADP ejpam-87 167 21 their their PRON ejpam-87 167 22 independence independence NOUN ejpam-87 167 23 as as ADV ejpam-87 167 24 well well ADV ejpam-87 167 25 . . PUNCT ejpam-87 168 1 3 3 X ejpam-87 168 2 . . NUM ejpam-87 168 3 closed close VERB ejpam-87 168 4 form form NOUN ejpam-87 168 5 bounds bound NOUN ejpam-87 168 6 for for ADP ejpam-87 168 7 the the DET ejpam-87 168 8 probability probability NOUN ejpam-87 168 9 of of ADP ejpam-87 168 10 the the DET ejpam-87 168 11 union union NOUN ejpam-87 168 12 based base VERB ejpam-87 168 13 on on ADP ejpam-87 168 14 sk1 sk1 PROPN ejpam-87 168 15 , , PUNCT ejpam-87 168 16 sk2 sk2 NOUN ejpam-87 168 17 let let VERB ejpam-87 168 18 m m NOUN ejpam-87 168 19 = = SYM ejpam-87 168 20 2 2 NUM ejpam-87 168 21 and and CCONJ ejpam-87 168 22 assume assume VERB ejpam-87 168 23 that that SCONJ ejpam-87 168 24 the the DET ejpam-87 168 25 binomial binomial ADJ ejpam-87 168 26 moments moment NOUN ejpam-87 168 27 sk1 sk1 PROPN ejpam-87 168 28 , , PUNCT ejpam-87 168 29 sk2 sk2 NOUN ejpam-87 168 30 , , PUNCT ejpam-87 168 31 1 1 NUM ejpam-87 168 32 ≤ ≤ NUM ejpam-87 168 33 k1 k1 NOUN ejpam-87 168 34 < < X ejpam-87 168 35 k2 k2 PROPN ejpam-87 168 36 ≤ ≤ PROPN ejpam-87 168 37 n n CCONJ ejpam-87 168 38 , , PUNCT ejpam-87 168 39 are be AUX ejpam-87 168 40 known know VERB ejpam-87 168 41 . . PUNCT ejpam-87 169 1 if if SCONJ ejpam-87 169 2 c(n c(n PROPN ejpam-87 169 3 , , PUNCT ejpam-87 169 4 k k NOUN ejpam-87 169 5 ) ) PUNCT ejpam-87 169 6 = = SYM ejpam-87 169 7 ( ( PUNCT ejpam-87 169 8 n n X ejpam-87 169 9 k k PROPN ejpam-87 169 10 ) ) PUNCT ejpam-87 169 11 , , PUNCT ejpam-87 169 12 then then ADV ejpam-87 169 13 we we PRON ejpam-87 169 14 have have VERB ejpam-87 169 15 the the DET ejpam-87 169 16 following follow VERB ejpam-87 169 17 recurrence recurrence NOUN ejpam-87 169 18 relation relation NOUN ejpam-87 169 19 known know VERB ejpam-87 169 20 as as ADP ejpam-87 169 21 pascal pascal PROPN ejpam-87 169 22 ’s ’s PART ejpam-87 169 23 rule rule NOUN ejpam-87 169 24 : : PUNCT ejpam-87 169 25 c(n c(n PROPN ejpam-87 169 26 + + CCONJ ejpam-87 169 27 1 1 NUM ejpam-87 169 28 , , PUNCT ejpam-87 169 29 k k PROPN ejpam-87 169 30 + + PROPN ejpam-87 169 31 1 1 X ejpam-87 169 32 ) ) PUNCT ejpam-87 169 33 = = SYM ejpam-87 170 1 c(n c(n PROPN ejpam-87 170 2 , , PUNCT ejpam-87 170 3 k k NOUN ejpam-87 170 4 ) ) PUNCT ejpam-87 171 1 + + CCONJ ejpam-87 171 2 c(n c(n PROPN ejpam-87 171 3 , , PUNCT ejpam-87 171 4 k k PROPN ejpam-87 171 5 + + PROPN ejpam-87 171 6 1 1 NUM ejpam-87 171 7 ) ) PUNCT ejpam-87 171 8 . . PUNCT ejpam-87 172 1 prékopa prékopa NOUN ejpam-87 172 2 , , PUNCT ejpam-87 172 3 m. m. NOUN ejpam-87 172 4 subasi subasi PROPN ejpam-87 172 5 , , PUNCT ejpam-87 172 6 e. e. PROPN ejpam-87 172 7 subasi subasi PROPN ejpam-87 172 8 / / SYM ejpam-87 172 9 eur eur PROPN ejpam-87 172 10 . . PUNCT ejpam-87 173 1 j. j. PROPN ejpam-87 173 2 pure pure PROPN ejpam-87 173 3 appl appl PROPN ejpam-87 173 4 . . PROPN ejpam-87 173 5 math math PROPN ejpam-87 173 6 , , PUNCT ejpam-87 173 7 1 1 NUM ejpam-87 173 8 ( ( PUNCT ejpam-87 173 9 2008 2008 NUM ejpam-87 173 10 ) ) PUNCT ejpam-87 173 11 , , PUNCT ejpam-87 173 12 ( ( PUNCT ejpam-87 173 13 60 60 NUM ejpam-87 173 14 - - SYM ejpam-87 173 15 81 81 NUM ejpam-87 173 16 ) ) PUNCT ejpam-87 173 17 67 67 NUM ejpam-87 173 18 by by ADP ejpam-87 173 19 the the DET ejpam-87 173 20 use use NOUN ejpam-87 173 21 of of ADP ejpam-87 173 22 these these PRON ejpam-87 173 23 the the DET ejpam-87 173 24 coefficients coefficient NOUN ejpam-87 173 25 of of ADP ejpam-87 173 26 the the DET ejpam-87 173 27 equality equality NOUN ejpam-87 173 28 constraints constraint NOUN ejpam-87 173 29 in in ADP ejpam-87 173 30 the the DET ejpam-87 173 31 relaxed relaxed ADJ ejpam-87 173 32 problems problem NOUN ejpam-87 173 33 can can AUX ejpam-87 173 34 be be AUX ejpam-87 173 35 given give VERB ejpam-87 173 36 as as SCONJ ejpam-87 173 37 follows follow VERB ejpam-87 173 38 : : PUNCT ejpam-87 173 39 j∑ j∑ PROPN ejpam-87 173 40 s s PROPN ejpam-87 174 1 = = VERB ejpam-87 174 2 i i PROPN ejpam-87 174 3 ( ( PUNCT ejpam-87 174 4 s s NOUN ejpam-87 174 5 k k X ejpam-87 174 6 ) ) PUNCT ejpam-87 174 7 = = SYM ejpam-87 175 1 ( ( PUNCT ejpam-87 175 2 j j PROPN ejpam-87 175 3 + + CCONJ ejpam-87 175 4 1 1 NUM ejpam-87 175 5 k k NOUN ejpam-87 175 6 + + NOUN ejpam-87 175 7 1 1 NUM ejpam-87 175 8 ) ) PUNCT ejpam-87 175 9 − − PROPN ejpam-87 176 1 ( ( PUNCT ejpam-87 176 2 i i PRON ejpam-87 176 3 k k PROPN ejpam-87 177 1 + + CCONJ ejpam-87 177 2 1 1 NUM ejpam-87 177 3 ) ) PUNCT ejpam-87 177 4 . . PUNCT ejpam-87 178 1 ( ( PUNCT ejpam-87 178 2 3.1 3.1 NUM ejpam-87 178 3 ) ) PUNCT ejpam-87 178 4 in in ADP ejpam-87 178 5 view view NOUN ejpam-87 178 6 of of ADP ejpam-87 178 7 ( ( PUNCT ejpam-87 178 8 3.1 3.1 NUM ejpam-87 178 9 ) ) PUNCT ejpam-87 178 10 the the DET ejpam-87 178 11 relaxed relaxed ADJ ejpam-87 178 12 version version NOUN ejpam-87 178 13 of of ADP ejpam-87 178 14 problem problem NOUN ejpam-87 178 15 ( ( PUNCT ejpam-87 178 16 1.5 1.5 NUM ejpam-87 178 17 ) ) PUNCT ejpam-87 178 18 can can AUX ejpam-87 178 19 be be AUX ejpam-87 178 20 written write VERB ejpam-87 178 21 in in ADP ejpam-87 178 22 the the DET ejpam-87 178 23 form form NOUN ejpam-87 178 24 : : PUNCT ejpam-87 178 25 min(max min(max X ejpam-87 178 26 ) ) PUNCT ejpam-87 178 27 { { PUNCT ejpam-87 178 28 mv0 mv0 NOUN ejpam-87 178 29 + + CCONJ ejpam-87 178 30 m∑ m∑ ADV ejpam-87 178 31 i=1 i=1 PROPN ejpam-87 178 32 ( ( PUNCT ejpam-87 178 33 m m VERB ejpam-87 178 34 − − NOUN ejpam-87 179 1 i i PRON ejpam-87 179 2 + + NUM ejpam-87 179 3 1)vi 1)vi PROPN ejpam-87 180 1 + + NUM ejpam-87 180 2 n∑ n∑ NOUN ejpam-87 180 3 i i NOUN ejpam-87 180 4 = = NOUN ejpam-87 180 5 m+1 m+1 X ejpam-87 180 6 ( ( PUNCT ejpam-87 180 7 i−m)vi i−m)vi PROPN ejpam-87 180 8 } } PUNCT ejpam-87 180 9 subject subject NOUN ejpam-87 180 10 to to ADP ejpam-87 180 11 m∑ m∑ CCONJ ejpam-87 180 12 i=0 i=0 VERB ejpam-87 180 13 ( ( PUNCT ejpam-87 180 14 m m VERB ejpam-87 180 15 − − NOUN ejpam-87 181 1 i i PRON ejpam-87 181 2 + + NUM ejpam-87 181 3 1)vi 1)vi PROPN ejpam-87 182 1 + + NUM ejpam-87 182 2 n∑ n∑ NOUN ejpam-87 182 3 i i NOUN ejpam-87 182 4 = = NOUN ejpam-87 182 5 m+1 m+1 X ejpam-87 182 6 ( ( PUNCT ejpam-87 182 7 i−m)vi i−m)vi PROPN ejpam-87 182 8 = = SYM ejpam-87 182 9 1 1 NUM ejpam-87 182 10 ( ( PUNCT ejpam-87 182 11 3.2 3.2 NUM ejpam-87 182 12 ) ) PUNCT ejpam-87 182 13 m∑ m∑ VERB ejpam-87 182 14 i=0 i=0 PROPN ejpam-87 183 1 [ [ X ejpam-87 183 2 ( ( PUNCT ejpam-87 183 3 m m VERB ejpam-87 183 4 + + ADJ ejpam-87 183 5 1 1 NUM ejpam-87 183 6 k1 k1 NOUN ejpam-87 183 7 + + CCONJ ejpam-87 183 8 1 1 NUM ejpam-87 183 9 ) ) PUNCT ejpam-87 183 10 − − PROPN ejpam-87 184 1 ( ( PUNCT ejpam-87 184 2 i i PRON ejpam-87 184 3 k1 k1 VERB ejpam-87 184 4 + + CCONJ ejpam-87 184 5 1 1 NUM ejpam-87 184 6 ) ) PUNCT ejpam-87 184 7 ] ] PUNCT ejpam-87 184 8 vi vi PROPN ejpam-87 185 1 + + CCONJ ejpam-87 185 2 n∑ n∑ NOUN ejpam-87 185 3 i i NOUN ejpam-87 185 4 = = NOUN ejpam-87 185 5 m+1 m+1 X ejpam-87 186 1 [ [ X ejpam-87 186 2 ( ( PUNCT ejpam-87 186 3 i i PRON ejpam-87 186 4 + + NOUN ejpam-87 186 5 1 1 NUM ejpam-87 186 6 k1 k1 NOUN ejpam-87 186 7 + + CCONJ ejpam-87 186 8 1 1 NUM ejpam-87 186 9 ) ) PUNCT ejpam-87 186 10 − − PROPN ejpam-87 187 1 ( ( PUNCT ejpam-87 187 2 m m VERB ejpam-87 187 3 + + NOUN ejpam-87 187 4 1 1 NUM ejpam-87 187 5 k1 k1 NOUN ejpam-87 187 6 + + CCONJ ejpam-87 187 7 1 1 NUM ejpam-87 187 8 ) ) PUNCT ejpam-87 187 9 ] ] PUNCT ejpam-87 187 10 vi vi PROPN ejpam-87 187 11 = = SYM ejpam-87 187 12 sk1 sk1 PROPN ejpam-87 187 13 m∑ m∑ VERB ejpam-87 187 14 i=0 i=0 PROPN ejpam-87 188 1 [ [ X ejpam-87 188 2 ( ( PUNCT ejpam-87 188 3 m m VERB ejpam-87 188 4 + + ADJ ejpam-87 188 5 1 1 NUM ejpam-87 188 6 k2 k2 NOUN ejpam-87 188 7 + + CCONJ ejpam-87 188 8 1 1 NUM ejpam-87 188 9 ) ) PUNCT ejpam-87 188 10 − − PROPN ejpam-87 189 1 ( ( PUNCT ejpam-87 189 2 i i PRON ejpam-87 189 3 k2 k2 PROPN ejpam-87 189 4 + + CCONJ ejpam-87 189 5 1 1 NUM ejpam-87 189 6 ) ) PUNCT ejpam-87 189 7 ] ] PUNCT ejpam-87 189 8 vi vi PROPN ejpam-87 190 1 + + CCONJ ejpam-87 190 2 n∑ n∑ NOUN ejpam-87 190 3 i i NOUN ejpam-87 190 4 = = NOUN ejpam-87 190 5 m+1 m+1 X ejpam-87 191 1 [ [ X ejpam-87 191 2 ( ( PUNCT ejpam-87 191 3 i i PRON ejpam-87 191 4 + + NOUN ejpam-87 191 5 1 1 NUM ejpam-87 191 6 k2 k2 NOUN ejpam-87 191 7 + + CCONJ ejpam-87 191 8 1 1 NUM ejpam-87 191 9 ) ) PUNCT ejpam-87 191 10 − − PROPN ejpam-87 191 11 ( ( PUNCT ejpam-87 191 12 m m VERB ejpam-87 191 13 + + PROPN ejpam-87 191 14 1 1 NUM ejpam-87 191 15 k2 k2 NOUN ejpam-87 191 16 + + CCONJ ejpam-87 191 17 1 1 NUM ejpam-87 191 18 ) ) PUNCT ejpam-87 191 19 ] ] PUNCT ejpam-87 191 20 vi vi NOUN ejpam-87 191 21 = = PUNCT ejpam-87 191 22 sk2 sk2 NOUN ejpam-87 191 23 vi vi PROPN ejpam-87 191 24 ≥ ≥ NOUN ejpam-87 191 25 0 0 NUM ejpam-87 191 26 , , PUNCT ejpam-87 191 27 i i PRON ejpam-87 191 28 = = NOUN ejpam-87 191 29 0 0 NUM ejpam-87 191 30 , , PUNCT ejpam-87 191 31 ... ... PUNCT ejpam-87 191 32 , , PUNCT ejpam-87 191 33 n n X ejpam-87 191 34 . . PUNCT ejpam-87 192 1 theorem theorem NOUN ejpam-87 192 2 2 2 NUM ejpam-87 192 3 provides provide VERB ejpam-87 192 4 us we PRON ejpam-87 192 5 with with ADP ejpam-87 192 6 the the DET ejpam-87 192 7 following follow VERB ejpam-87 192 8 dual dual ADJ ejpam-87 192 9 feasible feasible ADJ ejpam-87 192 10 bases basis NOUN ejpam-87 192 11 for for ADP ejpam-87 192 12 the the DET ejpam-87 192 13 above above ADJ ejpam-87 192 14 problem problem NOUN ejpam-87 192 15 : : PUNCT ejpam-87 192 16 bmin bmin PROPN ejpam-87 192 17 = = SYM ejpam-87 192 18 { { PUNCT ejpam-87 192 19 0 0 NUM ejpam-87 192 20 , , PUNCT ejpam-87 192 21 i i PRON ejpam-87 192 22 , , PUNCT ejpam-87 192 23 i i PRON ejpam-87 192 24 + + NOUN ejpam-87 192 25 1 1 NUM ejpam-87 192 26 } } PUNCT ejpam-87 192 27 , , PUNCT ejpam-87 192 28 1 1 NUM ejpam-87 192 29 ≤ ≤ NUM ejpam-87 193 1 i i PRON ejpam-87 193 2 ≤ ≤ ADJ ejpam-87 193 3 n− n− NOUN ejpam-87 193 4 1 1 NUM ejpam-87 193 5 , , PUNCT ejpam-87 193 6 bmax bmax VERB ejpam-87 193 7 = = PUNCT ejpam-87 193 8 { { PUNCT ejpam-87 193 9 0 0 NUM ejpam-87 193 10 , , PUNCT ejpam-87 193 11 1 1 NUM ejpam-87 193 12 , , PUNCT ejpam-87 193 13 n n CCONJ ejpam-87 193 14 } } PUNCT ejpam-87 193 15 or or CCONJ ejpam-87 193 16 bmax bmax VERB ejpam-87 193 17 ⊂ ⊂ PRON ejpam-87 193 18 { { PUNCT ejpam-87 193 19 1 1 NUM ejpam-87 193 20 , , PUNCT ejpam-87 193 21 ... ... PUNCT ejpam-87 193 22 , , PUNCT ejpam-87 193 23 n n CCONJ ejpam-87 193 24 } } PUNCT ejpam-87 193 25 . . PUNCT ejpam-87 194 1 prékopa prékopa ADJ ejpam-87 194 2 , , PUNCT ejpam-87 194 3 m. m. NOUN ejpam-87 194 4 subasi subasi PROPN ejpam-87 194 5 , , PUNCT ejpam-87 194 6 e. e. PROPN ejpam-87 194 7 subasi subasi PROPN ejpam-87 194 8 / / SYM ejpam-87 194 9 eur eur PROPN ejpam-87 194 10 . . PUNCT ejpam-87 195 1 j. j. PROPN ejpam-87 195 2 pure pure PROPN ejpam-87 195 3 appl appl PROPN ejpam-87 195 4 . . PROPN ejpam-87 195 5 math math PROPN ejpam-87 195 6 , , PUNCT ejpam-87 195 7 1 1 NUM ejpam-87 195 8 ( ( PUNCT ejpam-87 195 9 2008 2008 NUM ejpam-87 195 10 ) ) PUNCT ejpam-87 195 11 , , PUNCT ejpam-87 195 12 ( ( PUNCT ejpam-87 195 13 60 60 NUM ejpam-87 195 14 - - SYM ejpam-87 195 15 81 81 NUM ejpam-87 195 16 ) ) PUNCT ejpam-87 195 17 68 68 NUM ejpam-87 195 18 in in ADP ejpam-87 195 19 order order NOUN ejpam-87 195 20 to to PART ejpam-87 195 21 present present VERB ejpam-87 195 22 our our PRON ejpam-87 195 23 formulas formula NOUN ejpam-87 195 24 in in ADP ejpam-87 195 25 compact compact ADJ ejpam-87 195 26 forms form NOUN ejpam-87 195 27 we we PRON ejpam-87 195 28 introduce introduce VERB ejpam-87 195 29 the the DET ejpam-87 195 30 notations notation NOUN ejpam-87 195 31 : : PUNCT ejpam-87 195 32 σr σr PROPN ejpam-87 195 33 i i PROPN ejpam-87 195 34 , , PUNCT ejpam-87 195 35 j j PROPN ejpam-87 195 36 = = PRON ejpam-87 195 37 ( ( PUNCT ejpam-87 195 38 j j PROPN ejpam-87 195 39 − − PROPN ejpam-87 196 1 i i PRON ejpam-87 196 2 + + NOUN ejpam-87 196 3 1 1 X ejpam-87 196 4 ) ) PUNCT ejpam-87 196 5 ( ( PUNCT ejpam-87 196 6 i− i− PROPN ejpam-87 196 7 1 1 NUM ejpam-87 196 8 r r NOUN ejpam-87 196 9 ) ) PUNCT ejpam-87 196 10 − − PROPN ejpam-87 197 1 ( ( PUNCT ejpam-87 197 2 j j PROPN ejpam-87 197 3 + + CCONJ ejpam-87 197 4 1 1 NUM ejpam-87 197 5 r r NOUN ejpam-87 197 6 + + NOUN ejpam-87 197 7 1 1 NUM ejpam-87 197 8 ) ) PUNCT ejpam-87 197 9 + + CCONJ ejpam-87 197 10 ( ( PUNCT ejpam-87 197 11 i i PRON ejpam-87 197 12 r r VERB ejpam-87 197 13 + + NUM ejpam-87 197 14 1 1 NUM ejpam-87 197 15 ) ) PUNCT ejpam-87 197 16 σr σr PROPN ejpam-87 197 17 , , PUNCT ejpam-87 197 18 t t PROPN ejpam-87 198 1 i i PRON ejpam-87 198 2 , , PUNCT ejpam-87 198 3 j j PROPN ejpam-87 198 4 = = PRON ejpam-87 198 5 ( ( PUNCT ejpam-87 198 6 i− i− PROPN ejpam-87 198 7 1 1 NUM ejpam-87 198 8 r r NOUN ejpam-87 198 9 ) ) PUNCT ejpam-87 199 1 [ [ X ejpam-87 199 2 ( ( PUNCT ejpam-87 199 3 j j NOUN ejpam-87 199 4 + + CCONJ ejpam-87 199 5 1 1 NUM ejpam-87 199 6 t t NOUN ejpam-87 199 7 + + NOUN ejpam-87 199 8 1 1 NUM ejpam-87 199 9 ) ) PUNCT ejpam-87 199 10 − − PROPN ejpam-87 200 1 ( ( PUNCT ejpam-87 200 2 i i PRON ejpam-87 200 3 t t VERB ejpam-87 200 4 + + CCONJ ejpam-87 200 5 1 1 NUM ejpam-87 200 6 ) ) PUNCT ejpam-87 200 7 ] ] PUNCT ejpam-87 201 1 − − PROPN ejpam-87 201 2 ( ( PUNCT ejpam-87 201 3 i− i− PROPN ejpam-87 201 4 1 1 NUM ejpam-87 201 5 t t NOUN ejpam-87 201 6 ) ) PUNCT ejpam-87 202 1 [ [ X ejpam-87 202 2 ( ( PUNCT ejpam-87 202 3 j j NOUN ejpam-87 202 4 + + NOUN ejpam-87 202 5 1 1 NUM ejpam-87 202 6 r r NOUN ejpam-87 202 7 + + NOUN ejpam-87 202 8 1 1 NUM ejpam-87 202 9 ) ) PUNCT ejpam-87 202 10 − − PROPN ejpam-87 203 1 ( ( PUNCT ejpam-87 203 2 i i PRON ejpam-87 203 3 r r VERB ejpam-87 203 4 + + NOUN ejpam-87 203 5 1 1 NUM ejpam-87 203 6 ) ) PUNCT ejpam-87 203 7 ] ] PUNCT ejpam-87 204 1 γr γr PROPN ejpam-87 204 2 i i PRON ejpam-87 204 3 , , PUNCT ejpam-87 204 4 j j PROPN ejpam-87 204 5 = = SYM ejpam-87 204 6 i i PRON ejpam-87 205 1 [ [ X ejpam-87 205 2 ( ( PUNCT ejpam-87 205 3 j j NOUN ejpam-87 205 4 + + NOUN ejpam-87 205 5 1 1 NUM ejpam-87 205 6 r r NOUN ejpam-87 205 7 + + NOUN ejpam-87 205 8 1 1 NUM ejpam-87 205 9 ) ) PUNCT ejpam-87 205 10 − − PROPN ejpam-87 206 1 ( ( PUNCT ejpam-87 206 2 i i PRON ejpam-87 206 3 r r VERB ejpam-87 206 4 + + NOUN ejpam-87 206 5 1 1 NUM ejpam-87 206 6 ) ) PUNCT ejpam-87 206 7 ] ] PUNCT ejpam-87 207 1 − − PROPN ejpam-87 207 2 ( ( PUNCT ejpam-87 207 3 j j PROPN ejpam-87 207 4 − − PROPN ejpam-87 208 1 i i PRON ejpam-87 208 2 + + NOUN ejpam-87 208 3 1 1 X ejpam-87 208 4 ) ) PUNCT ejpam-87 208 5 ( ( PUNCT ejpam-87 208 6 i i PRON ejpam-87 208 7 r r VERB ejpam-87 208 8 + + NUM ejpam-87 208 9 1 1 NUM ejpam-87 208 10 ) ) PUNCT ejpam-87 208 11 γr γr PROPN ejpam-87 208 12 , , PUNCT ejpam-87 208 13 t t PROPN ejpam-87 208 14 i i PRON ejpam-87 208 15 , , PUNCT ejpam-87 208 16 j j PROPN ejpam-87 208 17 = = PRON ejpam-87 209 1 ( ( PUNCT ejpam-87 209 2 i i PRON ejpam-87 209 3 r r VERB ejpam-87 209 4 + + NOUN ejpam-87 209 5 1 1 NUM ejpam-87 209 6 ) ) PUNCT ejpam-87 210 1 [ [ X ejpam-87 210 2 ( ( PUNCT ejpam-87 210 3 j j NOUN ejpam-87 210 4 + + CCONJ ejpam-87 210 5 1 1 NUM ejpam-87 210 6 t t NOUN ejpam-87 210 7 + + NOUN ejpam-87 210 8 1 1 NUM ejpam-87 210 9 ) ) PUNCT ejpam-87 210 10 − − PROPN ejpam-87 211 1 ( ( PUNCT ejpam-87 211 2 i i PRON ejpam-87 211 3 t t VERB ejpam-87 211 4 + + CCONJ ejpam-87 211 5 1 1 NUM ejpam-87 211 6 ) ) PUNCT ejpam-87 211 7 ] ] PUNCT ejpam-87 212 1 − − PROPN ejpam-87 212 2 ( ( PUNCT ejpam-87 212 3 i i PRON ejpam-87 212 4 t t VERB ejpam-87 212 5 + + CCONJ ejpam-87 212 6 1 1 X ejpam-87 212 7 ) ) PUNCT ejpam-87 213 1 [ [ X ejpam-87 213 2 ( ( PUNCT ejpam-87 213 3 j j NOUN ejpam-87 213 4 + + NOUN ejpam-87 213 5 1 1 NUM ejpam-87 213 6 r r NOUN ejpam-87 213 7 + + NOUN ejpam-87 213 8 1 1 NUM ejpam-87 213 9 ) ) PUNCT ejpam-87 213 10 − − PROPN ejpam-87 214 1 ( ( PUNCT ejpam-87 214 2 i i PRON ejpam-87 214 3 r r VERB ejpam-87 214 4 + + NOUN ejpam-87 214 5 1 1 NUM ejpam-87 214 6 ) ) PUNCT ejpam-87 214 7 ] ] PUNCT ejpam-87 215 1 βr βr VERB ejpam-87 215 2 i i PRON ejpam-87 215 3 , , PUNCT ejpam-87 215 4 j j PROPN ejpam-87 215 5 = = PRON ejpam-87 215 6 ( ( PUNCT ejpam-87 215 7 j j PROPN ejpam-87 215 8 + + CCONJ ejpam-87 215 9 1 1 NUM ejpam-87 215 10 ) ) PUNCT ejpam-87 215 11 ( ( PUNCT ejpam-87 215 12 j j PROPN ejpam-87 215 13 + + CCONJ ejpam-87 215 14 1 1 NUM ejpam-87 215 15 r r NOUN ejpam-87 215 16 ) ) PUNCT ejpam-87 215 17 − − PROPN ejpam-87 216 1 ( ( PUNCT ejpam-87 216 2 j j PROPN ejpam-87 216 3 + + CCONJ ejpam-87 216 4 1 1 NUM ejpam-87 216 5 r r NOUN ejpam-87 216 6 + + NOUN ejpam-87 216 7 1 1 NUM ejpam-87 216 8 ) ) PUNCT ejpam-87 216 9 + + CCONJ ejpam-87 216 10 ( ( PUNCT ejpam-87 216 11 i i PRON ejpam-87 216 12 r r VERB ejpam-87 216 13 + + NUM ejpam-87 216 14 1 1 NUM ejpam-87 216 15 ) ) PUNCT ejpam-87 216 16 βr βr ADP ejpam-87 216 17 , , PUNCT ejpam-87 216 18 t t PROPN ejpam-87 216 19 i i PRON ejpam-87 216 20 , , PUNCT ejpam-87 216 21 j j PROPN ejpam-87 216 22 = = PRON ejpam-87 216 23 ( ( PUNCT ejpam-87 216 24 j j PROPN ejpam-87 217 1 + + CCONJ ejpam-87 217 2 1 1 NUM ejpam-87 217 3 r r NOUN ejpam-87 217 4 ) ) PUNCT ejpam-87 218 1 [ [ X ejpam-87 218 2 ( ( PUNCT ejpam-87 218 3 j j NOUN ejpam-87 218 4 + + CCONJ ejpam-87 218 5 1 1 NUM ejpam-87 218 6 t t NOUN ejpam-87 218 7 + + NOUN ejpam-87 218 8 1 1 NUM ejpam-87 218 9 ) ) PUNCT ejpam-87 218 10 − − PROPN ejpam-87 219 1 ( ( PUNCT ejpam-87 219 2 i i PRON ejpam-87 219 3 t t VERB ejpam-87 219 4 + + CCONJ ejpam-87 219 5 1 1 NUM ejpam-87 219 6 ) ) PUNCT ejpam-87 219 7 ] ] PUNCT ejpam-87 220 1 − − PROPN ejpam-87 220 2 ( ( PUNCT ejpam-87 220 3 j j PROPN ejpam-87 220 4 + + CCONJ ejpam-87 220 5 1 1 NUM ejpam-87 220 6 t t NOUN ejpam-87 220 7 ) ) PUNCT ejpam-87 221 1 [ [ X ejpam-87 221 2 ( ( PUNCT ejpam-87 221 3 j j NOUN ejpam-87 221 4 + + NOUN ejpam-87 221 5 1 1 NUM ejpam-87 221 6 r r NOUN ejpam-87 221 7 + + NOUN ejpam-87 221 8 1 1 NUM ejpam-87 221 9 ) ) PUNCT ejpam-87 221 10 − − PROPN ejpam-87 222 1 ( ( PUNCT ejpam-87 222 2 i i PRON ejpam-87 222 3 r r VERB ejpam-87 222 4 + + NOUN ejpam-87 222 5 1 1 NUM ejpam-87 222 6 ) ) PUNCT ejpam-87 222 7 ] ] PUNCT ejpam-87 223 1 αr αr X ejpam-87 223 2 i i PRON ejpam-87 223 3 , , PUNCT ejpam-87 223 4 j j PROPN ejpam-87 223 5 = = PRON ejpam-87 223 6 ( ( PUNCT ejpam-87 223 7 j j PROPN ejpam-87 223 8 − − PROPN ejpam-87 224 1 i i PRON ejpam-87 224 2 + + NOUN ejpam-87 224 3 1 1 X ejpam-87 224 4 ) ) PUNCT ejpam-87 224 5 ( ( PUNCT ejpam-87 224 6 j j PROPN ejpam-87 224 7 + + CCONJ ejpam-87 224 8 1 1 NUM ejpam-87 224 9 r r NOUN ejpam-87 224 10 ) ) PUNCT ejpam-87 224 11 − − PROPN ejpam-87 225 1 ( ( PUNCT ejpam-87 225 2 j j PROPN ejpam-87 225 3 + + CCONJ ejpam-87 225 4 1 1 NUM ejpam-87 225 5 r r NOUN ejpam-87 225 6 + + NOUN ejpam-87 225 7 1 1 NUM ejpam-87 225 8 ) ) PUNCT ejpam-87 225 9 + + CCONJ ejpam-87 225 10 ( ( PUNCT ejpam-87 225 11 i i PRON ejpam-87 225 12 r r VERB ejpam-87 225 13 + + CCONJ ejpam-87 225 14 1 1 NUM ejpam-87 225 15 ) ) PUNCT ejpam-87 225 16 αr αr PROPN ejpam-87 225 17 , , PUNCT ejpam-87 225 18 t t PROPN ejpam-87 226 1 i i PRON ejpam-87 226 2 , , PUNCT ejpam-87 226 3 j j PROPN ejpam-87 226 4 = = PRON ejpam-87 226 5 ( ( PUNCT ejpam-87 226 6 j j PROPN ejpam-87 226 7 + + CCONJ ejpam-87 226 8 1 1 NUM ejpam-87 226 9 r r NOUN ejpam-87 226 10 ) ) PUNCT ejpam-87 227 1 [ [ X ejpam-87 227 2 ( ( PUNCT ejpam-87 227 3 j j NOUN ejpam-87 227 4 + + CCONJ ejpam-87 227 5 1 1 NUM ejpam-87 227 6 t t NOUN ejpam-87 227 7 + + NOUN ejpam-87 227 8 1 1 NUM ejpam-87 227 9 ) ) PUNCT ejpam-87 227 10 − − PROPN ejpam-87 228 1 ( ( PUNCT ejpam-87 228 2 i i PRON ejpam-87 228 3 t t VERB ejpam-87 228 4 + + CCONJ ejpam-87 228 5 1 1 NUM ejpam-87 228 6 ) ) PUNCT ejpam-87 228 7 ] ] PUNCT ejpam-87 229 1 − − PROPN ejpam-87 229 2 ( ( PUNCT ejpam-87 229 3 j j PROPN ejpam-87 229 4 + + CCONJ ejpam-87 229 5 1 1 NUM ejpam-87 229 6 t t NOUN ejpam-87 229 7 ) ) PUNCT ejpam-87 230 1 [ [ X ejpam-87 230 2 ( ( PUNCT ejpam-87 230 3 j j NOUN ejpam-87 230 4 + + NOUN ejpam-87 230 5 1 1 NUM ejpam-87 230 6 r r NOUN ejpam-87 230 7 + + NOUN ejpam-87 230 8 1 1 NUM ejpam-87 230 9 ) ) PUNCT ejpam-87 230 10 − − PROPN ejpam-87 231 1 ( ( PUNCT ejpam-87 231 2 i i PRON ejpam-87 231 3 r r VERB ejpam-87 231 4 + + NOUN ejpam-87 231 5 1 1 NUM ejpam-87 231 6 ) ) PUNCT ejpam-87 231 7 ] ] PUNCT ejpam-87 231 8 δr δr ADP ejpam-87 231 9 i i PRON ejpam-87 231 10 , , PUNCT ejpam-87 231 11 j j PROPN ejpam-87 231 12 = = PRON ejpam-87 231 13 ( ( PUNCT ejpam-87 231 14 i− i− PROPN ejpam-87 231 15 1 1 NUM ejpam-87 231 16 ) ) PUNCT ejpam-87 232 1 [ [ X ejpam-87 232 2 ( ( PUNCT ejpam-87 232 3 j j NOUN ejpam-87 232 4 + + NOUN ejpam-87 232 5 1 1 NUM ejpam-87 232 6 r r NOUN ejpam-87 232 7 + + NOUN ejpam-87 232 8 1 1 NUM ejpam-87 232 9 ) ) PUNCT ejpam-87 232 10 − − PROPN ejpam-87 233 1 ( ( PUNCT ejpam-87 233 2 i i PRON ejpam-87 233 3 r r VERB ejpam-87 233 4 + + NOUN ejpam-87 233 5 1 1 NUM ejpam-87 233 6 ) ) PUNCT ejpam-87 233 7 ] ] PUNCT ejpam-87 234 1 − − PROPN ejpam-87 234 2 ( ( PUNCT ejpam-87 234 3 j j PROPN ejpam-87 234 4 − − PROPN ejpam-87 234 5 i i PROPN ejpam-87 234 6 ) ) PUNCT ejpam-87 235 1 ( ( PUNCT ejpam-87 235 2 i i PRON ejpam-87 235 3 r r VERB ejpam-87 235 4 + + NOUN ejpam-87 235 5 1 1 NUM ejpam-87 235 6 ) ) PUNCT ejpam-87 235 7 . . PUNCT ejpam-87 236 1 ( ( PUNCT ejpam-87 236 2 3.3 3.3 NUM ejpam-87 236 3 ) ) PUNCT ejpam-87 236 4 we we PRON ejpam-87 236 5 use use VERB ejpam-87 236 6 problem problem NOUN ejpam-87 236 7 ( ( PUNCT ejpam-87 236 8 3.2 3.2 NUM ejpam-87 236 9 ) ) PUNCT ejpam-87 236 10 to to PART ejpam-87 236 11 present present VERB ejpam-87 236 12 lower low ADJ ejpam-87 236 13 and and CCONJ ejpam-87 236 14 upper upper ADJ ejpam-87 236 15 bounds bound NOUN ejpam-87 236 16 for for ADP ejpam-87 236 17 p p PROPN ejpam-87 236 18 ( ( PUNCT ejpam-87 236 19 ν ν X ejpam-87 236 20 ≥ ≥ NOUN ejpam-87 236 21 1 1 NUM ejpam-87 236 22 ) ) PUNCT ejpam-87 236 23 . . PUNCT ejpam-87 237 1 to to PART ejpam-87 237 2 do do VERB ejpam-87 237 3 this this PRON ejpam-87 237 4 we we PRON ejpam-87 237 5 find find VERB ejpam-87 237 6 the the DET ejpam-87 237 7 optimal optimal ADJ ejpam-87 237 8 bases basis NOUN ejpam-87 237 9 for for ADP ejpam-87 237 10 the the DET ejpam-87 237 11 minimization minimization NOUN ejpam-87 237 12 and and CCONJ ejpam-87 237 13 maximization maximization NOUN ejpam-87 237 14 problems problem NOUN ejpam-87 237 15 , , PUNCT ejpam-87 237 16 respectively respectively ADV ejpam-87 237 17 . . PUNCT ejpam-87 238 1 we we PRON ejpam-87 238 2 already already ADV ejpam-87 238 3 have have VERB ejpam-87 238 4 a a DET ejpam-87 238 5 full full ADJ ejpam-87 238 6 description description NOUN ejpam-87 238 7 of of ADP ejpam-87 238 8 the the DET ejpam-87 238 9 dual dual ADJ ejpam-87 238 10 feasible feasible ADJ ejpam-87 238 11 bases basis NOUN ejpam-87 238 12 . . PUNCT ejpam-87 239 1 what what PRON ejpam-87 239 2 we we PRON ejpam-87 239 3 need need VERB ejpam-87 239 4 is be AUX ejpam-87 239 5 to to PART ejpam-87 239 6 find find VERB ejpam-87 239 7 those those PRON ejpam-87 239 8 ( ( PUNCT ejpam-87 239 9 one one NUM ejpam-87 239 10 for for ADP ejpam-87 239 11 the the DET ejpam-87 239 12 min min PROPN ejpam-87 239 13 problem problem NOUN ejpam-87 239 14 and and CCONJ ejpam-87 239 15 one one NUM ejpam-87 239 16 for for ADP ejpam-87 239 17 the the DET ejpam-87 239 18 max max PROPN ejpam-87 239 19 problem problem NOUN ejpam-87 239 20 ) ) PUNCT ejpam-87 239 21 that that PRON ejpam-87 239 22 are be AUX ejpam-87 239 23 also also ADV ejpam-87 239 24 primal primal ADJ ejpam-87 239 25 feasible feasible ADJ ejpam-87 239 26 . . PUNCT ejpam-87 240 1 three three NUM ejpam-87 240 2 cases case NOUN ejpam-87 240 3 will will AUX ejpam-87 240 4 be be AUX ejpam-87 240 5 considered consider VERB ejpam-87 240 6 . . PUNCT ejpam-87 241 1 case case NOUN ejpam-87 241 2 1 1 X ejpam-87 241 3 . . PUNCT ejpam-87 242 1 let let VERB ejpam-87 242 2 1 1 NUM ejpam-87 242 3 ≤ ≤ NUM ejpam-87 242 4 i i PRON ejpam-87 242 5 ≤ ≤ NOUN ejpam-87 242 6 m m VERB ejpam-87 242 7 − − PROPN ejpam-87 242 8 1 1 NUM ejpam-87 242 9 . . PUNCT ejpam-87 243 1 the the DET ejpam-87 243 2 primal primal ADJ ejpam-87 243 3 feasibility feasibility NOUN ejpam-87 243 4 conditions condition NOUN ejpam-87 243 5 for for ADP ejpam-87 243 6 bmin bmin NOUN ejpam-87 243 7 are be AUX ejpam-87 243 8 given give VERB ejpam-87 243 9 below below ADP ejpam-87 243 10 : : PUNCT ejpam-87 243 11 sk1σ sk1σ ADV ejpam-87 243 12 k2 k2 ADJ ejpam-87 243 13 i+1,m i+1,m NOUN ejpam-87 243 14 − − PROPN ejpam-87 243 15 sk2σ sk2σ ADJ ejpam-87 243 16 k1 k1 NOUN ejpam-87 243 17 i+1,m i+1,m NOUN ejpam-87 243 18 + + CCONJ ejpam-87 243 19 σk1,k2 σk1,k2 X ejpam-87 243 20 i+1,m i+1,m NOUN ejpam-87 243 21 ≥ ≥ PROPN ejpam-87 243 22 0 0 NUM ejpam-87 243 23 , , PUNCT ejpam-87 243 24 sk1γ sk1γ PROPN ejpam-87 243 25 k2 k2 PROPN ejpam-87 243 26 i+1,m i+1,m PROPN ejpam-87 243 27 − − PROPN ejpam-87 243 28 sk2γ sk2γ PROPN ejpam-87 243 29 k1 k1 PROPN ejpam-87 243 30 i+1,m i+1,m NOUN ejpam-87 243 31 − − PROPN ejpam-87 243 32 γk1,k2 γk1,k2 NOUN ejpam-87 243 33 i+1,m i+1,m ADJ ejpam-87 243 34 ≥ ≥ PROPN ejpam-87 243 35 0 0 NUM ejpam-87 243 36 , , PUNCT ejpam-87 243 37 sk1γ sk1γ PROPN ejpam-87 243 38 k2 k2 PROPN ejpam-87 243 39 i i PROPN ejpam-87 243 40 , , PUNCT ejpam-87 243 41 m m VERB ejpam-87 243 42 − − NOUN ejpam-87 243 43 sk2γ sk2γ PROPN ejpam-87 243 44 k1 k1 NOUN ejpam-87 244 1 i i PROPN ejpam-87 244 2 , , PUNCT ejpam-87 244 3 m m VERB ejpam-87 244 4 + + X ejpam-87 244 5 γk1,k2 γk1,k2 NOUN ejpam-87 244 6 i i PRON ejpam-87 244 7 , , PUNCT ejpam-87 244 8 m m VERB ejpam-87 244 9 ≤ ≤ ADJ ejpam-87 244 10 0 0 NUM ejpam-87 244 11 . . PUNCT ejpam-87 245 1 in in ADP ejpam-87 245 2 this this DET ejpam-87 245 3 case case NOUN ejpam-87 245 4 the the DET ejpam-87 245 5 closed closed ADJ ejpam-87 245 6 form form NOUN ejpam-87 245 7 lower lower ADV ejpam-87 245 8 bound bind VERB ejpam-87 245 9 for for ADP ejpam-87 245 10 p p PROPN ejpam-87 245 11 ( ( PUNCT ejpam-87 245 12 ν ν X ejpam-87 245 13 ≥ ≥ NOUN ejpam-87 245 14 1 1 NUM ejpam-87 245 15 ) ) PUNCT ejpam-87 245 16 is be AUX ejpam-87 245 17 expressed express VERB ejpam-87 245 18 by by ADP ejpam-87 245 19 1 1 NUM ejpam-87 245 20 − − PROPN ejpam-87 245 21 sk1σ sk1σ ADV ejpam-87 245 22 k2 k2 ADJ ejpam-87 245 23 i+1,m i+1,m NOUN ejpam-87 245 24 − − PROPN ejpam-87 245 25 sk2σ sk2σ ADJ ejpam-87 245 26 k1 k1 NOUN ejpam-87 245 27 i+1,m i+1,m NOUN ejpam-87 246 1 + + CCONJ ejpam-87 246 2 σk1,k2 σk1,k2 X ejpam-87 246 3 i+1,k i+1,k PROPN ejpam-87 246 4 i i PRON ejpam-87 246 5 σk1,k2 σk1,k2 VERB ejpam-87 246 6 i+1,m i+1,m VERB ejpam-87 247 1 + + CCONJ ejpam-87 247 2 ( ( PUNCT ejpam-87 247 3 i i PRON ejpam-87 247 4 k1 k1 VERB ejpam-87 247 5 + + CCONJ ejpam-87 247 6 1 1 X ejpam-87 247 7 ) ) PUNCT ejpam-87 247 8 σk2 σk2 NOUN ejpam-87 247 9 i+1,m i+1,m NOUN ejpam-87 247 10 − − PROPN ejpam-87 248 1 ( ( PUNCT ejpam-87 248 2 i i PRON ejpam-87 248 3 k2 k2 PROPN ejpam-87 248 4 + + CCONJ ejpam-87 248 5 1 1 X ejpam-87 248 6 ) ) PUNCT ejpam-87 248 7 σk1 σk1 AUX ejpam-87 248 8 i+1,m i+1,m NOUN ejpam-87 248 9 ≤ ≤ ADJ ejpam-87 249 1 p p NOUN ejpam-87 249 2 ( ( PUNCT ejpam-87 249 3 ν ν X ejpam-87 249 4 ≥ ≥ NOUN ejpam-87 249 5 1 1 NUM ejpam-87 249 6 ) ) PUNCT ejpam-87 249 7 , , PUNCT ejpam-87 249 8 ( ( PUNCT ejpam-87 249 9 3.4 3.4 NUM ejpam-87 249 10 ) ) PUNCT ejpam-87 249 11 where where SCONJ ejpam-87 249 12 σr σr PROPN ejpam-87 249 13 i i PROPN ejpam-87 249 14 , , PUNCT ejpam-87 249 15 j j PROPN ejpam-87 249 16 , , PUNCT ejpam-87 249 17 σ σ PROPN ejpam-87 249 18 r r PROPN ejpam-87 249 19 , , PUNCT ejpam-87 249 20 t t PROPN ejpam-87 249 21 i i PRON ejpam-87 249 22 , , PUNCT ejpam-87 249 23 j j PROPN ejpam-87 249 24 , , PUNCT ejpam-87 249 25 γ γ X ejpam-87 249 26 r r NOUN ejpam-87 249 27 i i PROPN ejpam-87 249 28 , , PUNCT ejpam-87 249 29 j j PROPN ejpam-87 249 30 , , PUNCT ejpam-87 249 31 γ γ X ejpam-87 249 32 r r PROPN ejpam-87 249 33 , , PUNCT ejpam-87 249 34 t t PROPN ejpam-87 250 1 i i PRON ejpam-87 250 2 , , PUNCT ejpam-87 250 3 j j PROPN ejpam-87 250 4 are be AUX ejpam-87 250 5 given give VERB ejpam-87 250 6 in in ADP ejpam-87 250 7 ( ( PUNCT ejpam-87 250 8 3.3 3.3 NUM ejpam-87 250 9 ) ) PUNCT ejpam-87 250 10 . . PUNCT ejpam-87 251 1 case case NOUN ejpam-87 251 2 2 2 X ejpam-87 251 3 . . PUNCT ejpam-87 252 1 let let VERB ejpam-87 252 2 i i PRON ejpam-87 252 3 = = NOUN ejpam-87 252 4 m m VERB ejpam-87 252 5 . . PUNCT ejpam-87 253 1 the the DET ejpam-87 253 2 conditions condition NOUN ejpam-87 253 3 that that PRON ejpam-87 253 4 ensure ensure VERB ejpam-87 253 5 the the DET ejpam-87 253 6 primal primal ADJ ejpam-87 253 7 feasibility feasibility NOUN ejpam-87 253 8 of of ADP ejpam-87 253 9 bmin bmin PROPN ejpam-87 253 10 = = SYM ejpam-87 253 11 { { PUNCT ejpam-87 253 12 0,m 0,m PROPN ejpam-87 253 13 , , PUNCT ejpam-87 253 14 m m VERB ejpam-87 253 15 + + ADJ ejpam-87 253 16 1 1 NUM ejpam-87 253 17 } } PUNCT ejpam-87 253 18 are be AUX ejpam-87 253 19 as as SCONJ ejpam-87 253 20 follows follow VERB ejpam-87 253 21 : : PUNCT ejpam-87 254 1 sk1 sk1 PROPN ejpam-87 254 2 ( ( PUNCT ejpam-87 254 3 m m PROPN ejpam-87 254 4 k2 k2 ADJ ejpam-87 254 5 − − PROPN ejpam-87 254 6 1 1 NUM ejpam-87 254 7 ) ) PUNCT ejpam-87 254 8 − − PROPN ejpam-87 255 1 sk2 sk2 PROPN ejpam-87 255 2 ( ( PUNCT ejpam-87 255 3 m m PROPN ejpam-87 255 4 k1 k1 NOUN ejpam-87 255 5 − − PROPN ejpam-87 255 6 1 1 NUM ejpam-87 255 7 ) ) PUNCT ejpam-87 255 8 − − PROPN ejpam-87 255 9 k2 k2 PROPN ejpam-87 255 10 − − PROPN ejpam-87 255 11 k1 k1 PROPN ejpam-87 255 12 m m PROPN ejpam-87 255 13 − − PROPN ejpam-87 255 14 k2 k2 NOUN ejpam-87 255 15 + + CCONJ ejpam-87 255 16 1 1 NUM ejpam-87 255 17 ( ( PUNCT ejpam-87 255 18 m m VERB ejpam-87 255 19 + + NOUN ejpam-87 255 20 1 1 NUM ejpam-87 255 21 k1 k1 NOUN ejpam-87 255 22 ) ) PUNCT ejpam-87 255 23 ( ( PUNCT ejpam-87 255 24 m m PROPN ejpam-87 255 25 k2 k2 PROPN ejpam-87 255 26 ) ) PUNCT ejpam-87 255 27 ≥ ≥ NOUN ejpam-87 255 28 0 0 NUM ejpam-87 255 29 , , PUNCT ejpam-87 255 30 prékopa prékopa NOUN ejpam-87 255 31 , , PUNCT ejpam-87 255 32 m. m. NOUN ejpam-87 255 33 subasi subasi PROPN ejpam-87 255 34 , , PUNCT ejpam-87 255 35 e. e. PROPN ejpam-87 255 36 subasi subasi PROPN ejpam-87 255 37 / / SYM ejpam-87 255 38 eur eur PROPN ejpam-87 255 39 . . PUNCT ejpam-87 256 1 j. j. PROPN ejpam-87 256 2 pure pure PROPN ejpam-87 256 3 appl appl PROPN ejpam-87 256 4 . . PROPN ejpam-87 256 5 math math PROPN ejpam-87 256 6 , , PUNCT ejpam-87 256 7 1 1 NUM ejpam-87 256 8 ( ( PUNCT ejpam-87 256 9 2008 2008 NUM ejpam-87 256 10 ) ) PUNCT ejpam-87 256 11 , , PUNCT ejpam-87 256 12 ( ( PUNCT ejpam-87 256 13 60 60 NUM ejpam-87 256 14 - - SYM ejpam-87 256 15 81 81 NUM ejpam-87 256 16 ) ) PUNCT ejpam-87 256 17 69 69 NUM ejpam-87 256 18 sk1β sk1β PROPN ejpam-87 256 19 k2 k2 PROPN ejpam-87 256 20 1,m 1,m PROPN ejpam-87 256 21 − − PROPN ejpam-87 256 22 sk2β sk2β PROPN ejpam-87 256 23 k1 k1 X ejpam-87 256 24 1,m 1,m NOUN ejpam-87 257 1 + + CCONJ ejpam-87 257 2 βk1,k2 βk1,k2 PROPN ejpam-87 257 3 1,m 1,m X ejpam-87 257 4 ≥ ≥ X ejpam-87 257 5 0 0 NUM ejpam-87 257 6 , , PUNCT ejpam-87 257 7 sk1β sk1β PROPN ejpam-87 257 8 k2 k2 PROPN ejpam-87 257 9 1,m−1 1,m−1 PROPN ejpam-87 257 10 − − PROPN ejpam-87 257 11 sk2β sk2β PROPN ejpam-87 257 12 k1 k1 X ejpam-87 257 13 1,m−1 1,m−1 PROPN ejpam-87 257 14 + + CCONJ ejpam-87 257 15 βk1,k2 βk1,k2 PROPN ejpam-87 257 16 1,m−1 1,m−1 NUM ejpam-87 257 17 ≥ ≥ NOUN ejpam-87 257 18 0 0 NUM ejpam-87 257 19 . . PUNCT ejpam-87 258 1 the the DET ejpam-87 258 2 corresponding corresponding ADJ ejpam-87 258 3 closed closed ADJ ejpam-87 258 4 form form NOUN ejpam-87 258 5 lower lower ADV ejpam-87 258 6 bound bind VERB ejpam-87 258 7 for for ADP ejpam-87 258 8 p p PROPN ejpam-87 258 9 ( ( PUNCT ejpam-87 258 10 ν ν X ejpam-87 258 11 ≥ ≥ NOUN ejpam-87 258 12 1 1 NUM ejpam-87 258 13 ) ) PUNCT ejpam-87 258 14 is be AUX ejpam-87 258 15 given give VERB ejpam-87 258 16 by by ADP ejpam-87 258 17 1 1 NUM ejpam-87 258 18 − − PROPN ejpam-87 258 19 sk1 sk1 PROPN ejpam-87 258 20 ( ( PUNCT ejpam-87 258 21 m m PROPN ejpam-87 258 22 k2 k2 ADJ ejpam-87 258 23 − − PROPN ejpam-87 258 24 1 1 NUM ejpam-87 258 25 ) ) PUNCT ejpam-87 258 26 − − PROPN ejpam-87 258 27 sk2 sk2 PROPN ejpam-87 258 28 ( ( PUNCT ejpam-87 258 29 m m PROPN ejpam-87 258 30 k1 k1 NOUN ejpam-87 258 31 − − PROPN ejpam-87 258 32 1 1 NUM ejpam-87 258 33 ) ) PUNCT ejpam-87 258 34 − − PROPN ejpam-87 258 35 k2−k1 k2−k1 PROPN ejpam-87 258 36 m−k2 m−k2 PROPN ejpam-87 258 37 + + NOUN ejpam-87 258 38 1 1 NUM ejpam-87 258 39 ( ( PUNCT ejpam-87 258 40 m m VERB ejpam-87 258 41 + + NOUN ejpam-87 258 42 1 1 NUM ejpam-87 258 43 k1 k1 NOUN ejpam-87 258 44 ) ) PUNCT ejpam-87 258 45 ( ( PUNCT ejpam-87 258 46 m m PROPN ejpam-87 258 47 k2 k2 ADJ ejpam-87 258 48 ) ) PUNCT ejpam-87 258 49 βk1,k2 βk1,k2 NOUN ejpam-87 258 50 1,m−1 1,m−1 NUM ejpam-87 258 51 − − PROPN ejpam-87 258 52 m m PROPN ejpam-87 258 53 ( ( PUNCT ejpam-87 258 54 k2−k1 k2−k1 NOUN ejpam-87 258 55 ) ) PUNCT ejpam-87 258 56 m−k2 m−k2 NOUN ejpam-87 259 1 + + NOUN ejpam-87 259 2 1 1 NUM ejpam-87 259 3 ( ( PUNCT ejpam-87 259 4 m m VERB ejpam-87 259 5 + + NOUN ejpam-87 259 6 1 1 NUM ejpam-87 259 7 k1 k1 NOUN ejpam-87 259 8 ) ) PUNCT ejpam-87 259 9 ( ( PUNCT ejpam-87 259 10 m m PROPN ejpam-87 259 11 k2 k2 ADJ ejpam-87 259 12 ) ) PUNCT ejpam-87 259 13 ≤ ≤ NOUN ejpam-87 259 14 p p NOUN ejpam-87 259 15 ( ( PUNCT ejpam-87 259 16 ν ν X ejpam-87 259 17 ≥ ≥ NOUN ejpam-87 259 18 1 1 NUM ejpam-87 259 19 ) ) PUNCT ejpam-87 259 20 , , PUNCT ejpam-87 259 21 ( ( PUNCT ejpam-87 259 22 3.5 3.5 NUM ejpam-87 259 23 ) ) PUNCT ejpam-87 259 24 where where SCONJ ejpam-87 259 25 βr βr ADP ejpam-87 259 26 i i PRON ejpam-87 259 27 , , PUNCT ejpam-87 259 28 j j PROPN ejpam-87 259 29 , , PUNCT ejpam-87 259 30 β β X ejpam-87 259 31 r r NOUN ejpam-87 259 32 , , PUNCT ejpam-87 259 33 t t PROPN ejpam-87 260 1 i i PRON ejpam-87 260 2 , , PUNCT ejpam-87 260 3 j j PROPN ejpam-87 260 4 are be AUX ejpam-87 260 5 given give VERB ejpam-87 260 6 in in ADP ejpam-87 260 7 ( ( PUNCT ejpam-87 260 8 3.3 3.3 NUM ejpam-87 260 9 ) ) PUNCT ejpam-87 260 10 . . PUNCT ejpam-87 261 1 case case NOUN ejpam-87 261 2 3 3 X ejpam-87 261 3 . . PUNCT ejpam-87 262 1 let let VERB ejpam-87 262 2 m m PRON ejpam-87 262 3 + + NOUN ejpam-87 262 4 1 1 NUM ejpam-87 262 5 ≤ ≤ NUM ejpam-87 262 6 i i PRON ejpam-87 262 7 ≤ ≤ NOUN ejpam-87 262 8 n n CCONJ ejpam-87 262 9 − − PROPN ejpam-87 262 10 1 1 NUM ejpam-87 262 11 . . PUNCT ejpam-87 262 12 bmin bmin PROPN ejpam-87 262 13 is be AUX ejpam-87 262 14 primal primal ADJ ejpam-87 262 15 feasible feasible ADJ ejpam-87 262 16 if if SCONJ ejpam-87 263 1 and and CCONJ ejpam-87 263 2 only only ADV ejpam-87 263 3 if if SCONJ ejpam-87 263 4 i i PRON ejpam-87 263 5 is be AUX ejpam-87 263 6 determined determine VERB ejpam-87 263 7 by by ADP ejpam-87 263 8 the the DET ejpam-87 263 9 following following ADJ ejpam-87 263 10 conditions condition NOUN ejpam-87 263 11 : : PUNCT ejpam-87 264 1 sk1α sk1α PROPN ejpam-87 264 2 k2 k2 PROPN ejpam-87 264 3 m+1,i m+1,i PROPN ejpam-87 264 4 − − PROPN ejpam-87 264 5 sk2α sk2α PROPN ejpam-87 264 6 k1 k1 PROPN ejpam-87 264 7 m+1,i m+1,i NOUN ejpam-87 264 8 + + CCONJ ejpam-87 264 9 αk1,k2 αk1,k2 NOUN ejpam-87 264 10 m+1,i m+1,i NOUN ejpam-87 264 11 ≥ ≥ NUM ejpam-87 264 12 0 0 NUM ejpam-87 264 13 , , PUNCT ejpam-87 264 14 sk1γ sk1γ PROPN ejpam-87 264 15 k2 k2 PROPN ejpam-87 264 16 m+1,i+1 m+1,i+1 NOUN ejpam-87 264 17 − − PROPN ejpam-87 264 18 sk2γ sk2γ PROPN ejpam-87 264 19 k1 k1 PROPN ejpam-87 264 20 m+1,i+1 m+1,i+1 NOUN ejpam-87 264 21 − − PROPN ejpam-87 264 22 γk1,k2 γk1,k2 NOUN ejpam-87 264 23 m+1,i+1 m+1,i+1 NOUN ejpam-87 264 24 ≥ ≥ NOUN ejpam-87 264 25 0 0 NUM ejpam-87 264 26 , , PUNCT ejpam-87 264 27 sk1γ sk1γ PROPN ejpam-87 264 28 k2 k2 PROPN ejpam-87 264 29 m+1,i m+1,i PROPN ejpam-87 264 30 − − PROPN ejpam-87 264 31 sk2γ sk2γ PROPN ejpam-87 264 32 k1 k1 PROPN ejpam-87 264 33 m+1,i m+1,i PROPN ejpam-87 264 34 − − PROPN ejpam-87 264 35 γk1,k2 γk1,k2 NOUN ejpam-87 264 36 m+1,i m+1,i NOUN ejpam-87 264 37 ≥ ≥ NUM ejpam-87 264 38 0 0 NUM ejpam-87 264 39 . . PUNCT ejpam-87 265 1 then then ADV ejpam-87 265 2 the the DET ejpam-87 265 3 closed closed ADJ ejpam-87 265 4 form form NOUN ejpam-87 265 5 lower lower ADV ejpam-87 265 6 bound bind VERB ejpam-87 265 7 is be AUX ejpam-87 265 8 the the DET ejpam-87 265 9 following following NOUN ejpam-87 265 10 : : PUNCT ejpam-87 265 11 1 1 NUM ejpam-87 265 12 − − PROPN ejpam-87 265 13 sk1α sk1α PROPN ejpam-87 265 14 k2 k2 PROPN ejpam-87 265 15 m+1,i m+1,i PROPN ejpam-87 265 16 − − PROPN ejpam-87 265 17 sk2α sk2α PROPN ejpam-87 265 18 k1 k1 PROPN ejpam-87 265 19 m+1,i m+1,i NOUN ejpam-87 265 20 + + CCONJ ejpam-87 265 21 αk1,k2 αk1,k2 PROPN ejpam-87 265 22 m+1,i m+1,i NOUN ejpam-87 265 23 ( ( PUNCT ejpam-87 265 24 m m PROPN ejpam-87 265 25 + + X ejpam-87 265 26 1 1 NUM ejpam-87 265 27 k1 k1 NOUN ejpam-87 265 28 + + CCONJ ejpam-87 265 29 1 1 NUM ejpam-87 265 30 ) ) PUNCT ejpam-87 265 31 αk2 αk2 NOUN ejpam-87 265 32 m+1,i m+1,i NOUN ejpam-87 265 33 − − PROPN ejpam-87 266 1 ( ( PUNCT ejpam-87 266 2 m m VERB ejpam-87 266 3 + + PROPN ejpam-87 266 4 1 1 NUM ejpam-87 266 5 k2 k2 NOUN ejpam-87 266 6 + + CCONJ ejpam-87 266 7 1 1 NUM ejpam-87 266 8 ) ) PUNCT ejpam-87 266 9 αk1 αk1 VERB ejpam-87 266 10 m+1,i m+1,i NOUN ejpam-87 266 11 + + CCONJ ejpam-87 267 1 ( ( PUNCT ejpam-87 267 2 m m VERB ejpam-87 267 3 + + ADJ ejpam-87 267 4 1 1 X ejpam-87 267 5 ) ) PUNCT ejpam-87 267 6 αk1,k2 αk1,k2 NOUN ejpam-87 267 7 m+1,i m+1,i NOUN ejpam-87 267 8 ≤ ≤ NUM ejpam-87 268 1 p p X ejpam-87 268 2 ( ( PUNCT ejpam-87 268 3 ν ν X ejpam-87 268 4 ≥ ≥ NOUN ejpam-87 268 5 1 1 NUM ejpam-87 268 6 ) ) PUNCT ejpam-87 268 7 , , PUNCT ejpam-87 268 8 ( ( PUNCT ejpam-87 268 9 3.6 3.6 NUM ejpam-87 268 10 ) ) PUNCT ejpam-87 268 11 where where SCONJ ejpam-87 268 12 αr αr ADP ejpam-87 268 13 i i PRON ejpam-87 268 14 , , PUNCT ejpam-87 268 15 j j PROPN ejpam-87 268 16 , , PUNCT ejpam-87 268 17 α α PROPN ejpam-87 268 18 r r PROPN ejpam-87 268 19 , , PUNCT ejpam-87 268 20 t t PROPN ejpam-87 269 1 i i PRON ejpam-87 269 2 , , PUNCT ejpam-87 269 3 j j PROPN ejpam-87 269 4 , , PUNCT ejpam-87 269 5 γ γ X ejpam-87 269 6 r r NOUN ejpam-87 269 7 i i PROPN ejpam-87 269 8 , , PUNCT ejpam-87 269 9 j j PROPN ejpam-87 269 10 , , PUNCT ejpam-87 269 11 γ γ X ejpam-87 269 12 r r PROPN ejpam-87 269 13 , , PUNCT ejpam-87 269 14 t t PROPN ejpam-87 270 1 i i PRON ejpam-87 270 2 , , PUNCT ejpam-87 270 3 j j PROPN ejpam-87 270 4 are be AUX ejpam-87 270 5 given give VERB ejpam-87 270 6 in in ADP ejpam-87 270 7 ( ( PUNCT ejpam-87 270 8 3.3 3.3 NUM ejpam-87 270 9 ) ) PUNCT ejpam-87 270 10 . . PUNCT ejpam-87 271 1 if if SCONJ ejpam-87 271 2 bmax bmax VERB ejpam-87 271 3 ⊂ ⊂ PRON ejpam-87 271 4 { { PUNCT ejpam-87 271 5 1 1 NUM ejpam-87 271 6 , , PUNCT ejpam-87 271 7 ... ... PUNCT ejpam-87 271 8 , , PUNCT ejpam-87 271 9 n n CCONJ ejpam-87 271 10 } } PUNCT ejpam-87 271 11 is be AUX ejpam-87 271 12 primal primal ADJ ejpam-87 271 13 feasible feasible ADJ ejpam-87 271 14 in in ADP ejpam-87 271 15 the the DET ejpam-87 271 16 relaxed relaxed ADJ ejpam-87 271 17 version version NOUN ejpam-87 271 18 of of ADP ejpam-87 271 19 the the DET ejpam-87 271 20 maximization maximization NOUN ejpam-87 271 21 problem problem NOUN ejpam-87 271 22 ( ( PUNCT ejpam-87 271 23 3.2 3.2 NUM ejpam-87 271 24 ) ) PUNCT ejpam-87 271 25 , , PUNCT ejpam-87 271 26 then then ADV ejpam-87 271 27 the the DET ejpam-87 271 28 upper upper ADJ ejpam-87 271 29 bound bind VERB ejpam-87 271 30 for for ADP ejpam-87 271 31 the the DET ejpam-87 271 32 probability probability NOUN ejpam-87 271 33 of of ADP ejpam-87 271 34 the the DET ejpam-87 271 35 union union NOUN ejpam-87 271 36 is be AUX ejpam-87 271 37 equal equal ADJ ejpam-87 271 38 to to ADP ejpam-87 271 39 1 1 NUM ejpam-87 271 40 . . PUNCT ejpam-87 272 1 the the DET ejpam-87 272 2 basis basis NOUN ejpam-87 272 3 bmax bmax NOUN ejpam-87 272 4 = = PUNCT ejpam-87 272 5 { { PUNCT ejpam-87 272 6 0 0 NUM ejpam-87 272 7 , , PUNCT ejpam-87 272 8 1 1 NUM ejpam-87 272 9 , , PUNCT ejpam-87 272 10 n n CCONJ ejpam-87 272 11 } } PUNCT ejpam-87 272 12 is be AUX ejpam-87 272 13 primal primal ADJ ejpam-87 272 14 feasible feasible ADJ ejpam-87 272 15 if if SCONJ ejpam-87 272 16 and and CCONJ ejpam-87 272 17 only only ADV ejpam-87 272 18 if if SCONJ ejpam-87 272 19 the the DET ejpam-87 272 20 following follow VERB ejpam-87 272 21 conditions condition NOUN ejpam-87 272 22 hold hold VERB ejpam-87 272 23 : : PUNCT ejpam-87 273 1 sk1δ sk1δ PROPN ejpam-87 273 2 k2 k2 PROPN ejpam-87 273 3 m+1,n m+1,n PROPN ejpam-87 273 4 − − NOUN ejpam-87 273 5 sk2δ sk2δ PROPN ejpam-87 273 6 k1 k1 PROPN ejpam-87 273 7 m+1,n m+1,n PROPN ejpam-87 273 8 − − PROPN ejpam-87 273 9 γk1,k2 γk1,k2 NOUN ejpam-87 273 10 m+1,n m+1,n NOUN ejpam-87 273 11 ≤ ≤ NOUN ejpam-87 273 12 0 0 NUM ejpam-87 273 13 , , PUNCT ejpam-87 273 14 sk1γ sk1γ PROPN ejpam-87 273 15 k2 k2 PROPN ejpam-87 273 16 m+1,n m+1,n PROPN ejpam-87 273 17 − − PROPN ejpam-87 273 18 sk2γ sk2γ PROPN ejpam-87 273 19 k1 k1 PROPN ejpam-87 273 20 m+1,n m+1,n PROPN ejpam-87 273 21 − − PROPN ejpam-87 273 22 γk1,k2 γk1,k2 NOUN ejpam-87 273 23 m+1,n m+1,n PROPN ejpam-87 273 24 ≥ ≥ NOUN ejpam-87 273 25 0 0 NUM ejpam-87 273 26 , , PUNCT ejpam-87 273 27 sk1 sk1 PROPN ejpam-87 273 28 ( ( PUNCT ejpam-87 273 29 m m PROPN ejpam-87 273 30 + + PROPN ejpam-87 273 31 1 1 NUM ejpam-87 273 32 k2 k2 NOUN ejpam-87 273 33 + + CCONJ ejpam-87 273 34 1 1 NUM ejpam-87 273 35 ) ) PUNCT ejpam-87 273 36 ≤ ≤ NOUN ejpam-87 273 37 sk2 sk2 NOUN ejpam-87 273 38 ( ( PUNCT ejpam-87 273 39 m m VERB ejpam-87 273 40 + + X ejpam-87 273 41 1 1 NUM ejpam-87 273 42 k1 k1 NOUN ejpam-87 273 43 + + CCONJ ejpam-87 273 44 1 1 NUM ejpam-87 273 45 ) ) PUNCT ejpam-87 273 46 . . PUNCT ejpam-87 274 1 the the DET ejpam-87 274 2 corresponding corresponding ADJ ejpam-87 274 3 closed close VERB ejpam-87 274 4 form form NOUN ejpam-87 274 5 upper upper ADJ ejpam-87 274 6 bound bind VERB ejpam-87 274 7 for for ADP ejpam-87 274 8 p p PROPN ejpam-87 274 9 ( ( PUNCT ejpam-87 274 10 ν ν X ejpam-87 274 11 ≥ ≥ NOUN ejpam-87 274 12 1 1 NUM ejpam-87 274 13 ) ) PUNCT ejpam-87 274 14 is be AUX ejpam-87 274 15 given give VERB ejpam-87 274 16 below below ADV ejpam-87 274 17 : : PUNCT ejpam-87 274 18 p p X ejpam-87 274 19 ( ( PUNCT ejpam-87 274 20 ν ν X ejpam-87 274 21 ≥ ≥ NUM ejpam-87 274 22 1 1 NUM ejpam-87 274 23 ) ) PUNCT ejpam-87 274 24 ≤ ≤ NOUN ejpam-87 274 25 sk1δ sk1δ PROPN ejpam-87 274 26 k2 k2 PROPN ejpam-87 274 27 m+1,n m+1,n PROPN ejpam-87 274 28 − − NOUN ejpam-87 274 29 sk2δ sk2δ PROPN ejpam-87 274 30 k1 k1 NOUN ejpam-87 274 31 m+1,n m+1,n NOUN ejpam-87 274 32 γk1,k2 γk1,k2 NOUN ejpam-87 274 33 m+1,n m+1,n NUM ejpam-87 274 34 , , PUNCT ejpam-87 274 35 ( ( PUNCT ejpam-87 274 36 3.7 3.7 NUM ejpam-87 274 37 ) ) PUNCT ejpam-87 274 38 where where SCONJ ejpam-87 274 39 δr δr ADP ejpam-87 274 40 i i PROPN ejpam-87 274 41 , , PUNCT ejpam-87 274 42 j j PROPN ejpam-87 274 43 , , PUNCT ejpam-87 274 44 γ γ X ejpam-87 274 45 r r NOUN ejpam-87 274 46 i i PROPN ejpam-87 274 47 , , PUNCT ejpam-87 274 48 j j PROPN ejpam-87 274 49 , , PUNCT ejpam-87 274 50 γ γ X ejpam-87 274 51 r r PROPN ejpam-87 274 52 , , PUNCT ejpam-87 274 53 t t PROPN ejpam-87 274 54 i i PRON ejpam-87 274 55 , , PUNCT ejpam-87 274 56 j j PROPN ejpam-87 274 57 are be AUX ejpam-87 274 58 given give VERB ejpam-87 274 59 in in ADP ejpam-87 274 60 ( ( PUNCT ejpam-87 274 61 3.3 3.3 NUM ejpam-87 274 62 ) ) PUNCT ejpam-87 274 63 . . PUNCT ejpam-87 275 1 if if SCONJ ejpam-87 275 2 we we PRON ejpam-87 275 3 use use VERB ejpam-87 275 4 the the DET ejpam-87 275 5 relaxed relaxed ADJ ejpam-87 275 6 version version NOUN ejpam-87 275 7 of of ADP ejpam-87 275 8 problem problem NOUN ejpam-87 275 9 ( ( PUNCT ejpam-87 275 10 1.6 1.6 NUM ejpam-87 275 11 ) ) PUNCT ejpam-87 275 12 , , PUNCT ejpam-87 275 13 rather rather ADV ejpam-87 275 14 than than ADP ejpam-87 275 15 that that PRON ejpam-87 275 16 of of ADP ejpam-87 275 17 problem problem NOUN ejpam-87 275 18 ( ( PUNCT ejpam-87 275 19 1.5 1.5 NUM ejpam-87 275 20 ) ) PUNCT ejpam-87 275 21 , , PUNCT ejpam-87 275 22 then then ADV ejpam-87 275 23 the the DET ejpam-87 275 24 lower low ADJ ejpam-87 275 25 and and CCONJ ejpam-87 275 26 upper upper ADJ ejpam-87 275 27 bounds bound NOUN ejpam-87 275 28 change change VERB ejpam-87 275 29 in in ADP ejpam-87 275 30 such such DET ejpam-87 275 31 a a DET ejpam-87 275 32 way way NOUN ejpam-87 275 33 that that PRON ejpam-87 275 34 we we PRON ejpam-87 275 35 have have VERB ejpam-87 275 36 to to PART ejpam-87 275 37 replace replace VERB ejpam-87 275 38 m m PROPN ejpam-87 275 39 − − PROPN ejpam-87 275 40 1 1 NUM ejpam-87 275 41 for for ADP ejpam-87 275 42 m m PROPN ejpam-87 275 43 prékopa prékopa ADJ ejpam-87 275 44 , , PUNCT ejpam-87 275 45 m. m. NOUN ejpam-87 275 46 subasi subasi PROPN ejpam-87 275 47 , , PUNCT ejpam-87 275 48 e. e. PROPN ejpam-87 275 49 subasi subasi PROPN ejpam-87 275 50 / / SYM ejpam-87 275 51 eur eur PROPN ejpam-87 275 52 . . PUNCT ejpam-87 276 1 j. j. PROPN ejpam-87 276 2 pure pure PROPN ejpam-87 276 3 appl appl PROPN ejpam-87 276 4 . . PROPN ejpam-87 276 5 math math PROPN ejpam-87 276 6 , , PUNCT ejpam-87 276 7 1 1 NUM ejpam-87 276 8 ( ( PUNCT ejpam-87 276 9 2008 2008 NUM ejpam-87 276 10 ) ) PUNCT ejpam-87 276 11 , , PUNCT ejpam-87 276 12 ( ( PUNCT ejpam-87 276 13 60 60 NUM ejpam-87 276 14 - - SYM ejpam-87 276 15 81 81 NUM ejpam-87 276 16 ) ) PUNCT ejpam-87 276 17 70 70 NUM ejpam-87 276 18 in in ADP ejpam-87 276 19 the the DET ejpam-87 276 20 formulas formula NOUN ejpam-87 276 21 of of ADP ejpam-87 276 22 section section NOUN ejpam-87 276 23 3 3 NUM ejpam-87 276 24 . . NOUN ejpam-87 276 25 4 4 NUM ejpam-87 276 26 . . NUM ejpam-87 276 27 closed close VERB ejpam-87 276 28 form form NOUN ejpam-87 276 29 bounds bound NOUN ejpam-87 276 30 for for ADP ejpam-87 276 31 the the DET ejpam-87 276 32 probability probability NOUN ejpam-87 276 33 of of ADP ejpam-87 276 34 the the DET ejpam-87 276 35 union union NOUN ejpam-87 276 36 based base VERB ejpam-87 276 37 on on ADP ejpam-87 276 38 s1 s1 PROPN ejpam-87 276 39 , , PUNCT ejpam-87 276 40 s2 s2 PROPN ejpam-87 276 41 , , PUNCT ejpam-87 276 42 s3 s3 PROPN ejpam-87 276 43 we we PRON ejpam-87 276 44 look look VERB ejpam-87 276 45 at at ADP ejpam-87 276 46 the the DET ejpam-87 276 47 relaxed relaxed ADJ ejpam-87 276 48 versions version NOUN ejpam-87 276 49 of of ADP ejpam-87 276 50 problems problem NOUN ejpam-87 276 51 ( ( PUNCT ejpam-87 276 52 1.5 1.5 NUM ejpam-87 276 53 ) ) PUNCT ejpam-87 276 54 , , PUNCT ejpam-87 276 55 ( ( PUNCT ejpam-87 276 56 1.6 1.6 NUM ejpam-87 276 57 ) ) PUNCT ejpam-87 276 58 and and CCONJ ejpam-87 276 59 create create VERB ejpam-87 276 60 bounds bound NOUN ejpam-87 276 61 for for ADP ejpam-87 276 62 the the DET ejpam-87 276 63 probability probability NOUN ejpam-87 276 64 of of ADP ejpam-87 276 65 the the DET ejpam-87 276 66 union union NOUN ejpam-87 276 67 , , PUNCT ejpam-87 276 68 based base VERB ejpam-87 276 69 on on ADP ejpam-87 276 70 the the DET ejpam-87 276 71 knowledge knowledge NOUN ejpam-87 276 72 of of ADP ejpam-87 276 73 the the DET ejpam-87 276 74 binomial binomial ADJ ejpam-87 276 75 moments moment NOUN ejpam-87 276 76 s1 s1 PROPN ejpam-87 276 77 , , PUNCT ejpam-87 276 78 s2 s2 PROPN ejpam-87 276 79 , , PUNCT ejpam-87 276 80 s3 s3 PROPN ejpam-87 276 81 . . PUNCT ejpam-87 277 1 since since SCONJ ejpam-87 277 2 m m PROPN ejpam-87 277 3 + + NUM ejpam-87 277 4 1 1 NUM ejpam-87 277 5 is be AUX ejpam-87 277 6 even even ADV ejpam-87 277 7 , , PUNCT ejpam-87 277 8 then then ADV ejpam-87 277 9 by by ADP ejpam-87 277 10 the the DET ejpam-87 277 11 use use NOUN ejpam-87 277 12 of of ADP ejpam-87 277 13 theorem theorem NOUN ejpam-87 277 14 2 2 NUM ejpam-87 277 15 , , PUNCT ejpam-87 277 16 we we PRON ejpam-87 277 17 derive derive VERB ejpam-87 277 18 that that SCONJ ejpam-87 277 19 any any DET ejpam-87 277 20 dual dual ADJ ejpam-87 277 21 feasible feasible ADJ ejpam-87 277 22 basis basis NOUN ejpam-87 277 23 bmin bmin NOUN ejpam-87 277 24 of of ADP ejpam-87 277 25 the the DET ejpam-87 277 26 relaxed relaxed ADJ ejpam-87 277 27 version version NOUN ejpam-87 277 28 of of ADP ejpam-87 277 29 the the DET ejpam-87 277 30 minimization minimization NOUN ejpam-87 277 31 problem problem NOUN ejpam-87 277 32 ( ( PUNCT ejpam-87 277 33 1.5 1.5 NUM ejpam-87 277 34 ) ) PUNCT ejpam-87 277 35 has have VERB ejpam-87 277 36 the the DET ejpam-87 277 37 form form NOUN ejpam-87 277 38 : : PUNCT ejpam-87 277 39 bmin bmin PROPN ejpam-87 277 40 = = SYM ejpam-87 277 41 { { PUNCT ejpam-87 277 42 0 0 NUM ejpam-87 277 43 , , PUNCT ejpam-87 277 44 i i PRON ejpam-87 277 45 , , PUNCT ejpam-87 277 46 i i PRON ejpam-87 277 47 + + NOUN ejpam-87 277 48 1 1 NUM ejpam-87 277 49 , , PUNCT ejpam-87 277 50 n n CCONJ ejpam-87 277 51 } } PUNCT ejpam-87 277 52 , , PUNCT ejpam-87 277 53 i i PRON ejpam-87 277 54 = = NOUN ejpam-87 277 55 1 1 NUM ejpam-87 277 56 , , PUNCT ejpam-87 277 57 ... ... PUNCT ejpam-87 277 58 , , PUNCT ejpam-87 277 59 n− n− NOUN ejpam-87 277 60 2 2 NUM ejpam-87 277 61 . . PUNCT ejpam-87 278 1 similarly similarly ADV ejpam-87 278 2 , , PUNCT ejpam-87 278 3 any any DET ejpam-87 278 4 dual dual ADJ ejpam-87 278 5 feasible feasible ADJ ejpam-87 278 6 basis basis NOUN ejpam-87 278 7 bmax bmax NOUN ejpam-87 278 8 of of ADP ejpam-87 278 9 relaxed relaxed ADJ ejpam-87 278 10 version version NOUN ejpam-87 278 11 of of ADP ejpam-87 278 12 the the DET ejpam-87 278 13 maximization maximization NOUN ejpam-87 278 14 problem problem NOUN ejpam-87 278 15 has have VERB ejpam-87 278 16 the the DET ejpam-87 278 17 form form NOUN ejpam-87 278 18 : : PUNCT ejpam-87 278 19 bmax bmax X ejpam-87 278 20 = = PUNCT ejpam-87 278 21 { { PUNCT ejpam-87 278 22 0 0 NUM ejpam-87 278 23 , , PUNCT ejpam-87 278 24 1 1 NUM ejpam-87 278 25 , , PUNCT ejpam-87 278 26 i i PRON ejpam-87 278 27 , , PUNCT ejpam-87 278 28 i i PRON ejpam-87 278 29 + + NOUN ejpam-87 278 30 1 1 NUM ejpam-87 278 31 } } PUNCT ejpam-87 278 32 , , PUNCT ejpam-87 278 33 i i PRON ejpam-87 278 34 = = NOUN ejpam-87 278 35 2 2 NUM ejpam-87 278 36 , , PUNCT ejpam-87 278 37 ... ... PUNCT ejpam-87 278 38 , , PUNCT ejpam-87 278 39 n− n− NOUN ejpam-87 278 40 1 1 NUM ejpam-87 278 41 , , PUNCT ejpam-87 278 42 or or CCONJ ejpam-87 278 43 bmax bmax VERB ejpam-87 278 44 ⊂ ⊂ PRON ejpam-87 278 45 { { PUNCT ejpam-87 278 46 1 1 NUM ejpam-87 278 47 , , PUNCT ejpam-87 278 48 ... ... PUNCT ejpam-87 278 49 , , PUNCT ejpam-87 278 50 n n CCONJ ejpam-87 278 51 } } PUNCT ejpam-87 278 52 . . PUNCT ejpam-87 279 1 below below ADP ejpam-87 279 2 we we PRON ejpam-87 279 3 present present VERB ejpam-87 279 4 conditions condition NOUN ejpam-87 279 5 that that PRON ejpam-87 279 6 ensure ensure VERB ejpam-87 279 7 the the DET ejpam-87 279 8 primal primal ADJ ejpam-87 279 9 feasibility feasibility NOUN ejpam-87 279 10 of of ADP ejpam-87 279 11 bmin bmin PROPN ejpam-87 279 12 as as ADV ejpam-87 279 13 well well ADV ejpam-87 279 14 as as ADP ejpam-87 279 15 the the DET ejpam-87 279 16 corresponding corresponding ADJ ejpam-87 279 17 lower low ADJ ejpam-87 279 18 bounds bound NOUN ejpam-87 279 19 for for ADP ejpam-87 279 20 p p PROPN ejpam-87 279 21 ( ( PUNCT ejpam-87 279 22 ν ν X ejpam-87 279 23 ≥ ≥ NOUN ejpam-87 279 24 1 1 NUM ejpam-87 279 25 ) ) PUNCT ejpam-87 279 26 , , PUNCT ejpam-87 279 27 i.e. i.e. X ejpam-87 279 28 , , PUNCT ejpam-87 279 29 the the DET ejpam-87 279 30 probability probability NOUN ejpam-87 279 31 of of ADP ejpam-87 279 32 the the DET ejpam-87 279 33 union union NOUN ejpam-87 279 34 of of ADP ejpam-87 279 35 the the DET ejpam-87 279 36 events event NOUN ejpam-87 279 37 . . PUNCT ejpam-87 280 1 case case NOUN ejpam-87 280 2 1 1 X ejpam-87 280 3 . . PUNCT ejpam-87 281 1 let let VERB ejpam-87 281 2 1 1 NUM ejpam-87 281 3 ≤ ≤ NUM ejpam-87 281 4 i i PRON ejpam-87 281 5 ≤ ≤ NOUN ejpam-87 281 6 m m VERB ejpam-87 281 7 − − PROPN ejpam-87 281 8 1 1 NUM ejpam-87 281 9 . . PUNCT ejpam-87 281 10 bmin bmin PROPN ejpam-87 281 11 is be AUX ejpam-87 281 12 primal primal ADJ ejpam-87 281 13 feasible feasible ADJ ejpam-87 281 14 if if SCONJ ejpam-87 282 1 and and CCONJ ejpam-87 282 2 only only ADV ejpam-87 282 3 if if SCONJ ejpam-87 282 4 i i PRON ejpam-87 282 5 is be AUX ejpam-87 282 6 determined determine VERB ejpam-87 282 7 by by ADP ejpam-87 282 8 the the DET ejpam-87 282 9 conditions condition NOUN ejpam-87 282 10 2[im 2[im NUM ejpam-87 283 1 + + CCONJ ejpam-87 283 2 ( ( PUNCT ejpam-87 283 3 n− n− NOUN ejpam-87 283 4 1)(i 1)(i NUM ejpam-87 283 5 + + NOUN ejpam-87 283 6 m m VERB ejpam-87 283 7 − − PROPN ejpam-87 283 8 1)]s1 1)]s1 NUM ejpam-87 283 9 − − NOUN ejpam-87 283 10 6(n 6(n NUM ejpam-87 284 1 + + CCONJ ejpam-87 284 2 i i PRON ejpam-87 284 3 + + NOUN ejpam-87 284 4 m m VERB ejpam-87 284 5 − − ADP ejpam-87 284 6 3)s2 3)s2 NUM ejpam-87 284 7 + + SYM ejpam-87 284 8 24s3 24s3 NUM ejpam-87 284 9 ≥ ≥ NOUN ejpam-87 284 10 mni mni PROPN ejpam-87 284 11 , , PUNCT ejpam-87 284 12 2[m(i− 2[m(i− NUM ejpam-87 284 13 1 1 NUM ejpam-87 284 14 ) ) PUNCT ejpam-87 284 15 + + CCONJ ejpam-87 284 16 ( ( PUNCT ejpam-87 284 17 n− n− NOUN ejpam-87 284 18 1)(i 1)(i NUM ejpam-87 284 19 + + CCONJ ejpam-87 284 20 m m VERB ejpam-87 284 21 − − NOUN ejpam-87 284 22 2)]s1 2)]s1 NUM ejpam-87 284 23 − − NOUN ejpam-87 284 24 6(n 6(n NUM ejpam-87 285 1 + + CCONJ ejpam-87 285 2 i i PRON ejpam-87 285 3 + + NOUN ejpam-87 285 4 m m VERB ejpam-87 285 5 − − NOUN ejpam-87 285 6 4)s2 4)s2 NUM ejpam-87 285 7 + + NOUN ejpam-87 285 8 24s3 24s3 NUM ejpam-87 285 9 ≤ ≤ NOUN ejpam-87 285 10 mn(i− mn(i− PROPN ejpam-87 285 11 1 1 NUM ejpam-87 285 12 ) ) PUNCT ejpam-87 285 13 , , PUNCT ejpam-87 285 14 2(i− 2(i− NUM ejpam-87 285 15 1)(i 1)(i NUM ejpam-87 285 16 + + CCONJ ejpam-87 285 17 2 2 NUM ejpam-87 285 18 m m NOUN ejpam-87 285 19 − − NUM ejpam-87 285 20 2)s1 2)s1 NUM ejpam-87 285 21 − − PROPN ejpam-87 286 1 6(2i 6(2i NUM ejpam-87 287 1 + + NOUN ejpam-87 287 2 m m VERB ejpam-87 287 3 − − NOUN ejpam-87 287 4 4)s2 4)s2 NUM ejpam-87 287 5 + + NOUN ejpam-87 287 6 24s3 24s3 NUM ejpam-87 287 7 ≥ ≥ NOUN ejpam-87 287 8 mi(i− mi(i− NUM ejpam-87 287 9 1 1 NUM ejpam-87 287 10 ) ) PUNCT ejpam-87 287 11 , , PUNCT ejpam-87 287 12 2[i(n 2[i(n NUM ejpam-87 287 13 + + CCONJ ejpam-87 287 14 2 2 NUM ejpam-87 287 15 m m NOUN ejpam-87 287 16 + + NUM ejpam-87 287 17 i i NOUN ejpam-87 287 18 ) ) PUNCT ejpam-87 288 1 + + CCONJ ejpam-87 288 2 ( ( PUNCT ejpam-87 288 3 n− n− NOUN ejpam-87 288 4 1)(i 1)(i NUM ejpam-87 288 5 + + NOUN ejpam-87 288 6 m m VERB ejpam-87 288 7 − − PROPN ejpam-87 288 8 1)]s1 1)]s1 NUM ejpam-87 288 9 − − NOUN ejpam-87 288 10 6(n 6(n NUM ejpam-87 288 11 + + NUM ejpam-87 288 12 2i 2i NUM ejpam-87 289 1 + + CCONJ ejpam-87 289 2 m m VERB ejpam-87 289 3 − − PROPN ejpam-87 289 4 3)s2 3)s2 NUM ejpam-87 289 5 + + SYM ejpam-87 289 6 24s3 24s3 NUM ejpam-87 289 7 ≤ ≤ NOUN ejpam-87 289 8 i[m(2n i[m(2n X ejpam-87 289 9 + + CCONJ ejpam-87 290 1 i i PRON ejpam-87 290 2 + + NOUN ejpam-87 290 3 1 1 X ejpam-87 290 4 ) ) PUNCT ejpam-87 290 5 + + CCONJ ejpam-87 290 6 ( ( PUNCT ejpam-87 290 7 i i PRON ejpam-87 290 8 + + NOUN ejpam-87 290 9 1)(n 1)(n NUM ejpam-87 290 10 + + CCONJ ejpam-87 290 11 1 1 NUM ejpam-87 290 12 ) ) PUNCT ejpam-87 290 13 ] ] PUNCT ejpam-87 290 14 . . PUNCT ejpam-87 291 1 in in ADP ejpam-87 291 2 this this DET ejpam-87 291 3 case case NOUN ejpam-87 291 4 the the DET ejpam-87 291 5 lower lower ADV ejpam-87 291 6 bound bind VERB ejpam-87 291 7 for for ADP ejpam-87 291 8 p p PROPN ejpam-87 291 9 ( ( PUNCT ejpam-87 291 10 ν ν X ejpam-87 291 11 ≥ ≥ NOUN ejpam-87 291 12 1 1 NUM ejpam-87 291 13 ) ) PUNCT ejpam-87 291 14 is be AUX ejpam-87 291 15 obtained obtain VERB ejpam-87 291 16 as as ADP ejpam-87 291 17 follows follow VERB ejpam-87 291 18 : : PUNCT ejpam-87 291 19 2[i(n 2[i(n NUM ejpam-87 291 20 + + CCONJ ejpam-87 291 21 2 2 NUM ejpam-87 291 22 m m NOUN ejpam-87 291 23 + + NUM ejpam-87 291 24 i i NOUN ejpam-87 291 25 ) ) PUNCT ejpam-87 292 1 + + CCONJ ejpam-87 292 2 ( ( PUNCT ejpam-87 292 3 n− n− NOUN ejpam-87 292 4 1)(i 1)(i NUM ejpam-87 292 5 + + NOUN ejpam-87 292 6 m m VERB ejpam-87 292 7 − − PROPN ejpam-87 292 8 1)]s1 1)]s1 NUM ejpam-87 292 9 − − NOUN ejpam-87 292 10 6(n 6(n NUM ejpam-87 292 11 + + NUM ejpam-87 292 12 2i 2i NUM ejpam-87 293 1 + + CCONJ ejpam-87 293 2 m m VERB ejpam-87 293 3 − − PROPN ejpam-87 293 4 3)s2 3)s2 NUM ejpam-87 293 5 + + SYM ejpam-87 293 6 24s3 24s3 NUM ejpam-87 293 7 ( ( PUNCT ejpam-87 293 8 n n PROPN ejpam-87 293 9 + + CCONJ ejpam-87 293 10 1)(m 1)(m NUM ejpam-87 294 1 + + SYM ejpam-87 294 2 1)(i 1)(i NUM ejpam-87 294 3 + + CCONJ ejpam-87 294 4 1)i 1)i NUM ejpam-87 294 5 + + CCONJ ejpam-87 294 6 mn(i− mn(i− PROPN ejpam-87 294 7 1 1 NUM ejpam-87 294 8 ) ) PUNCT ejpam-87 294 9 ( ( PUNCT ejpam-87 294 10 n n PROPN ejpam-87 294 11 + + CCONJ ejpam-87 294 12 1)(m 1)(m NUM ejpam-87 295 1 + + SYM ejpam-87 295 2 1)(i 1)(i NUM ejpam-87 295 3 + + CCONJ ejpam-87 295 4 1 1 NUM ejpam-87 295 5 ) ) PUNCT ejpam-87 295 6 ≤ ≤ NOUN ejpam-87 295 7 p p NOUN ejpam-87 295 8 ( ( PUNCT ejpam-87 295 9 ν ν X ejpam-87 295 10 ≥ ≥ NOUN ejpam-87 295 11 1 1 NUM ejpam-87 295 12 ) ) PUNCT ejpam-87 295 13 . . PUNCT ejpam-87 296 1 ( ( PUNCT ejpam-87 296 2 4.1 4.1 NUM ejpam-87 296 3 ) ) PUNCT ejpam-87 296 4 case case NOUN ejpam-87 296 5 2 2 X ejpam-87 296 6 . . PUNCT ejpam-87 297 1 let let VERB ejpam-87 297 2 i i PRON ejpam-87 297 3 = = NOUN ejpam-87 297 4 m m VERB ejpam-87 297 5 . . PUNCT ejpam-87 298 1 basis basis NOUN ejpam-87 298 2 bmin bmin NOUN ejpam-87 298 3 = = PRON ejpam-87 298 4 { { PUNCT ejpam-87 298 5 0,m 0,m PROPN ejpam-87 298 6 , , PUNCT ejpam-87 298 7 m m VERB ejpam-87 298 8 + + ADJ ejpam-87 298 9 1 1 NUM ejpam-87 298 10 , , PUNCT ejpam-87 298 11 n n CCONJ ejpam-87 298 12 } } PUNCT ejpam-87 298 13 is be AUX ejpam-87 298 14 primal primal ADJ ejpam-87 298 15 feasible feasible ADJ ejpam-87 298 16 if if SCONJ ejpam-87 298 17 and and CCONJ ejpam-87 298 18 only only ADV ejpam-87 298 19 if if SCONJ ejpam-87 298 20 the the DET ejpam-87 298 21 following follow VERB ejpam-87 298 22 conditions condition NOUN ejpam-87 298 23 are be AUX ejpam-87 298 24 satisfied satisfied ADJ ejpam-87 298 25 : : PUNCT ejpam-87 298 26 2m(2n 2m(2n NUM ejpam-87 298 27 + + NOUN ejpam-87 298 28 m m VERB ejpam-87 298 29 − − PROPN ejpam-87 298 30 1)s1 1)s1 NUM ejpam-87 298 31 − − PROPN ejpam-87 298 32 6(n 6(n NUM ejpam-87 298 33 + + CCONJ ejpam-87 298 34 2 2 NUM ejpam-87 298 35 m m NOUN ejpam-87 298 36 − − NOUN ejpam-87 298 37 2)s2 2)s2 ADJ ejpam-87 298 38 + + SYM ejpam-87 298 39 24s3 24s3 NUM ejpam-87 298 40 ≥ ≥ NOUN ejpam-87 298 41 m(m m(m NOUN ejpam-87 298 42 + + NUM ejpam-87 298 43 1)n 1)n NUM ejpam-87 298 44 , , PUNCT ejpam-87 298 45 2(m 2(m NUM ejpam-87 298 46 − − PROPN ejpam-87 298 47 1)(2n 1)(2n NUM ejpam-87 299 1 + + NUM ejpam-87 299 2 m m VERB ejpam-87 300 1 − − NUM ejpam-87 300 2 2)s1 2)s1 NUM ejpam-87 300 3 − − NOUN ejpam-87 300 4 6(n 6(n NUM ejpam-87 300 5 + + CCONJ ejpam-87 301 1 2 2 NUM ejpam-87 301 2 m m NOUN ejpam-87 301 3 − − NOUN ejpam-87 301 4 4)s2 4)s2 NUM ejpam-87 301 5 + + NOUN ejpam-87 301 6 24s3 24s3 NUM ejpam-87 301 7 ≤ ≤ NOUN ejpam-87 301 8 ( ( PUNCT ejpam-87 301 9 m m VERB ejpam-87 301 10 − − NOUN ejpam-87 302 1 1)mn 1)mn INTJ ejpam-87 302 2 , , PUNCT ejpam-87 303 1 6m(m 6m(m NUM ejpam-87 303 2 − − PROPN ejpam-87 304 1 1)s1 1)s1 NUM ejpam-87 304 2 − − PROPN ejpam-87 305 1 18s2 18s2 NUM ejpam-87 305 2 + + PROPN ejpam-87 305 3 24s3 24s3 NUM ejpam-87 305 4 ≥ ≥ NOUN ejpam-87 305 5 ( ( PUNCT ejpam-87 305 6 m m VERB ejpam-87 305 7 − − NOUN ejpam-87 305 8 1)m(m 1)m(m NUM ejpam-87 305 9 + + CCONJ ejpam-87 305 10 1 1 NUM ejpam-87 305 11 ) ) PUNCT ejpam-87 305 12 , , PUNCT ejpam-87 305 13 prékopa prékopa NOUN ejpam-87 305 14 , , PUNCT ejpam-87 305 15 m. m. NOUN ejpam-87 305 16 subasi subasi PROPN ejpam-87 305 17 , , PUNCT ejpam-87 305 18 e. e. PROPN ejpam-87 305 19 subasi subasi PROPN ejpam-87 305 20 / / SYM ejpam-87 305 21 eur eur PROPN ejpam-87 305 22 . . PUNCT ejpam-87 306 1 j. j. PROPN ejpam-87 306 2 pure pure PROPN ejpam-87 306 3 appl appl PROPN ejpam-87 306 4 . . PROPN ejpam-87 306 5 math math PROPN ejpam-87 306 6 , , PUNCT ejpam-87 306 7 1 1 NUM ejpam-87 306 8 ( ( PUNCT ejpam-87 306 9 2008 2008 NUM ejpam-87 306 10 ) ) PUNCT ejpam-87 306 11 , , PUNCT ejpam-87 306 12 ( ( PUNCT ejpam-87 306 13 60 60 NUM ejpam-87 306 14 - - SYM ejpam-87 306 15 81 81 NUM ejpam-87 306 16 ) ) PUNCT ejpam-87 306 17 71 71 NUM ejpam-87 306 18 6m(n 6m(n NUM ejpam-87 307 1 + + CCONJ ejpam-87 307 2 m)s1 m)s1 PROPN ejpam-87 307 3 − − NOUN ejpam-87 308 1 6(n 6(n NUM ejpam-87 308 2 + + CCONJ ejpam-87 308 3 3 3 NUM ejpam-87 308 4 m m NOUN ejpam-87 308 5 − − NOUN ejpam-87 308 6 2)s2 2)s2 ADJ ejpam-87 308 7 + + CCONJ ejpam-87 308 8 24s3 24s3 NUM ejpam-87 308 9 ≤ ≤ NOUN ejpam-87 308 10 ( ( PUNCT ejpam-87 308 11 m m VERB ejpam-87 308 12 + + X ejpam-87 308 13 1)(3n 1)(3n NUM ejpam-87 308 14 + + NUM ejpam-87 308 15 m m VERB ejpam-87 308 16 + + ADJ ejpam-87 308 17 2 2 NUM ejpam-87 308 18 ) ) PUNCT ejpam-87 308 19 . . PUNCT ejpam-87 309 1 the the DET ejpam-87 309 2 corresponding correspond VERB ejpam-87 309 3 lower lower ADV ejpam-87 309 4 bound bind VERB ejpam-87 309 5 for for ADP ejpam-87 309 6 p p PROPN ejpam-87 309 7 ( ( PUNCT ejpam-87 309 8 ν ν X ejpam-87 309 9 ≥ ≥ NOUN ejpam-87 309 10 1 1 NUM ejpam-87 309 11 ) ) PUNCT ejpam-87 309 12 is be AUX ejpam-87 309 13 given give VERB ejpam-87 309 14 below below ADP ejpam-87 309 15 : : PUNCT ejpam-87 309 16 6m(n 6m(n PROPN ejpam-87 309 17 + + CCONJ ejpam-87 309 18 m)s1 m)s1 PROPN ejpam-87 310 1 − − NOUN ejpam-87 310 2 6(n 6(n NUM ejpam-87 311 1 + + CCONJ ejpam-87 311 2 3 3 NUM ejpam-87 311 3 m m NOUN ejpam-87 311 4 − − NOUN ejpam-87 311 5 2)s2 2)s2 ADJ ejpam-87 311 6 + + CCONJ ejpam-87 311 7 24s3 24s3 NUM ejpam-87 311 8 m(m m(m NOUN ejpam-87 311 9 + + CCONJ ejpam-87 311 10 1)(m 1)(m NUM ejpam-87 311 11 + + NUM ejpam-87 311 12 2)(n 2)(n NUM ejpam-87 311 13 + + CCONJ ejpam-87 311 14 1 1 NUM ejpam-87 311 15 ) ) PUNCT ejpam-87 312 1 + + CCONJ ejpam-87 312 2 n(m n(m PROPN ejpam-87 312 3 − − NOUN ejpam-87 312 4 1 1 NUM ejpam-87 312 5 ) ) PUNCT ejpam-87 312 6 ( ( PUNCT ejpam-87 312 7 m m VERB ejpam-87 312 8 + + ADJ ejpam-87 312 9 2)(n 2)(n NUM ejpam-87 312 10 + + CCONJ ejpam-87 312 11 1 1 NUM ejpam-87 312 12 ) ) PUNCT ejpam-87 312 13 ≤ ≤ NOUN ejpam-87 312 14 p p NOUN ejpam-87 312 15 ( ( PUNCT ejpam-87 312 16 ν ν X ejpam-87 312 17 ≥ ≥ NOUN ejpam-87 312 18 1 1 NUM ejpam-87 312 19 ) ) PUNCT ejpam-87 312 20 . . PUNCT ejpam-87 313 1 ( ( PUNCT ejpam-87 313 2 4.2 4.2 NUM ejpam-87 313 3 ) ) PUNCT ejpam-87 313 4 case case NOUN ejpam-87 313 5 3 3 X ejpam-87 313 6 . . PUNCT ejpam-87 314 1 let let VERB ejpam-87 314 2 m m PRON ejpam-87 314 3 +1 +1 ADJ ejpam-87 314 4 ≤ ≤ X ejpam-87 315 1 i i PRON ejpam-87 315 2 ≤ ≤ PROPN ejpam-87 315 3 n−2 n−2 PROPN ejpam-87 315 4 . . PUNCT ejpam-87 315 5 bmin bmin PROPN ejpam-87 315 6 is be AUX ejpam-87 315 7 primal primal ADJ ejpam-87 315 8 feasible feasible ADJ ejpam-87 315 9 if if SCONJ ejpam-87 315 10 and and CCONJ ejpam-87 315 11 only only ADV ejpam-87 315 12 if if SCONJ ejpam-87 315 13 i i PRON ejpam-87 315 14 satisfies satisfy VERB ejpam-87 315 15 the the DET ejpam-87 315 16 following follow VERB ejpam-87 315 17 conditions condition NOUN ejpam-87 315 18 : : PUNCT ejpam-87 316 1 2[nm 2[nm NUM ejpam-87 316 2 + + NUM ejpam-87 316 3 i(n i(n NOUN ejpam-87 316 4 + + NOUN ejpam-87 316 5 m m VERB ejpam-87 316 6 − − NOUN ejpam-87 316 7 1)]s1 1)]s1 NUM ejpam-87 316 8 − − NOUN ejpam-87 316 9 6(n 6(n NUM ejpam-87 317 1 + + CCONJ ejpam-87 317 2 i i PRON ejpam-87 317 3 + + NOUN ejpam-87 317 4 m m VERB ejpam-87 317 5 − − ADP ejpam-87 317 6 2)s2 2)s2 ADJ ejpam-87 317 7 + + SYM ejpam-87 317 8 24s3 24s3 NUM ejpam-87 317 9 ≥ ≥ NOUN ejpam-87 317 10 mn(i mn(i VERB ejpam-87 317 11 + + CCONJ ejpam-87 317 12 1 1 NUM ejpam-87 317 13 ) ) PUNCT ejpam-87 317 14 , , PUNCT ejpam-87 317 15 2[im 2[im NUM ejpam-87 318 1 + + CCONJ ejpam-87 318 2 ( ( PUNCT ejpam-87 318 3 n− n− NOUN ejpam-87 318 4 1)(i 1)(i NUM ejpam-87 318 5 + + NOUN ejpam-87 318 6 m m VERB ejpam-87 318 7 − − PROPN ejpam-87 318 8 1)]s1 1)]s1 NUM ejpam-87 318 9 − − NOUN ejpam-87 318 10 6(n 6(n NUM ejpam-87 319 1 + + CCONJ ejpam-87 319 2 i i PRON ejpam-87 319 3 + + NOUN ejpam-87 319 4 m m VERB ejpam-87 319 5 − − ADP ejpam-87 319 6 3)s2 3)s2 NUM ejpam-87 319 7 + + NUM ejpam-87 319 8 24s3 24s3 NUM ejpam-87 319 9 ≤ ≤ NUM ejpam-87 319 10 mni mni X ejpam-87 319 11 , , PUNCT ejpam-87 319 12 2i(i 2i(i NUM ejpam-87 319 13 + + CCONJ ejpam-87 319 14 2 2 NUM ejpam-87 319 15 m m NOUN ejpam-87 319 16 − − NOUN ejpam-87 319 17 1)s1 1)s1 NUM ejpam-87 319 18 − − PROPN ejpam-87 320 1 6(2i 6(2i NUM ejpam-87 321 1 + + CCONJ ejpam-87 321 2 m m VERB ejpam-87 321 3 − − ADP ejpam-87 321 4 2)s2 2)s2 ADJ ejpam-87 321 5 + + SYM ejpam-87 321 6 24s3 24s3 NUM ejpam-87 321 7 ≥ ≥ NOUN ejpam-87 321 8 i(m i(m NOUN ejpam-87 321 9 + + CCONJ ejpam-87 321 10 1)m 1)m NUM ejpam-87 321 11 , , PUNCT ejpam-87 321 12 2[i(n 2[i(n NUM ejpam-87 321 13 + + CCONJ ejpam-87 321 14 2 2 NUM ejpam-87 321 15 m m NOUN ejpam-87 321 16 + + NUM ejpam-87 321 17 i i NOUN ejpam-87 321 18 ) ) PUNCT ejpam-87 322 1 + + CCONJ ejpam-87 322 2 ( ( PUNCT ejpam-87 322 3 n n PROPN ejpam-87 322 4 + + CCONJ ejpam-87 322 5 1)(i 1)(i NUM ejpam-87 322 6 + + NUM ejpam-87 322 7 m m VERB ejpam-87 322 8 + + NOUN ejpam-87 322 9 1)]s1 1)]s1 NUM ejpam-87 322 10 − − NOUN ejpam-87 322 11 6(n 6(n NUM ejpam-87 322 12 + + NUM ejpam-87 322 13 2i 2i NUM ejpam-87 322 14 + + NOUN ejpam-87 322 15 m m VERB ejpam-87 322 16 − − NOUN ejpam-87 322 17 1)s2 1)s2 NUM ejpam-87 322 18 + + SYM ejpam-87 322 19 24s3 24s3 NUM ejpam-87 322 20 ≤ ≤ NOUN ejpam-87 322 21 ( ( PUNCT ejpam-87 322 22 i i PRON ejpam-87 322 23 + + NUM ejpam-87 322 24 1)[im 1)[im NUM ejpam-87 323 1 + + CCONJ ejpam-87 323 2 ( ( PUNCT ejpam-87 323 3 n n PROPN ejpam-87 323 4 + + CCONJ ejpam-87 323 5 1)(i 1)(i NUM ejpam-87 323 6 + + CCONJ ejpam-87 323 7 2 2 NUM ejpam-87 323 8 m m NOUN ejpam-87 323 9 + + NOUN ejpam-87 323 10 2 2 NUM ejpam-87 323 11 ) ) PUNCT ejpam-87 323 12 ] ] PUNCT ejpam-87 323 13 . . PUNCT ejpam-87 324 1 in in ADP ejpam-87 324 2 this this DET ejpam-87 324 3 case case NOUN ejpam-87 324 4 the the DET ejpam-87 324 5 lower lower ADV ejpam-87 324 6 bound bind VERB ejpam-87 324 7 is be AUX ejpam-87 324 8 obtained obtain VERB ejpam-87 324 9 as as ADP ejpam-87 324 10 follows follow VERB ejpam-87 324 11 : : PUNCT ejpam-87 324 12 2[i(n 2[i(n NUM ejpam-87 324 13 + + CCONJ ejpam-87 324 14 2 2 NUM ejpam-87 324 15 m m NOUN ejpam-87 324 16 + + NUM ejpam-87 324 17 i i NOUN ejpam-87 324 18 ) ) PUNCT ejpam-87 325 1 + + CCONJ ejpam-87 325 2 ( ( PUNCT ejpam-87 325 3 n n PROPN ejpam-87 325 4 + + CCONJ ejpam-87 325 5 1)(i 1)(i NUM ejpam-87 325 6 + + NUM ejpam-87 325 7 m m VERB ejpam-87 325 8 + + NOUN ejpam-87 325 9 1)]s1 1)]s1 NUM ejpam-87 325 10 − − NOUN ejpam-87 325 11 6(n 6(n NUM ejpam-87 325 12 + + NUM ejpam-87 325 13 2i 2i NUM ejpam-87 325 14 + + NOUN ejpam-87 325 15 m m VERB ejpam-87 325 16 − − NOUN ejpam-87 325 17 1)s2 1)s2 NUM ejpam-87 325 18 + + SYM ejpam-87 325 19 24s3 24s3 NUM ejpam-87 325 20 ( ( PUNCT ejpam-87 325 21 i i PRON ejpam-87 325 22 + + NOUN ejpam-87 326 1 1)(i 1)(i NUM ejpam-87 327 1 + + CCONJ ejpam-87 327 2 2)(m 2)(m NUM ejpam-87 327 3 + + SYM ejpam-87 327 4 1)(n 1)(n NUM ejpam-87 327 5 + + CCONJ ejpam-87 327 6 1 1 NUM ejpam-87 327 7 ) ) PUNCT ejpam-87 328 1 + + CCONJ ejpam-87 328 2 nim nim PROPN ejpam-87 328 3 ( ( PUNCT ejpam-87 328 4 i i PRON ejpam-87 328 5 + + CCONJ ejpam-87 328 6 2)(m 2)(m NUM ejpam-87 328 7 + + SYM ejpam-87 328 8 1)(n 1)(n NUM ejpam-87 328 9 + + CCONJ ejpam-87 328 10 1 1 NUM ejpam-87 328 11 ) ) PUNCT ejpam-87 328 12 ≤ ≤ NOUN ejpam-87 328 13 p p NOUN ejpam-87 328 14 ( ( PUNCT ejpam-87 328 15 ν ν X ejpam-87 328 16 ≥ ≥ NOUN ejpam-87 328 17 1 1 NUM ejpam-87 328 18 ) ) PUNCT ejpam-87 328 19 . . PUNCT ejpam-87 329 1 ( ( PUNCT ejpam-87 329 2 4.3 4.3 NUM ejpam-87 329 3 ) ) PUNCT ejpam-87 329 4 in in ADP ejpam-87 329 5 order order NOUN ejpam-87 329 6 to to PART ejpam-87 329 7 obtain obtain VERB ejpam-87 329 8 an an DET ejpam-87 329 9 upper upper ADJ ejpam-87 329 10 bound bind VERB ejpam-87 329 11 for for ADP ejpam-87 329 12 p p PROPN ejpam-87 329 13 ( ( PUNCT ejpam-87 329 14 ν ν X ejpam-87 329 15 ≥ ≥ NUM ejpam-87 329 16 1 1 NUM ejpam-87 329 17 ) ) PUNCT ejpam-87 329 18 we we PRON ejpam-87 329 19 consider consider VERB ejpam-87 329 20 the the DET ejpam-87 329 21 relaxed relaxed ADJ ejpam-87 329 22 version version NOUN ejpam-87 329 23 of of ADP ejpam-87 329 24 the the DET ejpam-87 329 25 maximization maximization NOUN ejpam-87 329 26 problem problem NOUN ejpam-87 329 27 ( ( PUNCT ejpam-87 329 28 1.5 1.5 NUM ejpam-87 329 29 ) ) PUNCT ejpam-87 329 30 . . PUNCT ejpam-87 330 1 note note VERB ejpam-87 330 2 that that SCONJ ejpam-87 330 3 if if SCONJ ejpam-87 330 4 the the DET ejpam-87 330 5 dual dual ADJ ejpam-87 330 6 feasible feasible ADJ ejpam-87 330 7 basis basis NOUN ejpam-87 330 8 bmax bmax NOUN ejpam-87 330 9 ⊂ ⊂ X ejpam-87 330 10 { { PUNCT ejpam-87 330 11 1 1 NUM ejpam-87 330 12 , , PUNCT ejpam-87 330 13 ... ... PUNCT ejpam-87 330 14 , , PUNCT ejpam-87 330 15 n n CCONJ ejpam-87 330 16 } } PUNCT ejpam-87 330 17 is be AUX ejpam-87 330 18 also also ADV ejpam-87 330 19 primal primal ADJ ejpam-87 330 20 feasible feasible ADJ ejpam-87 330 21 , , PUNCT ejpam-87 330 22 then then ADV ejpam-87 330 23 the the DET ejpam-87 330 24 optimum optimum ADJ ejpam-87 330 25 value value NOUN ejpam-87 330 26 of of ADP ejpam-87 330 27 the the DET ejpam-87 330 28 maximization maximization NOUN ejpam-87 330 29 problem problem NOUN ejpam-87 330 30 , , PUNCT ejpam-87 330 31 i.e. i.e. X ejpam-87 330 32 , , PUNCT ejpam-87 330 33 the the DET ejpam-87 330 34 upper upper ADJ ejpam-87 330 35 bound bind VERB ejpam-87 330 36 for for ADP ejpam-87 330 37 the the DET ejpam-87 330 38 probability probability NOUN ejpam-87 330 39 of of ADP ejpam-87 330 40 the the DET ejpam-87 330 41 union union NOUN ejpam-87 330 42 , , PUNCT ejpam-87 330 43 is be AUX ejpam-87 330 44 equal equal ADJ ejpam-87 330 45 to to ADP ejpam-87 330 46 1 1 NUM ejpam-87 330 47 . . PUNCT ejpam-87 331 1 as as ADP ejpam-87 331 2 before before ADV ejpam-87 331 3 , , PUNCT ejpam-87 331 4 we we PRON ejpam-87 331 5 have have VERB ejpam-87 331 6 three three NUM ejpam-87 331 7 cases case NOUN ejpam-87 331 8 for for ADP ejpam-87 331 9 the the DET ejpam-87 331 10 choice choice NOUN ejpam-87 331 11 of of ADP ejpam-87 331 12 i. i. NOUN ejpam-87 331 13 case case PROPN ejpam-87 331 14 1 1 X ejpam-87 331 15 . . PUNCT ejpam-87 332 1 let let VERB ejpam-87 332 2 2 2 NUM ejpam-87 332 3 ≤ ≤ NOUN ejpam-87 333 1 i i PRON ejpam-87 333 2 ≤ ≤ NOUN ejpam-87 333 3 m m VERB ejpam-87 333 4 − − PROPN ejpam-87 333 5 1 1 NUM ejpam-87 333 6 . . PUNCT ejpam-87 334 1 the the DET ejpam-87 334 2 primal primal ADJ ejpam-87 334 3 feasibility feasibility NOUN ejpam-87 334 4 conditions condition NOUN ejpam-87 334 5 for for ADP ejpam-87 334 6 the the DET ejpam-87 334 7 basis basis NOUN ejpam-87 334 8 bmax bmax NOUN ejpam-87 334 9 = = PUNCT ejpam-87 334 10 { { PUNCT ejpam-87 334 11 0 0 NUM ejpam-87 334 12 , , PUNCT ejpam-87 334 13 1 1 NUM ejpam-87 334 14 , , PUNCT ejpam-87 334 15 i i PRON ejpam-87 334 16 , , PUNCT ejpam-87 334 17 i i PRON ejpam-87 334 18 + + CCONJ ejpam-87 334 19 1 1 X ejpam-87 334 20 } } PUNCT ejpam-87 334 21 are be AUX ejpam-87 334 22 as as SCONJ ejpam-87 334 23 follows follow VERB ejpam-87 334 24 : : PUNCT ejpam-87 334 25 2(i− 2(i− NUM ejpam-87 334 26 1)(i 1)(i NUM ejpam-87 334 27 + + CCONJ ejpam-87 334 28 2 2 NUM ejpam-87 334 29 m m NOUN ejpam-87 334 30 − − NUM ejpam-87 334 31 2)s1 2)s1 NUM ejpam-87 334 32 − − PROPN ejpam-87 335 1 6(2i 6(2i NUM ejpam-87 336 1 + + NOUN ejpam-87 336 2 m m VERB ejpam-87 336 3 − − NOUN ejpam-87 336 4 4)s2 4)s2 NUM ejpam-87 336 5 + + NOUN ejpam-87 336 6 24s3 24s3 NUM ejpam-87 336 7 ≥ ≥ NOUN ejpam-87 336 8 m(i− m(i− PROPN ejpam-87 336 9 1)i 1)i NUM ejpam-87 336 10 , , PUNCT ejpam-87 336 11 2(i− 2(i− NUM ejpam-87 336 12 1)(m 1)(m NUM ejpam-87 336 13 − − PROPN ejpam-87 336 14 1)s1 1)s1 NUM ejpam-87 337 1 − − NOUN ejpam-87 337 2 6(i 6(i NUM ejpam-87 338 1 + + CCONJ ejpam-87 338 2 m m VERB ejpam-87 338 3 − − PROPN ejpam-87 338 4 3)s2 3)s2 NUM ejpam-87 338 5 + + NUM ejpam-87 338 6 24s3 24s3 NUM ejpam-87 338 7 ≤ ≤ NOUN ejpam-87 338 8 0 0 NUM ejpam-87 338 9 , , PUNCT ejpam-87 338 10 2(i− 2(i− NUM ejpam-87 338 11 2)(m 2)(m NUM ejpam-87 338 12 − − NOUN ejpam-87 338 13 1)s1 1)s1 NUM ejpam-87 338 14 − − NOUN ejpam-87 338 15 6(i 6(i NUM ejpam-87 339 1 + + CCONJ ejpam-87 339 2 m m VERB ejpam-87 339 3 − − NOUN ejpam-87 339 4 4)s2 4)s2 NUM ejpam-87 339 5 + + NOUN ejpam-87 339 6 24s3 24s3 NUM ejpam-87 339 7 ≥ ≥ NOUN ejpam-87 339 8 0 0 NUM ejpam-87 339 9 , , PUNCT ejpam-87 339 10 2[i(i 2[i(i NUM ejpam-87 339 11 + + CCONJ ejpam-87 339 12 m m VERB ejpam-87 339 13 ) ) PUNCT ejpam-87 340 1 + + CCONJ ejpam-87 340 2 ( ( PUNCT ejpam-87 340 3 i− i− PROPN ejpam-87 340 4 1)(m 1)(m NUM ejpam-87 340 5 − − PROPN ejpam-87 340 6 1)]s1 1)]s1 NUM ejpam-87 340 7 − − PROPN ejpam-87 340 8 6(2i 6(2i NUM ejpam-87 341 1 + + CCONJ ejpam-87 341 2 m m VERB ejpam-87 341 3 − − PROPN ejpam-87 341 4 3)s2 3)s2 NUM ejpam-87 341 5 + + NUM ejpam-87 341 6 24s3 24s3 NUM ejpam-87 341 7 ≤ ≤ NOUN ejpam-87 341 8 i(i i(i PROPN ejpam-87 341 9 + + CCONJ ejpam-87 342 1 1)(m 1)(m NUM ejpam-87 342 2 + + CCONJ ejpam-87 342 3 1 1 NUM ejpam-87 342 4 ) ) PUNCT ejpam-87 342 5 . . PUNCT ejpam-87 343 1 the the DET ejpam-87 343 2 corresponding corresponding ADJ ejpam-87 343 3 upper upper ADJ ejpam-87 343 4 bound bind VERB ejpam-87 343 5 for for ADP ejpam-87 343 6 p p PROPN ejpam-87 343 7 ( ( PUNCT ejpam-87 343 8 ν ν X ejpam-87 343 9 ≥ ≥ NOUN ejpam-87 343 10 1 1 NUM ejpam-87 343 11 ) ) PUNCT ejpam-87 343 12 is be AUX ejpam-87 343 13 presented present VERB ejpam-87 343 14 below below ADP ejpam-87 343 15 : : PUNCT ejpam-87 343 16 p p X ejpam-87 343 17 ( ( PUNCT ejpam-87 343 18 ν ν X ejpam-87 343 19 ≥ ≥ NUM ejpam-87 343 20 1 1 NUM ejpam-87 343 21 ) ) PUNCT ejpam-87 343 22 ≤ ≤ NOUN ejpam-87 343 23 2[i(i 2[i(i NUM ejpam-87 344 1 + + NUM ejpam-87 344 2 m m X ejpam-87 344 3 ) ) PUNCT ejpam-87 345 1 + + CCONJ ejpam-87 345 2 ( ( PUNCT ejpam-87 345 3 i− i− PROPN ejpam-87 345 4 1)(m 1)(m NUM ejpam-87 345 5 − − PROPN ejpam-87 345 6 1)]s1 1)]s1 NUM ejpam-87 345 7 − − PROPN ejpam-87 345 8 6(2i 6(2i NUM ejpam-87 346 1 + + CCONJ ejpam-87 346 2 m m VERB ejpam-87 346 3 − − PROPN ejpam-87 346 4 3)s2 3)s2 NUM ejpam-87 346 5 + + SYM ejpam-87 346 6 24s3 24s3 NUM ejpam-87 346 7 i(i i(i PROPN ejpam-87 346 8 + + CCONJ ejpam-87 346 9 1)(m 1)(m NUM ejpam-87 346 10 + + CCONJ ejpam-87 346 11 1 1 NUM ejpam-87 346 12 ) ) PUNCT ejpam-87 346 13 . . PUNCT ejpam-87 347 1 ( ( PUNCT ejpam-87 347 2 4.4 4.4 NUM ejpam-87 347 3 ) ) PUNCT ejpam-87 347 4 prékopa prékopa PROPN ejpam-87 347 5 , , PUNCT ejpam-87 347 6 m. m. NOUN ejpam-87 347 7 subasi subasi PROPN ejpam-87 347 8 , , PUNCT ejpam-87 347 9 e. e. PROPN ejpam-87 347 10 subasi subasi PROPN ejpam-87 347 11 / / SYM ejpam-87 347 12 eur eur PROPN ejpam-87 347 13 . . PUNCT ejpam-87 348 1 j. j. PROPN ejpam-87 348 2 pure pure PROPN ejpam-87 348 3 appl appl PROPN ejpam-87 348 4 . . PROPN ejpam-87 348 5 math math PROPN ejpam-87 348 6 , , PUNCT ejpam-87 348 7 1 1 NUM ejpam-87 348 8 ( ( PUNCT ejpam-87 348 9 2008 2008 NUM ejpam-87 348 10 ) ) PUNCT ejpam-87 348 11 , , PUNCT ejpam-87 348 12 ( ( PUNCT ejpam-87 348 13 60 60 NUM ejpam-87 348 14 - - SYM ejpam-87 348 15 81 81 NUM ejpam-87 348 16 ) ) PUNCT ejpam-87 348 17 72 72 NUM ejpam-87 348 18 case case NOUN ejpam-87 348 19 2 2 X ejpam-87 348 20 . . PUNCT ejpam-87 349 1 let let VERB ejpam-87 349 2 i i PRON ejpam-87 349 3 = = NOUN ejpam-87 349 4 m m VERB ejpam-87 349 5 . . PUNCT ejpam-87 350 1 the the DET ejpam-87 350 2 basis basis NOUN ejpam-87 350 3 bmax bmax NOUN ejpam-87 350 4 = = PUNCT ejpam-87 350 5 { { PUNCT ejpam-87 350 6 0 0 NUM ejpam-87 350 7 , , PUNCT ejpam-87 350 8 1 1 NUM ejpam-87 350 9 , , PUNCT ejpam-87 350 10 m m PROPN ejpam-87 350 11 , , PUNCT ejpam-87 350 12 m m VERB ejpam-87 350 13 + + ADJ ejpam-87 350 14 1 1 NUM ejpam-87 350 15 } } PUNCT ejpam-87 350 16 is be AUX ejpam-87 350 17 primal primal ADJ ejpam-87 350 18 feasible feasible ADJ ejpam-87 350 19 if if SCONJ ejpam-87 350 20 and and CCONJ ejpam-87 350 21 only only ADV ejpam-87 350 22 if if SCONJ ejpam-87 350 23 6m(m 6m(m NUM ejpam-87 350 24 − − PROPN ejpam-87 350 25 1)s1 1)s1 NUM ejpam-87 350 26 − − PROPN ejpam-87 351 1 18(m 18(m NUM ejpam-87 352 1 − − PROPN ejpam-87 352 2 1)s2 1)s2 NUM ejpam-87 352 3 + + SYM ejpam-87 352 4 24s3 24s3 NUM ejpam-87 352 5 ≥ ≥ NOUN ejpam-87 352 6 ( ( PUNCT ejpam-87 352 7 m m VERB ejpam-87 352 8 − − NOUN ejpam-87 352 9 1)m(m 1)m(m NUM ejpam-87 352 10 + + CCONJ ejpam-87 352 11 1 1 NUM ejpam-87 352 12 ) ) PUNCT ejpam-87 352 13 , , PUNCT ejpam-87 352 14 2m(m 2m(m NUM ejpam-87 352 15 − − PROPN ejpam-87 353 1 1)s1 1)s1 NUM ejpam-87 353 2 − − NOUN ejpam-87 353 3 12(m 12(m NUM ejpam-87 354 1 − − PROPN ejpam-87 354 2 1)s2 1)s2 NUM ejpam-87 354 3 + + NOUN ejpam-87 354 4 24s3 24s3 NUM ejpam-87 354 5 ≤ ≤ NOUN ejpam-87 354 6 0 0 NUM ejpam-87 354 7 , , PUNCT ejpam-87 354 8 2(m 2(m NUM ejpam-87 354 9 − − NUM ejpam-87 354 10 1)(m 1)(m NUM ejpam-87 354 11 − − NUM ejpam-87 354 12 2)s1 2)s1 NUM ejpam-87 354 13 − − NOUN ejpam-87 354 14 12(m 12(m NUM ejpam-87 354 15 − − ADP ejpam-87 354 16 2)s2 2)s2 ADJ ejpam-87 354 17 + + SYM ejpam-87 354 18 24s3 24s3 NUM ejpam-87 354 19 ≥ ≥ NOUN ejpam-87 354 20 0 0 NUM ejpam-87 354 21 , , PUNCT ejpam-87 354 22 6m2s1 6m2s1 NUM ejpam-87 354 23 − − NOUN ejpam-87 354 24 6(3 6(3 NOUN ejpam-87 354 25 m m NOUN ejpam-87 354 26 − − NUM ejpam-87 354 27 2)s2 2)s2 ADJ ejpam-87 354 28 + + CCONJ ejpam-87 354 29 24s3 24s3 NUM ejpam-87 354 30 ≤ ≤ NUM ejpam-87 354 31 m(m m(m NOUN ejpam-87 354 32 − − NOUN ejpam-87 354 33 1)(m 1)(m NUM ejpam-87 354 34 + + NOUN ejpam-87 354 35 1 1 NUM ejpam-87 354 36 ) ) PUNCT ejpam-87 354 37 . . PUNCT ejpam-87 355 1 the the DET ejpam-87 355 2 corresponding corresponding ADJ ejpam-87 355 3 upper upper ADJ ejpam-87 355 4 bound bind VERB ejpam-87 355 5 for for ADP ejpam-87 355 6 p p PROPN ejpam-87 355 7 ( ( PUNCT ejpam-87 355 8 ν ν X ejpam-87 355 9 ≥ ≥ NOUN ejpam-87 355 10 1 1 NUM ejpam-87 355 11 ) ) PUNCT ejpam-87 355 12 is be AUX ejpam-87 355 13 given give VERB ejpam-87 355 14 below below ADV ejpam-87 355 15 : : PUNCT ejpam-87 355 16 p p X ejpam-87 355 17 ( ( PUNCT ejpam-87 355 18 ν ν X ejpam-87 355 19 ≥ ≥ NUM ejpam-87 355 20 1 1 NUM ejpam-87 355 21 ) ) PUNCT ejpam-87 355 22 ≤ ≤ NOUN ejpam-87 355 23 6m2s1 6m2s1 NUM ejpam-87 355 24 − − NOUN ejpam-87 355 25 6(3 6(3 NUM ejpam-87 355 26 m m NOUN ejpam-87 355 27 − − NUM ejpam-87 355 28 2)s2 2)s2 ADJ ejpam-87 355 29 + + CCONJ ejpam-87 355 30 24s3 24s3 NUM ejpam-87 355 31 m(m m(m NOUN ejpam-87 355 32 + + NOUN ejpam-87 356 1 1)(m 1)(m NUM ejpam-87 356 2 + + CCONJ ejpam-87 356 3 2 2 NUM ejpam-87 356 4 ) ) PUNCT ejpam-87 356 5 . . PUNCT ejpam-87 357 1 ( ( PUNCT ejpam-87 357 2 4.5 4.5 NUM ejpam-87 357 3 ) ) PUNCT ejpam-87 357 4 case case NOUN ejpam-87 357 5 3 3 X ejpam-87 357 6 . . PUNCT ejpam-87 358 1 let let VERB ejpam-87 358 2 m m PRON ejpam-87 358 3 + + NOUN ejpam-87 358 4 1 1 NUM ejpam-87 358 5 ≤ ≤ NUM ejpam-87 358 6 i i PRON ejpam-87 358 7 ≤ ≤ NOUN ejpam-87 358 8 n n CCONJ ejpam-87 358 9 − − PROPN ejpam-87 358 10 1 1 NUM ejpam-87 358 11 . . PUNCT ejpam-87 359 1 the the DET ejpam-87 359 2 basis basis NOUN ejpam-87 359 3 bmax bmax NOUN ejpam-87 359 4 is be AUX ejpam-87 359 5 primal primal ADJ ejpam-87 359 6 feasible feasible ADJ ejpam-87 359 7 if if SCONJ ejpam-87 359 8 and and CCONJ ejpam-87 359 9 only only ADV ejpam-87 359 10 if if SCONJ ejpam-87 359 11 i i PRON ejpam-87 359 12 is be AUX ejpam-87 359 13 determined determine VERB ejpam-87 359 14 by by ADP ejpam-87 359 15 the the DET ejpam-87 359 16 following following ADJ ejpam-87 359 17 conditions condition NOUN ejpam-87 359 18 : : PUNCT ejpam-87 359 19 2i(i 2i(i NUM ejpam-87 359 20 + + CCONJ ejpam-87 360 1 2 2 NUM ejpam-87 360 2 m m NOUN ejpam-87 360 3 − − NOUN ejpam-87 360 4 1)s1 1)s1 NUM ejpam-87 360 5 − − PROPN ejpam-87 361 1 6(2i 6(2i NUM ejpam-87 362 1 + + CCONJ ejpam-87 362 2 m m VERB ejpam-87 362 3 − − ADP ejpam-87 362 4 2)s2 2)s2 ADJ ejpam-87 362 5 + + SYM ejpam-87 362 6 24s3 24s3 NUM ejpam-87 362 7 ≥ ≥ NOUN ejpam-87 362 8 i(i i(i PROPN ejpam-87 362 9 + + CCONJ ejpam-87 362 10 1)m 1)m NUM ejpam-87 362 11 , , PUNCT ejpam-87 362 12 2i(m 2i(m NUM ejpam-87 362 13 − − NOUN ejpam-87 363 1 1)s1 1)s1 NUM ejpam-87 363 2 − − PROPN ejpam-87 363 3 6(i 6(i NUM ejpam-87 364 1 + + CCONJ ejpam-87 364 2 m m VERB ejpam-87 364 3 − − NUM ejpam-87 364 4 2)s2 2)s2 ADJ ejpam-87 364 5 + + CCONJ ejpam-87 364 6 24s3 24s3 NUM ejpam-87 364 7 ≤ ≤ NOUN ejpam-87 364 8 0 0 NUM ejpam-87 364 9 , , PUNCT ejpam-87 364 10 2(i− 2(i− NUM ejpam-87 364 11 1)(m 1)(m NUM ejpam-87 364 12 − − PROPN ejpam-87 364 13 1)s1 1)s1 NUM ejpam-87 364 14 − − NOUN ejpam-87 364 15 6(i 6(i NUM ejpam-87 365 1 + + CCONJ ejpam-87 365 2 m m VERB ejpam-87 365 3 − − PROPN ejpam-87 365 4 3)s2 3)s2 NUM ejpam-87 365 5 + + SYM ejpam-87 365 6 24s3 24s3 NUM ejpam-87 365 7 ≥ ≥ NOUN ejpam-87 365 8 0 0 NUM ejpam-87 365 9 , , PUNCT ejpam-87 365 10 2[i(i 2[i(i NUM ejpam-87 365 11 + + CCONJ ejpam-87 365 12 m m X ejpam-87 365 13 ) ) PUNCT ejpam-87 366 1 + + CCONJ ejpam-87 366 2 ( ( PUNCT ejpam-87 366 3 i i PRON ejpam-87 366 4 + + NOUN ejpam-87 366 5 1)(m 1)(m NUM ejpam-87 366 6 + + NUM ejpam-87 366 7 1)]s1 1)]s1 NUM ejpam-87 366 8 − − PROPN ejpam-87 366 9 6(2i 6(2i NUM ejpam-87 366 10 + + NOUN ejpam-87 366 11 m m VERB ejpam-87 366 12 − − NOUN ejpam-87 366 13 1)s2 1)s2 NUM ejpam-87 366 14 + + SYM ejpam-87 366 15 24s3 24s3 NUM ejpam-87 366 16 ≤ ≤ NOUN ejpam-87 366 17 ( ( PUNCT ejpam-87 366 18 i i PRON ejpam-87 366 19 + + NOUN ejpam-87 366 20 1)(i 1)(i NUM ejpam-87 366 21 + + CCONJ ejpam-87 366 22 2)(m 2)(m NUM ejpam-87 366 23 + + CCONJ ejpam-87 366 24 1 1 NUM ejpam-87 366 25 ) ) PUNCT ejpam-87 366 26 . . PUNCT ejpam-87 367 1 with with ADP ejpam-87 367 2 i i PRON ejpam-87 367 3 satisfying satisfy VERB ejpam-87 367 4 these these DET ejpam-87 367 5 inequalities inequality NOUN ejpam-87 367 6 we we PRON ejpam-87 367 7 have have VERB ejpam-87 367 8 the the DET ejpam-87 367 9 upper upper ADJ ejpam-87 367 10 bound bind VERB ejpam-87 367 11 given give VERB ejpam-87 367 12 by by ADP ejpam-87 367 13 : : PUNCT ejpam-87 367 14 p p X ejpam-87 367 15 ( ( PUNCT ejpam-87 367 16 ν ν X ejpam-87 367 17 ≥ ≥ NUM ejpam-87 367 18 1 1 NUM ejpam-87 367 19 ) ) PUNCT ejpam-87 367 20 ≤ ≤ NOUN ejpam-87 367 21 2[i(i 2[i(i NUM ejpam-87 368 1 + + NUM ejpam-87 368 2 m m X ejpam-87 368 3 ) ) PUNCT ejpam-87 369 1 + + CCONJ ejpam-87 369 2 ( ( PUNCT ejpam-87 369 3 i i PRON ejpam-87 369 4 + + NOUN ejpam-87 369 5 1)(m 1)(m NUM ejpam-87 369 6 + + NUM ejpam-87 369 7 1)]s1 1)]s1 NUM ejpam-87 369 8 − − PROPN ejpam-87 369 9 6(2i 6(2i NUM ejpam-87 369 10 + + NOUN ejpam-87 369 11 m m VERB ejpam-87 369 12 − − NOUN ejpam-87 369 13 1)s2 1)s2 NUM ejpam-87 369 14 + + SYM ejpam-87 369 15 24s3 24s3 NUM ejpam-87 369 16 ( ( PUNCT ejpam-87 369 17 i i PRON ejpam-87 369 18 + + NOUN ejpam-87 369 19 1)(i 1)(i NUM ejpam-87 369 20 + + CCONJ ejpam-87 369 21 2)(m 2)(m NUM ejpam-87 369 22 + + CCONJ ejpam-87 369 23 1 1 NUM ejpam-87 369 24 ) ) PUNCT ejpam-87 369 25 . . PUNCT ejpam-87 370 1 ( ( PUNCT ejpam-87 370 2 4.6 4.6 NUM ejpam-87 370 3 ) ) PUNCT ejpam-87 370 4 if if SCONJ ejpam-87 370 5 we we PRON ejpam-87 370 6 replace replace VERB ejpam-87 370 7 m m PRON ejpam-87 370 8 −1 −1 NOUN ejpam-87 370 9 for for ADP ejpam-87 370 10 m m PROPN ejpam-87 370 11 in in ADP ejpam-87 370 12 the the DET ejpam-87 370 13 formulas formula NOUN ejpam-87 370 14 of of ADP ejpam-87 370 15 section section NOUN ejpam-87 370 16 4 4 NUM ejpam-87 370 17 , , PUNCT ejpam-87 370 18 then then ADV ejpam-87 370 19 we we PRON ejpam-87 370 20 obtain obtain VERB ejpam-87 370 21 the the DET ejpam-87 370 22 closed close VERB ejpam-87 370 23 form form NOUN ejpam-87 370 24 bounds bound NOUN ejpam-87 370 25 that that PRON ejpam-87 370 26 come come VERB ejpam-87 370 27 out out ADP ejpam-87 370 28 of of ADP ejpam-87 370 29 the the DET ejpam-87 370 30 relaxed relaxed ADJ ejpam-87 370 31 version version NOUN ejpam-87 370 32 of of ADP ejpam-87 370 33 problem problem NOUN ejpam-87 370 34 ( ( PUNCT ejpam-87 370 35 1.6 1.6 NUM ejpam-87 370 36 ) ) PUNCT ejpam-87 370 37 . . PUNCT ejpam-87 371 1 5 5 X ejpam-87 371 2 . . NUM ejpam-87 371 3 closed close VERB ejpam-87 371 4 form form NOUN ejpam-87 371 5 upper upper ADJ ejpam-87 371 6 bounds bound NOUN ejpam-87 371 7 for for ADP ejpam-87 371 8 the the DET ejpam-87 371 9 probability probability NOUN ejpam-87 371 10 of of ADP ejpam-87 371 11 the the DET ejpam-87 371 12 union union NOUN ejpam-87 371 13 based base VERB ejpam-87 371 14 on on ADP ejpam-87 371 15 s1 s1 PROPN ejpam-87 371 16 , , PUNCT ejpam-87 371 17 s2 s2 PROPN ejpam-87 371 18 , , PUNCT ejpam-87 371 19 s3 s3 PROPN ejpam-87 371 20 , , PUNCT ejpam-87 371 21 s4 s4 PROPN ejpam-87 371 22 in in ADP ejpam-87 371 23 this this DET ejpam-87 371 24 section section NOUN ejpam-87 371 25 we we PRON ejpam-87 371 26 present present VERB ejpam-87 371 27 upper upper ADJ ejpam-87 371 28 bound bind VERB ejpam-87 371 29 formulas formula NOUN ejpam-87 371 30 for for ADP ejpam-87 371 31 the the DET ejpam-87 371 32 probability probability NOUN ejpam-87 371 33 of of ADP ejpam-87 371 34 the the DET ejpam-87 371 35 union union NOUN ejpam-87 371 36 of of ADP ejpam-87 371 37 events event NOUN ejpam-87 371 38 based base VERB ejpam-87 371 39 on on ADP ejpam-87 371 40 the the DET ejpam-87 371 41 first first ADJ ejpam-87 371 42 four four NUM ejpam-87 371 43 binomial binomial ADJ ejpam-87 371 44 moments moment NOUN ejpam-87 371 45 . . PUNCT ejpam-87 372 1 since since SCONJ ejpam-87 372 2 m m PROPN ejpam-87 372 3 + + NUM ejpam-87 372 4 1 1 NUM ejpam-87 372 5 is be AUX ejpam-87 372 6 odd odd ADJ ejpam-87 372 7 , , PUNCT ejpam-87 372 8 then then ADV ejpam-87 372 9 by by ADP ejpam-87 372 10 theorem theorem NOUN ejpam-87 372 11 2 2 NUM ejpam-87 372 12 , , PUNCT ejpam-87 372 13 any any DET ejpam-87 372 14 dual dual ADJ ejpam-87 372 15 feasible feasible ADJ ejpam-87 372 16 basis basis NOUN ejpam-87 372 17 bmax bmax NOUN ejpam-87 372 18 of of ADP ejpam-87 372 19 the the DET ejpam-87 372 20 relaxed relaxed ADJ ejpam-87 372 21 version version NOUN ejpam-87 372 22 of of ADP ejpam-87 372 23 the the DET ejpam-87 372 24 maximization maximization NOUN ejpam-87 372 25 problem problem NOUN ejpam-87 372 26 ( ( PUNCT ejpam-87 372 27 1.5 1.5 NUM ejpam-87 372 28 ) ) PUNCT ejpam-87 372 29 or or CCONJ ejpam-87 372 30 ( ( PUNCT ejpam-87 372 31 1.6 1.6 NUM ejpam-87 372 32 ) ) PUNCT ejpam-87 372 33 is be AUX ejpam-87 372 34 of of ADP ejpam-87 372 35 the the DET ejpam-87 372 36 form form NOUN ejpam-87 372 37 : : PUNCT ejpam-87 372 38 bmax bmax X ejpam-87 372 39 = = PUNCT ejpam-87 372 40 { { PUNCT ejpam-87 372 41 0 0 NUM ejpam-87 372 42 , , PUNCT ejpam-87 372 43 1 1 NUM ejpam-87 372 44 , , PUNCT ejpam-87 372 45 i i PRON ejpam-87 372 46 , , PUNCT ejpam-87 372 47 i i PRON ejpam-87 372 48 + + NOUN ejpam-87 372 49 1 1 NUM ejpam-87 372 50 , , PUNCT ejpam-87 372 51 n n CCONJ ejpam-87 372 52 } } PUNCT ejpam-87 372 53 , , PUNCT ejpam-87 372 54 i i PRON ejpam-87 372 55 = = NOUN ejpam-87 372 56 2 2 NUM ejpam-87 372 57 , , PUNCT ejpam-87 372 58 ... ... PUNCT ejpam-87 372 59 , , PUNCT ejpam-87 372 60 n− n− NOUN ejpam-87 372 61 1 1 NUM ejpam-87 372 62 , , PUNCT ejpam-87 372 63 or or CCONJ ejpam-87 372 64 bmax bmax VERB ejpam-87 372 65 ⊂ ⊂ PRON ejpam-87 372 66 { { PUNCT ejpam-87 372 67 1 1 NUM ejpam-87 372 68 , , PUNCT ejpam-87 372 69 ... ... PUNCT ejpam-87 372 70 , , PUNCT ejpam-87 372 71 n n CCONJ ejpam-87 372 72 } } PUNCT ejpam-87 372 73 . . PUNCT ejpam-87 373 1 if if SCONJ ejpam-87 373 2 bmax bmax VERB ejpam-87 373 3 ⊂ ⊂ PRON ejpam-87 373 4 { { PUNCT ejpam-87 373 5 1 1 NUM ejpam-87 373 6 , , PUNCT ejpam-87 373 7 ... ... PUNCT ejpam-87 373 8 , , PUNCT ejpam-87 373 9 n n CCONJ ejpam-87 373 10 } } PUNCT ejpam-87 373 11 , , PUNCT ejpam-87 373 12 then then ADV ejpam-87 373 13 the the DET ejpam-87 373 14 upper upper ADJ ejpam-87 373 15 bound bind VERB ejpam-87 373 16 for for ADP ejpam-87 373 17 p p PROPN ejpam-87 373 18 ( ( PUNCT ejpam-87 373 19 ν ν X ejpam-87 373 20 ≥ ≥ NOUN ejpam-87 373 21 1 1 NUM ejpam-87 373 22 ) ) PUNCT ejpam-87 373 23 is be AUX ejpam-87 373 24 1 1 NUM ejpam-87 373 25 . . PUNCT ejpam-87 374 1 in in ADP ejpam-87 374 2 order order NOUN ejpam-87 374 3 to to PART ejpam-87 374 4 determine determine VERB ejpam-87 374 5 the the DET ejpam-87 374 6 index index NOUN ejpam-87 374 7 i i PRON ejpam-87 374 8 that that PRON ejpam-87 374 9 ensures ensure VERB ejpam-87 374 10 the the DET ejpam-87 374 11 primal primal ADJ ejpam-87 374 12 feasibility feasibility NOUN ejpam-87 374 13 of of ADP ejpam-87 374 14 the the DET ejpam-87 374 15 basis basis NOUN ejpam-87 374 16 of of ADP ejpam-87 374 17 the the DET ejpam-87 374 18 form form NOUN ejpam-87 374 19 bmax bmax AUX ejpam-87 374 20 = = PUNCT ejpam-87 374 21 { { PUNCT ejpam-87 374 22 0 0 NUM ejpam-87 374 23 , , PUNCT ejpam-87 374 24 1 1 NUM ejpam-87 374 25 , , PUNCT ejpam-87 374 26 i i PRON ejpam-87 374 27 , , PUNCT ejpam-87 374 28 i i PRON ejpam-87 374 29 + + NOUN ejpam-87 374 30 1 1 NUM ejpam-87 374 31 , , PUNCT ejpam-87 374 32 n n CCONJ ejpam-87 374 33 } } PUNCT ejpam-87 374 34 we we PRON ejpam-87 374 35 consider consider VERB ejpam-87 374 36 the the DET ejpam-87 374 37 following follow VERB ejpam-87 374 38 cases case NOUN ejpam-87 374 39 . . PUNCT ejpam-87 375 1 prékopa prékopa ADJ ejpam-87 375 2 , , PUNCT ejpam-87 375 3 m. m. NOUN ejpam-87 375 4 subasi subasi PROPN ejpam-87 375 5 , , PUNCT ejpam-87 375 6 e. e. PROPN ejpam-87 375 7 subasi subasi PROPN ejpam-87 375 8 / / SYM ejpam-87 375 9 eur eur PROPN ejpam-87 375 10 . . PUNCT ejpam-87 376 1 j. j. PROPN ejpam-87 376 2 pure pure PROPN ejpam-87 376 3 appl appl PROPN ejpam-87 376 4 . . PROPN ejpam-87 376 5 math math PROPN ejpam-87 376 6 , , PUNCT ejpam-87 376 7 1 1 NUM ejpam-87 376 8 ( ( PUNCT ejpam-87 376 9 2008 2008 NUM ejpam-87 376 10 ) ) PUNCT ejpam-87 376 11 , , PUNCT ejpam-87 376 12 ( ( PUNCT ejpam-87 376 13 60 60 NUM ejpam-87 376 14 - - SYM ejpam-87 376 15 81 81 NUM ejpam-87 376 16 ) ) PUNCT ejpam-87 376 17 73 73 NUM ejpam-87 376 18 case case NOUN ejpam-87 376 19 1 1 X ejpam-87 376 20 . . PUNCT ejpam-87 377 1 let let VERB ejpam-87 377 2 2 2 NUM ejpam-87 377 3 ≤ ≤ NOUN ejpam-87 378 1 i i PRON ejpam-87 378 2 ≤ ≤ NOUN ejpam-87 378 3 m m VERB ejpam-87 378 4 − − PROPN ejpam-87 378 5 1 1 NUM ejpam-87 378 6 . . PUNCT ejpam-87 379 1 the the DET ejpam-87 379 2 primal primal ADJ ejpam-87 379 3 feasibility feasibility NOUN ejpam-87 379 4 conditions condition NOUN ejpam-87 379 5 are be AUX ejpam-87 379 6 : : PUNCT ejpam-87 379 7 2[i(i− 2[i(i− NUM ejpam-87 379 8 1)(n 1)(n NUM ejpam-87 379 9 + + X ejpam-87 379 10 m)− m)− PROPN ejpam-87 379 11 ( ( PUNCT ejpam-87 379 12 m m VERB ejpam-87 379 13 − − NUM ejpam-87 379 14 1)(n− 1)(n− NUM ejpam-87 379 15 1 1 NUM ejpam-87 379 16 ) ) PUNCT ejpam-87 379 17 + + CCONJ ejpam-87 379 18 2i(nm 2i(nm NUM ejpam-87 379 19 + + SYM ejpam-87 379 20 1)]s1 1)]s1 NUM ejpam-87 379 21 −6(3n −6(3n NOUN ejpam-87 379 22 + + CCONJ ejpam-87 379 23 3 3 NUM ejpam-87 379 24 m m NOUN ejpam-87 379 25 − − NOUN ejpam-87 379 26 nm nm PUNCT ejpam-87 379 27 − − PROPN ejpam-87 379 28 i2 i2 NOUN ejpam-87 379 29 + + CCONJ ejpam-87 379 30 5i− 5i− NUM ejpam-87 379 31 2im 2im NOUN ejpam-87 379 32 − − PROPN ejpam-87 379 33 2ni− 2ni− NUM ejpam-87 379 34 7)s2 7)s2 NUM ejpam-87 379 35 + + SYM ejpam-87 379 36 24(n 24(n NUM ejpam-87 380 1 + + NUM ejpam-87 380 2 2i 2i NUM ejpam-87 380 3 + + NOUN ejpam-87 380 4 m m VERB ejpam-87 380 5 − − NOUN ejpam-87 381 1 6)s3 6)s3 NOUN ejpam-87 381 2 − − NUM ejpam-87 381 3 120s4 120s4 NUM ejpam-87 381 4 ≤ ≤ X ejpam-87 381 5 i(i i(i PROPN ejpam-87 382 1 + + CCONJ ejpam-87 383 1 1)(n 1)(n NUM ejpam-87 383 2 + + CCONJ ejpam-87 383 3 1)(m 1)(m NUM ejpam-87 383 4 + + CCONJ ejpam-87 383 5 1 1 NUM ejpam-87 383 6 ) ) PUNCT ejpam-87 383 7 , , PUNCT ejpam-87 383 8 2(i− 2(i− NUM ejpam-87 383 9 1)(ni 1)(ni NUM ejpam-87 384 1 + + CCONJ ejpam-87 385 1 i i PRON ejpam-87 385 2 m m VERB ejpam-87 385 3 + + ADJ ejpam-87 385 4 2 2 NUM ejpam-87 385 5 nm nm NOUN ejpam-87 385 6 − − NOUN ejpam-87 386 1 2n− 2n− NUM ejpam-87 386 2 i− i− PROPN ejpam-87 386 3 2 2 NUM ejpam-87 386 4 m m NOUN ejpam-87 386 5 + + NUM ejpam-87 386 6 2)s1 2)s1 NUM ejpam-87 386 7 − − NOUN ejpam-87 387 1 6[nm 6[nm NUM ejpam-87 387 2 + + CCONJ ejpam-87 387 3 ( ( PUNCT ejpam-87 387 4 i− i− PROPN ejpam-87 387 5 2)(2n 2)(2n NUM ejpam-87 387 6 + + CCONJ ejpam-87 387 7 2 2 NUM ejpam-87 387 8 m m NOUN ejpam-87 387 9 + + NUM ejpam-87 387 10 i− i− PROPN ejpam-87 387 11 5)]s2 5)]s2 NUM ejpam-87 387 12 +24(n +24(n NOUN ejpam-87 388 1 + + CCONJ ejpam-87 388 2 2i 2i NUM ejpam-87 388 3 + + NOUN ejpam-87 388 4 m m VERB ejpam-87 388 5 − − NOUN ejpam-87 389 1 7)s3 7)s3 NUM ejpam-87 389 2 − − PROPN ejpam-87 389 3 120s4 120s4 NUM ejpam-87 389 4 ≥ ≥ NOUN ejpam-87 389 5 ( ( PUNCT ejpam-87 389 6 i− i− PROPN ejpam-87 389 7 1)inm 1)inm NUM ejpam-87 389 8 , , PUNCT ejpam-87 389 9 2(m 2(m NUM ejpam-87 389 10 − − PROPN ejpam-87 389 11 1)(n− 1)(n− NUM ejpam-87 389 12 1)(i− 1)(i− NUM ejpam-87 389 13 1)s1 1)s1 NUM ejpam-87 390 1 − − NOUN ejpam-87 390 2 6(ni 6(ni NOUN ejpam-87 390 3 + + CCONJ ejpam-87 391 1 i i NOUN ejpam-87 391 2 m m VERB ejpam-87 391 3 + + NOUN ejpam-87 391 4 nm nm ADV ejpam-87 391 5 − − PROPN ejpam-87 392 1 3n− 3n− PROPN ejpam-87 392 2 3i− 3i− NUM ejpam-87 392 3 3 3 NUM ejpam-87 392 4 m m NOUN ejpam-87 392 5 + + NOUN ejpam-87 392 6 7)s2 7)s2 NUM ejpam-87 392 7 +24(n +24(n NOUN ejpam-87 393 1 + + X ejpam-87 393 2 i i PRON ejpam-87 393 3 + + NOUN ejpam-87 393 4 m m VERB ejpam-87 393 5 − − NOUN ejpam-87 394 1 6)s3 6)s3 NOUN ejpam-87 394 2 − − NUM ejpam-87 394 3 120s4 120s4 NUM ejpam-87 394 4 ≤ ≤ NUM ejpam-87 394 5 0 0 NUM ejpam-87 394 6 , , PUNCT ejpam-87 394 7 2(m 2(m NUM ejpam-87 394 8 − − PROPN ejpam-87 395 1 1)(n− 1)(n− NUM ejpam-87 395 2 1)(i− 1)(i− NUM ejpam-87 395 3 2)s1 2)s1 NUM ejpam-87 396 1 − − NOUN ejpam-87 396 2 6(ni 6(ni NOUN ejpam-87 396 3 + + CCONJ ejpam-87 397 1 i i NOUN ejpam-87 397 2 m m VERB ejpam-87 397 3 + + X ejpam-87 397 4 nm nm VERB ejpam-87 397 5 − − PROPN ejpam-87 398 1 4n− 4n− NUM ejpam-87 398 2 3i− 3i− NUM ejpam-87 398 3 4 4 NUM ejpam-87 398 4 m m NOUN ejpam-87 398 5 + + NOUN ejpam-87 398 6 10)s2 10)s2 NUM ejpam-87 398 7 +24(n +24(n NOUN ejpam-87 399 1 + + X ejpam-87 399 2 i i PRON ejpam-87 399 3 + + NOUN ejpam-87 399 4 m m VERB ejpam-87 399 5 − − NOUN ejpam-87 400 1 7)s3 7)s3 NUM ejpam-87 400 2 − − PROPN ejpam-87 400 3 120s4 120s4 NUM ejpam-87 400 4 ≥ ≥ NOUN ejpam-87 400 5 0 0 NUM ejpam-87 400 6 , , PUNCT ejpam-87 400 7 2(m 2(m NUM ejpam-87 401 1 − − PROPN ejpam-87 401 2 1)(i− 1)(i− NUM ejpam-87 401 3 1)(i− 1)(i− NUM ejpam-87 401 4 2)s1 2)s1 NUM ejpam-87 401 5 − − PROPN ejpam-87 402 1 6(i− 6(i− NUM ejpam-87 402 2 2)(i 2)(i NUM ejpam-87 402 3 + + CCONJ ejpam-87 402 4 2 2 NUM ejpam-87 402 5 m m NOUN ejpam-87 402 6 − − NOUN ejpam-87 402 7 5)s2 5)s2 NUM ejpam-87 402 8 +24(2i +24(2i NOUN ejpam-87 402 9 + + NOUN ejpam-87 402 10 m m VERB ejpam-87 402 11 − − NOUN ejpam-87 403 1 7)s3 7)s3 NUM ejpam-87 403 2 − − PROPN ejpam-87 403 3 120s4 120s4 NUM ejpam-87 403 4 ≤ ≤ NOUN ejpam-87 403 5 0 0 NUM ejpam-87 403 6 . . PUNCT ejpam-87 404 1 under under ADP ejpam-87 404 2 these these DET ejpam-87 404 3 conditions condition NOUN ejpam-87 404 4 the the DET ejpam-87 404 5 corresponding corresponding ADJ ejpam-87 404 6 upper upper ADJ ejpam-87 404 7 bound bind VERB ejpam-87 404 8 is be AUX ejpam-87 404 9 given give VERB ejpam-87 404 10 below below ADV ejpam-87 404 11 : : PUNCT ejpam-87 404 12 p p X ejpam-87 404 13 ( ( PUNCT ejpam-87 404 14 ν ν X ejpam-87 404 15 ≥ ≥ NUM ejpam-87 404 16 1 1 NUM ejpam-87 404 17 ) ) PUNCT ejpam-87 404 18 ≤ ≤ NOUN ejpam-87 404 19 2[i(i− 2[i(i− NUM ejpam-87 404 20 1)(n 1)(n NUM ejpam-87 404 21 + + PROPN ejpam-87 404 22 m)− m)− PROPN ejpam-87 404 23 ( ( PUNCT ejpam-87 404 24 m m VERB ejpam-87 404 25 − − NUM ejpam-87 404 26 1)(n− 1)(n− NUM ejpam-87 404 27 1 1 NUM ejpam-87 404 28 ) ) PUNCT ejpam-87 404 29 + + CCONJ ejpam-87 404 30 2i(nm 2i(nm NUM ejpam-87 404 31 + + SYM ejpam-87 404 32 1)]s1 1)]s1 NUM ejpam-87 404 33 ( ( PUNCT ejpam-87 404 34 m m PROPN ejpam-87 404 35 + + NUM ejpam-87 404 36 1)i(i 1)i(i NUM ejpam-87 404 37 + + CCONJ ejpam-87 404 38 1)(n 1)(n NUM ejpam-87 404 39 + + CCONJ ejpam-87 404 40 1 1 NUM ejpam-87 404 41 ) ) PUNCT ejpam-87 404 42 − − PROPN ejpam-87 405 1 6(3n 6(3n NUM ejpam-87 405 2 + + CCONJ ejpam-87 405 3 3 3 NUM ejpam-87 405 4 m m NOUN ejpam-87 405 5 − − NOUN ejpam-87 405 6 nm nm PUNCT ejpam-87 405 7 − − PROPN ejpam-87 405 8 i2 i2 NOUN ejpam-87 405 9 + + CCONJ ejpam-87 405 10 5i− 5i− NUM ejpam-87 405 11 2im 2im NOUN ejpam-87 405 12 − − PROPN ejpam-87 405 13 2ni− 2ni− NUM ejpam-87 405 14 7)s2 7)s2 NUM ejpam-87 405 15 ( ( PUNCT ejpam-87 405 16 m m PROPN ejpam-87 405 17 + + NUM ejpam-87 405 18 1)i(i 1)i(i NUM ejpam-87 406 1 + + CCONJ ejpam-87 406 2 1)(n 1)(n NUM ejpam-87 406 3 + + CCONJ ejpam-87 406 4 1 1 NUM ejpam-87 406 5 ) ) PUNCT ejpam-87 406 6 ( ( PUNCT ejpam-87 406 7 5.1 5.1 NUM ejpam-87 406 8 ) ) PUNCT ejpam-87 407 1 + + NUM ejpam-87 407 2 24(n 24(n NUM ejpam-87 408 1 + + NUM ejpam-87 408 2 2i 2i NUM ejpam-87 408 3 + + NOUN ejpam-87 408 4 m m VERB ejpam-87 408 5 − − NOUN ejpam-87 409 1 6)s3 6)s3 NOUN ejpam-87 409 2 − − NUM ejpam-87 409 3 120s4 120s4 NUM ejpam-87 409 4 ( ( PUNCT ejpam-87 409 5 m m VERB ejpam-87 409 6 + + NUM ejpam-87 409 7 1)i(i 1)i(i NUM ejpam-87 410 1 + + CCONJ ejpam-87 410 2 1)(n 1)(n NUM ejpam-87 410 3 + + CCONJ ejpam-87 410 4 1 1 NUM ejpam-87 410 5 ) ) PUNCT ejpam-87 410 6 . . PUNCT ejpam-87 411 1 case case NOUN ejpam-87 411 2 2 2 X ejpam-87 411 3 . . PUNCT ejpam-87 412 1 let let VERB ejpam-87 412 2 i i PRON ejpam-87 412 3 = = NOUN ejpam-87 412 4 m m VERB ejpam-87 412 5 . . PUNCT ejpam-87 413 1 the the DET ejpam-87 413 2 basis basis NOUN ejpam-87 413 3 bmax bmax NOUN ejpam-87 413 4 = = PUNCT ejpam-87 413 5 { { PUNCT ejpam-87 413 6 0 0 NUM ejpam-87 413 7 , , PUNCT ejpam-87 413 8 1 1 NUM ejpam-87 413 9 , , PUNCT ejpam-87 413 10 m m PROPN ejpam-87 413 11 , , PUNCT ejpam-87 413 12 m m VERB ejpam-87 413 13 + + NOUN ejpam-87 413 14 1 1 NUM ejpam-87 413 15 , , PUNCT ejpam-87 413 16 n n CCONJ ejpam-87 413 17 } } PUNCT ejpam-87 413 18 is be AUX ejpam-87 413 19 primal primal ADJ ejpam-87 413 20 feasible feasible ADJ ejpam-87 413 21 if if SCONJ ejpam-87 413 22 and and CCONJ ejpam-87 413 23 only only ADV ejpam-87 413 24 if if SCONJ ejpam-87 413 25 2m(3 2m(3 NUM ejpam-87 413 26 nm nm VERB ejpam-87 413 27 + + NUM ejpam-87 413 28 m2 m2 PROPN ejpam-87 413 29 + + CCONJ ejpam-87 413 30 2)s1 2)s1 NUM ejpam-87 414 1 − − NOUN ejpam-87 414 2 6(3m2 6(3m2 NOUN ejpam-87 415 1 + + CCONJ ejpam-87 415 2 ( ( PUNCT ejpam-87 415 3 3 3 NUM ejpam-87 415 4 m m NOUN ejpam-87 415 5 − − NOUN ejpam-87 415 6 2)(n− 2)(n− NUM ejpam-87 415 7 2))s2 2))s2 NUM ejpam-87 415 8 + + SYM ejpam-87 415 9 24(n 24(n NUM ejpam-87 415 10 + + CCONJ ejpam-87 415 11 3 3 NUM ejpam-87 415 12 m m NOUN ejpam-87 415 13 − − NOUN ejpam-87 415 14 5)s3 5)s3 NUM ejpam-87 415 15 − − PROPN ejpam-87 415 16 120s4 120s4 NUM ejpam-87 415 17 ≤ ≤ NUM ejpam-87 415 18 m(m m(m NOUN ejpam-87 415 19 + + CCONJ ejpam-87 415 20 1)(m 1)(m NUM ejpam-87 415 21 + + NUM ejpam-87 415 22 2)(n 2)(n NUM ejpam-87 415 23 + + CCONJ ejpam-87 415 24 1 1 NUM ejpam-87 415 25 ) ) PUNCT ejpam-87 415 26 , , PUNCT ejpam-87 415 27 2m(m 2m(m NUM ejpam-87 415 28 − − PROPN ejpam-87 416 1 1)(3n 1)(3n NUM ejpam-87 416 2 + + NUM ejpam-87 416 3 m m VERB ejpam-87 416 4 − − NUM ejpam-87 416 5 2)s1 2)s1 NUM ejpam-87 417 1 − − PROPN ejpam-87 418 1 18(m 18(m NUM ejpam-87 419 1 − − NOUN ejpam-87 419 2 1)(n 1)(n NUM ejpam-87 420 1 + + NOUN ejpam-87 420 2 m m VERB ejpam-87 420 3 − − NUM ejpam-87 420 4 2)s2 2)s2 ADJ ejpam-87 420 5 + + SYM ejpam-87 420 6 24(n 24(n NUM ejpam-87 421 1 + + CCONJ ejpam-87 421 2 3 3 NUM ejpam-87 421 3 m m NOUN ejpam-87 421 4 − − NOUN ejpam-87 422 1 6)s3 6)s3 NOUN ejpam-87 422 2 − − NUM ejpam-87 422 3 120s4 120s4 NUM ejpam-87 422 4 ≥ ≥ NOUN ejpam-87 422 5 m(m m(m NOUN ejpam-87 422 6 − − NOUN ejpam-87 422 7 1)(m 1)(m NUM ejpam-87 423 1 + + CCONJ ejpam-87 423 2 1)n 1)n NUM ejpam-87 423 3 , , PUNCT ejpam-87 423 4 2m(m 2m(m NUM ejpam-87 423 5 − − PROPN ejpam-87 424 1 1)(n− 1)(n− NUM ejpam-87 424 2 1)s1 1)s1 NUM ejpam-87 424 3 − − NOUN ejpam-87 424 4 6(m 6(m NUM ejpam-87 424 5 − − PROPN ejpam-87 424 6 1)(2n 1)(2n NUM ejpam-87 424 7 + + CCONJ ejpam-87 424 8 m m VERB ejpam-87 424 9 − − NOUN ejpam-87 424 10 4)s2 4)s2 NUM ejpam-87 424 11 + + X ejpam-87 424 12 24(n 24(n NUM ejpam-87 425 1 + + CCONJ ejpam-87 425 2 2 2 NUM ejpam-87 425 3 m m NOUN ejpam-87 425 4 − − NOUN ejpam-87 425 5 5)s3 5)s3 NUM ejpam-87 425 6 − − PROPN ejpam-87 425 7 120s4 120s4 NUM ejpam-87 425 8 ≤ ≤ NUM ejpam-87 425 9 0 0 NUM ejpam-87 425 10 , , PUNCT ejpam-87 425 11 2(m 2(m NUM ejpam-87 425 12 − − NUM ejpam-87 426 1 1)(m 1)(m NUM ejpam-87 426 2 − − NUM ejpam-87 426 3 2)(n− 2)(n− NUM ejpam-87 426 4 1)s1− 1)s1− NOUN ejpam-87 426 5 6(m 6(m NUM ejpam-87 426 6 − − PROPN ejpam-87 426 7 2)(2n 2)(2n NUM ejpam-87 427 1 + + NOUN ejpam-87 428 1 m m VERB ejpam-87 428 2 − − NOUN ejpam-87 429 1 5)s2 5)s2 PRON ejpam-87 429 2 + + NOUN ejpam-87 429 3 24(n 24(n NUM ejpam-87 430 1 + + CCONJ ejpam-87 430 2 2 2 NUM ejpam-87 430 3 m m NUM ejpam-87 430 4 − − NOUN ejpam-87 430 5 7)s3− 7)s3− NUM ejpam-87 430 6 120s4 120s4 NUM ejpam-87 430 7 ≥ ≥ NOUN ejpam-87 430 8 0 0 NUM ejpam-87 430 9 , , PUNCT ejpam-87 430 10 2m(m 2m(m NUM ejpam-87 430 11 − − PUNCT ejpam-87 431 1 1)(m 1)(m NUM ejpam-87 431 2 − − PROPN ejpam-87 431 3 2)s1 2)s1 NUM ejpam-87 431 4 − − PROPN ejpam-87 432 1 18(m 18(m NUM ejpam-87 433 1 − − NUM ejpam-87 433 2 1)(m 1)(m NUM ejpam-87 434 1 − − NOUN ejpam-87 434 2 2)s2 2)s2 ADJ ejpam-87 434 3 + + NUM ejpam-87 434 4 72(m 72(m NUM ejpam-87 434 5 − − PROPN ejpam-87 434 6 2)s3 2)s3 NUM ejpam-87 434 7 − − PROPN ejpam-87 434 8 120s4 120s4 NUM ejpam-87 434 9 ≤ ≤ NOUN ejpam-87 434 10 0 0 NUM ejpam-87 434 11 . . PUNCT ejpam-87 435 1 prékopa prékopa ADJ ejpam-87 435 2 , , PUNCT ejpam-87 435 3 m. m. NOUN ejpam-87 435 4 subasi subasi PROPN ejpam-87 435 5 , , PUNCT ejpam-87 435 6 e. e. PROPN ejpam-87 435 7 subasi subasi PROPN ejpam-87 435 8 / / SYM ejpam-87 435 9 eur eur PROPN ejpam-87 435 10 . . PUNCT ejpam-87 436 1 j. j. PROPN ejpam-87 436 2 pure pure PROPN ejpam-87 436 3 appl appl PROPN ejpam-87 436 4 . . PROPN ejpam-87 436 5 math math PROPN ejpam-87 436 6 , , PUNCT ejpam-87 436 7 1 1 NUM ejpam-87 436 8 ( ( PUNCT ejpam-87 436 9 2008 2008 NUM ejpam-87 436 10 ) ) PUNCT ejpam-87 436 11 , , PUNCT ejpam-87 436 12 ( ( PUNCT ejpam-87 436 13 60 60 NUM ejpam-87 436 14 - - SYM ejpam-87 436 15 81 81 NUM ejpam-87 436 16 ) ) PUNCT ejpam-87 436 17 74 74 NUM ejpam-87 436 18 the the DET ejpam-87 436 19 closed closed ADJ ejpam-87 436 20 form form NOUN ejpam-87 436 21 upper upper ADJ ejpam-87 436 22 bound bind VERB ejpam-87 436 23 for for ADP ejpam-87 436 24 p p PROPN ejpam-87 436 25 ( ( PUNCT ejpam-87 436 26 ν ν X ejpam-87 436 27 ≥ ≥ NOUN ejpam-87 436 28 1 1 NUM ejpam-87 436 29 ) ) PUNCT ejpam-87 436 30 is be AUX ejpam-87 436 31 given give VERB ejpam-87 436 32 by by ADP ejpam-87 436 33 p p PROPN ejpam-87 436 34 ( ( PUNCT ejpam-87 436 35 ν ν X ejpam-87 436 36 ≥ ≥ NUM ejpam-87 436 37 1 1 NUM ejpam-87 436 38 ) ) PUNCT ejpam-87 436 39 ≤ ≤ NOUN ejpam-87 436 40 2m(3 2m(3 NUM ejpam-87 436 41 nm nm NOUN ejpam-87 436 42 + + NUM ejpam-87 437 1 m2 m2 PROPN ejpam-87 438 1 + + CCONJ ejpam-87 439 1 2)s1 2)s1 NUM ejpam-87 440 1 − − NOUN ejpam-87 441 1 6(3m2 6(3m2 NOUN ejpam-87 442 1 + + CCONJ ejpam-87 442 2 ( ( PUNCT ejpam-87 442 3 3 3 NUM ejpam-87 442 4 m m NOUN ejpam-87 442 5 − − NOUN ejpam-87 442 6 2)(n− 2)(n− NUM ejpam-87 442 7 2))s2 2))s2 NUM ejpam-87 442 8 m(m m(m NOUN ejpam-87 442 9 + + CCONJ ejpam-87 442 10 1)(m 1)(m NUM ejpam-87 442 11 + + NUM ejpam-87 442 12 2)(n 2)(n NUM ejpam-87 442 13 + + CCONJ ejpam-87 442 14 1 1 NUM ejpam-87 442 15 ) ) PUNCT ejpam-87 442 16 + + NUM ejpam-87 442 17 24(n 24(n NUM ejpam-87 443 1 + + CCONJ ejpam-87 443 2 3 3 NUM ejpam-87 443 3 m m NOUN ejpam-87 443 4 − − NOUN ejpam-87 443 5 5)s3 5)s3 NUM ejpam-87 443 6 − − PROPN ejpam-87 443 7 120s4 120s4 NUM ejpam-87 443 8 m(m m(m NOUN ejpam-87 443 9 + + CCONJ ejpam-87 443 10 1)(m 1)(m NUM ejpam-87 443 11 + + NUM ejpam-87 443 12 2)(n 2)(n NUM ejpam-87 443 13 + + CCONJ ejpam-87 443 14 1 1 NUM ejpam-87 443 15 ) ) PUNCT ejpam-87 443 16 . . PUNCT ejpam-87 444 1 ( ( PUNCT ejpam-87 444 2 5.2 5.2 NUM ejpam-87 444 3 ) ) PUNCT ejpam-87 444 4 case case NOUN ejpam-87 444 5 3 3 X ejpam-87 444 6 . . PUNCT ejpam-87 445 1 let let VERB ejpam-87 445 2 m m PRON ejpam-87 445 3 + + NOUN ejpam-87 445 4 1 1 NUM ejpam-87 445 5 ≤ ≤ NUM ejpam-87 445 6 i i PRON ejpam-87 445 7 ≤ ≤ NOUN ejpam-87 445 8 n n CCONJ ejpam-87 446 1 − − PROPN ejpam-87 446 2 2 2 NUM ejpam-87 446 3 . . PUNCT ejpam-87 446 4 bmax bmax PROPN ejpam-87 446 5 is be AUX ejpam-87 446 6 primal primal ADJ ejpam-87 446 7 feasible feasible ADJ ejpam-87 446 8 if if SCONJ ejpam-87 446 9 and and CCONJ ejpam-87 446 10 only only ADV ejpam-87 446 11 if if SCONJ ejpam-87 446 12 i i PRON ejpam-87 446 13 satisfies satisfy VERB ejpam-87 446 14 the the DET ejpam-87 446 15 conditions condition NOUN ejpam-87 446 16 : : PUNCT ejpam-87 446 17 2[(i+1)(ni+nm 2[(i+1)(ni+nm NUM ejpam-87 446 18 + + NUM ejpam-87 447 1 i i NOUN ejpam-87 447 2 m m VERB ejpam-87 447 3 +1)+n+ +1)+n+ PUNCT ejpam-87 447 4 i+m i+m X ejpam-87 447 5 ] ] PUNCT ejpam-87 447 6 s1−6[(i−1)(i−2)+2i(n+m)+(n−1)(m−1)]s2 s1−6[(i−1)(i−2)+2i(n+m)+(n−1)(m−1)]s2 X ejpam-87 447 7 +24(n +24(n NOUN ejpam-87 447 8 + + NUM ejpam-87 447 9 2i 2i NUM ejpam-87 447 10 + + NOUN ejpam-87 447 11 m m VERB ejpam-87 447 12 − − NOUN ejpam-87 447 13 4)s3 4)s3 NUM ejpam-87 447 14 − − PROPN ejpam-87 447 15 120s4 120s4 NUM ejpam-87 447 16 ≤ ≤ NOUN ejpam-87 448 1 ( ( PUNCT ejpam-87 448 2 i i PRON ejpam-87 448 3 + + NOUN ejpam-87 448 4 1)(i 1)(i NUM ejpam-87 448 5 + + CCONJ ejpam-87 448 6 2)(m 2)(m NUM ejpam-87 448 7 + + SYM ejpam-87 448 8 1)(n 1)(n NUM ejpam-87 448 9 + + CCONJ ejpam-87 448 10 1 1 NUM ejpam-87 448 11 ) ) PUNCT ejpam-87 448 12 , , PUNCT ejpam-87 448 13 2i[n(i 2i[n(i NUM ejpam-87 448 14 + + CCONJ ejpam-87 448 15 m m VERB ejpam-87 448 16 ) ) PUNCT ejpam-87 449 1 + + CCONJ ejpam-87 449 2 ( ( PUNCT ejpam-87 449 3 m m VERB ejpam-87 449 4 − − NOUN ejpam-87 449 5 1)(n 1)(n NUM ejpam-87 449 6 + + NUM ejpam-87 449 7 i− i− PROPN ejpam-87 449 8 1)]s1 1)]s1 NUM ejpam-87 449 9 − − PROPN ejpam-87 449 10 6[(i− 6[(i− NUM ejpam-87 449 11 1)(2n 1)(2n NUM ejpam-87 449 12 + + CCONJ ejpam-87 449 13 2 2 NUM ejpam-87 449 14 m m NOUN ejpam-87 449 15 + + NUM ejpam-87 449 16 i− i− PROPN ejpam-87 449 17 4 4 NUM ejpam-87 449 18 ) ) PUNCT ejpam-87 449 19 + + NUM ejpam-87 449 20 nm nm ADJ ejpam-87 449 21 ] ] X ejpam-87 449 22 s2 s2 NOUN ejpam-87 449 23 +24(n +24(n NOUN ejpam-87 450 1 + + CCONJ ejpam-87 450 2 2i 2i NUM ejpam-87 450 3 + + NOUN ejpam-87 450 4 m m VERB ejpam-87 450 5 − − PROPN ejpam-87 450 6 5)s3 5)s3 NUM ejpam-87 450 7 − − PROPN ejpam-87 450 8 120s4 120s4 NUM ejpam-87 450 9 ≥ ≥ NOUN ejpam-87 450 10 ni(i ni(i NOUN ejpam-87 450 11 + + CCONJ ejpam-87 450 12 1)m 1)m NUM ejpam-87 450 13 , , PUNCT ejpam-87 450 14 2i(m 2i(m NUM ejpam-87 450 15 − − NOUN ejpam-87 450 16 1)(n− 1)(n− NUM ejpam-87 450 17 1)s1 1)s1 NUM ejpam-87 450 18 − − NOUN ejpam-87 451 1 6[(n− 6[(n− NUM ejpam-87 451 2 2)(i 2)(i NUM ejpam-87 451 3 + + CCONJ ejpam-87 451 4 m m VERB ejpam-87 451 5 − − NOUN ejpam-87 451 6 2 2 NUM ejpam-87 451 7 ) ) PUNCT ejpam-87 452 1 + + CCONJ ejpam-87 452 2 i(m i(m NOUN ejpam-87 452 3 − − PUNCT ejpam-87 452 4 1)]s2 1)]s2 NUM ejpam-87 452 5 +24(n +24(n NOUN ejpam-87 453 1 + + CCONJ ejpam-87 453 2 i i PRON ejpam-87 453 3 + + NOUN ejpam-87 453 4 m m VERB ejpam-87 453 5 − − PROPN ejpam-87 453 6 5)s3 5)s3 NUM ejpam-87 453 7 − − PROPN ejpam-87 453 8 120s4 120s4 NUM ejpam-87 453 9 ≤ ≤ NUM ejpam-87 453 10 0 0 NUM ejpam-87 453 11 , , PUNCT ejpam-87 453 12 2(i− 2(i− NUM ejpam-87 453 13 1)(m 1)(m NUM ejpam-87 453 14 − − NUM ejpam-87 453 15 1)(n− 1)(n− NUM ejpam-87 453 16 1)s1 1)s1 NUM ejpam-87 453 17 − − NOUN ejpam-87 454 1 6[(n− 6[(n− NUM ejpam-87 454 2 3)(i 3)(i NUM ejpam-87 454 3 + + CCONJ ejpam-87 454 4 m m VERB ejpam-87 454 5 − − NOUN ejpam-87 454 6 1 1 NUM ejpam-87 454 7 ) ) PUNCT ejpam-87 454 8 + + CCONJ ejpam-87 455 1 i i PRON ejpam-87 455 2 m m VERB ejpam-87 455 3 − − NOUN ejpam-87 455 4 2n 2n NUM ejpam-87 456 1 + + CCONJ ejpam-87 456 2 4]s2 4]s2 PRON ejpam-87 456 3 +24(ni +24(ni PROPN ejpam-87 456 4 + + NUM ejpam-87 456 5 m m VERB ejpam-87 456 6 − − NOUN ejpam-87 457 1 6)s3 6)s3 NOUN ejpam-87 457 2 − − NUM ejpam-87 457 3 120s4 120s4 NUM ejpam-87 457 4 ≥ ≥ NOUN ejpam-87 457 5 0 0 NUM ejpam-87 457 6 , , PUNCT ejpam-87 457 7 2i(i− 2i(i− NUM ejpam-87 457 8 1)(m 1)(m NUM ejpam-87 457 9 − − NOUN ejpam-87 457 10 1)s1 1)s1 NUM ejpam-87 457 11 − − PROPN ejpam-87 458 1 6(i− 6(i− NUM ejpam-87 458 2 1)(i 1)(i NUM ejpam-87 458 3 + + CCONJ ejpam-87 458 4 2 2 NUM ejpam-87 458 5 m m NOUN ejpam-87 458 6 − − NOUN ejpam-87 458 7 4)s2 4)s2 NUM ejpam-87 458 8 + + NOUN ejpam-87 458 9 24(2i 24(2i NUM ejpam-87 459 1 + + CCONJ ejpam-87 459 2 m m VERB ejpam-87 459 3 − − PROPN ejpam-87 459 4 5)s3 5)s3 NUM ejpam-87 459 5 − − PROPN ejpam-87 459 6 120s4 120s4 NUM ejpam-87 459 7 ≤ ≤ NOUN ejpam-87 459 8 0 0 NUM ejpam-87 459 9 . . PUNCT ejpam-87 460 1 the the DET ejpam-87 460 2 corresponding corresponding ADJ ejpam-87 460 3 upper upper ADJ ejpam-87 460 4 bound bind VERB ejpam-87 460 5 for for ADP ejpam-87 460 6 p p PROPN ejpam-87 460 7 ( ( PUNCT ejpam-87 460 8 ν ν X ejpam-87 460 9 ≥ ≥ NOUN ejpam-87 460 10 1 1 NUM ejpam-87 460 11 ) ) PUNCT ejpam-87 460 12 is be AUX ejpam-87 460 13 given give VERB ejpam-87 460 14 by by ADP ejpam-87 460 15 p p PROPN ejpam-87 460 16 ( ( PUNCT ejpam-87 460 17 ν ν X ejpam-87 460 18 ≥ ≥ NUM ejpam-87 460 19 1 1 NUM ejpam-87 460 20 ) ) PUNCT ejpam-87 460 21 ≤ ≤ NOUN ejpam-87 461 1 2[(i 2[(i NUM ejpam-87 461 2 + + CCONJ ejpam-87 461 3 1)(ni 1)(ni NUM ejpam-87 461 4 + + SYM ejpam-87 461 5 nm nm VERB ejpam-87 462 1 + + X ejpam-87 462 2 i i PRON ejpam-87 462 3 m m VERB ejpam-87 462 4 + + ADJ ejpam-87 462 5 1 1 NUM ejpam-87 462 6 ) ) PUNCT ejpam-87 462 7 + + NUM ejpam-87 462 8 n n CCONJ ejpam-87 462 9 + + CCONJ ejpam-87 462 10 i i PRON ejpam-87 462 11 + + NOUN ejpam-87 462 12 m m VERB ejpam-87 462 13 ] ] X ejpam-87 462 14 s1 s1 NOUN ejpam-87 462 15 ( ( PUNCT ejpam-87 462 16 i i PRON ejpam-87 462 17 + + PUNCT ejpam-87 462 18 1)(i 1)(i NUM ejpam-87 462 19 + + SYM ejpam-87 462 20 2)(n 2)(n NUM ejpam-87 463 1 + + CCONJ ejpam-87 463 2 1)(m 1)(m NUM ejpam-87 463 3 + + CCONJ ejpam-87 463 4 1 1 NUM ejpam-87 463 5 ) ) PUNCT ejpam-87 463 6 − − PROPN ejpam-87 463 7 6[(i− 6[(i− NUM ejpam-87 463 8 1)(i− 1)(i− NUM ejpam-87 463 9 2 2 NUM ejpam-87 463 10 ) ) PUNCT ejpam-87 464 1 + + CCONJ ejpam-87 464 2 2i(n 2i(n NUM ejpam-87 464 3 + + SYM ejpam-87 464 4 m m VERB ejpam-87 464 5 ) ) PUNCT ejpam-87 465 1 + + CCONJ ejpam-87 465 2 ( ( PUNCT ejpam-87 465 3 n− n− NOUN ejpam-87 465 4 1)(m 1)(m NUM ejpam-87 465 5 − − NOUN ejpam-87 465 6 1)]s2 1)]s2 NUM ejpam-87 465 7 ( ( PUNCT ejpam-87 465 8 i i PRON ejpam-87 465 9 + + NOUN ejpam-87 465 10 1)(i 1)(i NUM ejpam-87 465 11 + + SYM ejpam-87 465 12 2)(n 2)(n NUM ejpam-87 465 13 + + CCONJ ejpam-87 465 14 1)(m 1)(m NUM ejpam-87 465 15 + + CCONJ ejpam-87 465 16 1 1 NUM ejpam-87 465 17 ) ) PUNCT ejpam-87 465 18 ( ( PUNCT ejpam-87 465 19 5.3 5.3 NUM ejpam-87 465 20 ) ) PUNCT ejpam-87 465 21 + + NUM ejpam-87 465 22 24(n 24(n NUM ejpam-87 466 1 + + NUM ejpam-87 466 2 2i 2i NUM ejpam-87 466 3 + + NOUN ejpam-87 466 4 m m VERB ejpam-87 466 5 − − NOUN ejpam-87 466 6 4)s3 4)s3 NUM ejpam-87 466 7 − − PROPN ejpam-87 466 8 120s4 120s4 NUM ejpam-87 466 9 ( ( PUNCT ejpam-87 466 10 i i PRON ejpam-87 466 11 + + NOUN ejpam-87 466 12 1)(i 1)(i NUM ejpam-87 466 13 + + SYM ejpam-87 466 14 2)(n 2)(n NUM ejpam-87 466 15 + + CCONJ ejpam-87 466 16 1)(m 1)(m NUM ejpam-87 466 17 + + CCONJ ejpam-87 466 18 1 1 NUM ejpam-87 466 19 ) ) PUNCT ejpam-87 466 20 . . PUNCT ejpam-87 467 1 as as ADP ejpam-87 467 2 before before ADV ejpam-87 467 3 , , PUNCT ejpam-87 467 4 if if SCONJ ejpam-87 467 5 we we PRON ejpam-87 467 6 apply apply VERB ejpam-87 467 7 our our PRON ejpam-87 467 8 bounding bounding NOUN ejpam-87 467 9 technique technique NOUN ejpam-87 467 10 on on ADP ejpam-87 467 11 the the DET ejpam-87 467 12 relaxed relaxed ADJ ejpam-87 467 13 problem problem NOUN ejpam-87 467 14 ( ( PUNCT ejpam-87 467 15 1.6 1.6 NUM ejpam-87 467 16 ) ) PUNCT ejpam-87 467 17 , , PUNCT ejpam-87 467 18 rather rather ADV ejpam-87 467 19 than than ADP ejpam-87 467 20 ( ( PUNCT ejpam-87 467 21 1.5 1.5 NUM ejpam-87 467 22 ) ) PUNCT ejpam-87 467 23 , , PUNCT ejpam-87 467 24 then then ADV ejpam-87 467 25 the the DET ejpam-87 467 26 just just ADV ejpam-87 467 27 derived derive VERB ejpam-87 467 28 formulas formula NOUN ejpam-87 467 29 provide provide VERB ejpam-87 467 30 us we PRON ejpam-87 467 31 with with ADP ejpam-87 467 32 the the DET ejpam-87 467 33 upper upper ADJ ejpam-87 467 34 bounds bound NOUN ejpam-87 467 35 if if SCONJ ejpam-87 467 36 we we PRON ejpam-87 467 37 replace replace VERB ejpam-87 467 38 m m PROPN ejpam-87 467 39 − − PROPN ejpam-87 467 40 1 1 NUM ejpam-87 467 41 for for ADP ejpam-87 467 42 m m PRON ejpam-87 467 43 . . PUNCT ejpam-87 468 1 6 6 X ejpam-87 468 2 . . X ejpam-87 468 3 algorithmic algorithmic ADJ ejpam-87 468 4 bounds bound NOUN ejpam-87 468 5 in in ADP ejpam-87 468 6 sections section NOUN ejpam-87 468 7 3 3 NUM ejpam-87 468 8 , , PUNCT ejpam-87 468 9 4 4 NUM ejpam-87 468 10 and and CCONJ ejpam-87 468 11 5 5 NUM ejpam-87 468 12 we we PRON ejpam-87 468 13 have have AUX ejpam-87 468 14 derived derive VERB ejpam-87 468 15 closed closed ADJ ejpam-87 468 16 form form NOUN ejpam-87 468 17 bounds bound NOUN ejpam-87 468 18 for for ADP ejpam-87 468 19 the the DET ejpam-87 468 20 probability probability NOUN ejpam-87 468 21 of of ADP ejpam-87 468 22 the the DET ejpam-87 468 23 union union NOUN ejpam-87 468 24 , , PUNCT ejpam-87 468 25 by by ADP ejpam-87 468 26 the the DET ejpam-87 468 27 use use NOUN ejpam-87 468 28 of of ADP ejpam-87 468 29 the the DET ejpam-87 468 30 relaxed relaxed ADJ ejpam-87 468 31 problems problem NOUN ejpam-87 468 32 ( ( PUNCT ejpam-87 468 33 1.5 1.5 NUM ejpam-87 468 34 ) ) PUNCT ejpam-87 468 35 , , PUNCT ejpam-87 468 36 ( ( PUNCT ejpam-87 468 37 1.6 1.6 NUM ejpam-87 468 38 ) ) PUNCT ejpam-87 468 39 for for ADP ejpam-87 468 40 the the DET ejpam-87 468 41 cases case NOUN ejpam-87 468 42 of of ADP ejpam-87 468 43 m m NOUN ejpam-87 468 44 = = SYM ejpam-87 468 45 2 2 NUM ejpam-87 468 46 , , PUNCT ejpam-87 468 47 3 3 NUM ejpam-87 468 48 , , PUNCT ejpam-87 468 49 4 4 NUM ejpam-87 468 50 . . X ejpam-87 469 1 for for ADP ejpam-87 469 2 larger large ADJ ejpam-87 469 3 m m NOUN ejpam-87 469 4 values value NOUN ejpam-87 469 5 the the DET ejpam-87 469 6 solution solution NOUN ejpam-87 469 7 of of ADP ejpam-87 469 8 the the DET ejpam-87 469 9 relaxed relaxed ADJ ejpam-87 469 10 problems problem NOUN ejpam-87 469 11 can can AUX ejpam-87 469 12 be be AUX ejpam-87 469 13 obtained obtain VERB ejpam-87 469 14 by by ADP ejpam-87 469 15 specially specially ADV ejpam-87 469 16 designed design VERB ejpam-87 469 17 dual dual ADJ ejpam-87 469 18 algorithms algorithm NOUN ejpam-87 469 19 of of ADP ejpam-87 469 20 linear linear PROPN ejpam-87 469 21 programming programming NOUN ejpam-87 469 22 . . PUNCT ejpam-87 470 1 once once ADV ejpam-87 470 2 an an DET ejpam-87 470 3 algorithm algorithm NOUN ejpam-87 470 4 of of ADP ejpam-87 470 5 this this DET ejpam-87 470 6 kind kind NOUN ejpam-87 470 7 terminates terminate VERB ejpam-87 470 8 , , PUNCT ejpam-87 470 9 the the DET ejpam-87 470 10 prékopa prékopa PROPN ejpam-87 470 11 , , PUNCT ejpam-87 470 12 m. m. NOUN ejpam-87 470 13 subasi subasi PROPN ejpam-87 470 14 , , PUNCT ejpam-87 470 15 e. e. PROPN ejpam-87 470 16 subasi subasi PROPN ejpam-87 470 17 / / SYM ejpam-87 470 18 eur eur PROPN ejpam-87 470 19 . . PUNCT ejpam-87 471 1 j. j. PROPN ejpam-87 471 2 pure pure PROPN ejpam-87 471 3 appl appl PROPN ejpam-87 471 4 . . PROPN ejpam-87 471 5 math math PROPN ejpam-87 471 6 , , PUNCT ejpam-87 471 7 1 1 NUM ejpam-87 471 8 ( ( PUNCT ejpam-87 471 9 2008 2008 NUM ejpam-87 471 10 ) ) PUNCT ejpam-87 471 11 , , PUNCT ejpam-87 471 12 ( ( PUNCT ejpam-87 471 13 60 60 NUM ejpam-87 471 14 - - SYM ejpam-87 471 15 81 81 NUM ejpam-87 471 16 ) ) PUNCT ejpam-87 471 17 75 75 NUM ejpam-87 471 18 solutions solution NOUN ejpam-87 471 19 for for ADP ejpam-87 471 20 the the DET ejpam-87 471 21 non non ADJ ejpam-87 471 22 - - ADJ ejpam-87 471 23 relaxed relaxed ADJ ejpam-87 471 24 problem problem NOUN ejpam-87 471 25 can can AUX ejpam-87 471 26 be be AUX ejpam-87 471 27 continued continue VERB ejpam-87 471 28 again again ADV ejpam-87 471 29 by by ADP ejpam-87 471 30 the the DET ejpam-87 471 31 dual dual ADJ ejpam-87 471 32 algorithm algorithm NOUN ejpam-87 471 33 . . PUNCT ejpam-87 472 1 in in ADP ejpam-87 472 2 fact fact NOUN ejpam-87 472 3 , , PUNCT ejpam-87 472 4 as as SCONJ ejpam-87 472 5 it it PRON ejpam-87 472 6 is be AUX ejpam-87 472 7 well well ADV ejpam-87 472 8 known know VERB ejpam-87 472 9 in in ADP ejpam-87 472 10 linear linear PROPN ejpam-87 472 11 programming programming NOUN ejpam-87 472 12 , , PUNCT ejpam-87 472 13 the the DET ejpam-87 472 14 dual dual ADJ ejpam-87 472 15 algorithm algorithm NOUN ejpam-87 472 16 can can AUX ejpam-87 472 17 efficiently efficiently ADV ejpam-87 472 18 be be AUX ejpam-87 472 19 used use VERB ejpam-87 472 20 , , PUNCT ejpam-87 472 21 as as ADP ejpam-87 472 22 a a DET ejpam-87 472 23 reoptimization reoptimization NOUN ejpam-87 472 24 technique technique NOUN ejpam-87 472 25 , , PUNCT ejpam-87 472 26 whenever whenever SCONJ ejpam-87 472 27 the the DET ejpam-87 472 28 optimal optimal ADJ ejpam-87 472 29 basis basis NOUN ejpam-87 472 30 has have AUX ejpam-87 472 31 already already ADV ejpam-87 472 32 been be AUX ejpam-87 472 33 found find VERB ejpam-87 472 34 but but CCONJ ejpam-87 472 35 a a DET ejpam-87 472 36 further further ADJ ejpam-87 472 37 constraint constraint NOUN ejpam-87 472 38 is be AUX ejpam-87 472 39 introduced introduce VERB ejpam-87 472 40 into into ADP ejpam-87 472 41 the the DET ejpam-87 472 42 problem problem NOUN ejpam-87 472 43 . . PUNCT ejpam-87 473 1 the the DET ejpam-87 473 2 algorithm algorithm NOUN ejpam-87 473 3 presented present VERB ejpam-87 473 4 below below ADP ejpam-87 473 5 works work NOUN ejpam-87 473 6 in in ADP ejpam-87 473 7 this this DET ejpam-87 473 8 way way NOUN ejpam-87 473 9 and and CCONJ ejpam-87 473 10 is be AUX ejpam-87 473 11 applicable applicable ADJ ejpam-87 473 12 to to ADP ejpam-87 473 13 cases case NOUN ejpam-87 473 14 with with ADP ejpam-87 473 15 consecutive consecutive ADJ ejpam-87 473 16 and and CCONJ ejpam-87 473 17 non non ADJ ejpam-87 473 18 - - ADJ ejpam-87 473 19 consecutive consecutive ADJ ejpam-87 473 20 moments moment NOUN ejpam-87 473 21 . . PUNCT ejpam-87 474 1 we we PRON ejpam-87 474 2 remark remark VERB ejpam-87 474 3 that that SCONJ ejpam-87 474 4 it it PRON ejpam-87 474 5 is be AUX ejpam-87 474 6 more more ADV ejpam-87 474 7 practical practical ADJ ejpam-87 474 8 to to PART ejpam-87 474 9 carry carry VERB ejpam-87 474 10 out out ADP ejpam-87 474 11 the the DET ejpam-87 474 12 algorithms algorithm NOUN ejpam-87 474 13 to to PART ejpam-87 474 14 obtain obtain VERB ejpam-87 474 15 the the DET ejpam-87 474 16 bound bind VERB ejpam-87 474 17 , , PUNCT ejpam-87 474 18 rather rather ADV ejpam-87 474 19 than than SCONJ ejpam-87 474 20 to to PART ejpam-87 474 21 apply apply VERB ejpam-87 474 22 a a DET ejpam-87 474 23 complicated complicated ADJ ejpam-87 474 24 closed closed ADJ ejpam-87 474 25 form form NOUN ejpam-87 474 26 formula formula NOUN ejpam-87 474 27 . . PUNCT ejpam-87 475 1 algorithmic algorithmic ADJ ejpam-87 475 2 solutions solution NOUN ejpam-87 475 3 of of ADP ejpam-87 475 4 problems problem NOUN ejpam-87 475 5 ( ( PUNCT ejpam-87 475 6 1.5 1.5 NUM ejpam-87 475 7 ) ) PUNCT ejpam-87 475 8 , , PUNCT ejpam-87 475 9 ( ( PUNCT ejpam-87 475 10 1.6 1.6 NUM ejpam-87 475 11 ) ) PUNCT ejpam-87 475 12 step step NOUN ejpam-87 475 13 0 0 NUM ejpam-87 475 14 . . PUNCT ejpam-87 475 15 find find VERB ejpam-87 475 16 an an DET ejpam-87 475 17 initial initial ADJ ejpam-87 475 18 dual dual ADJ ejpam-87 475 19 feasible feasible ADJ ejpam-87 475 20 basis basis NOUN ejpam-87 475 21 b b NOUN ejpam-87 475 22 to to ADP ejpam-87 475 23 the the DET ejpam-87 475 24 relaxed relaxed ADJ ejpam-87 475 25 problem problem NOUN ejpam-87 475 26 . . PUNCT ejpam-87 476 1 any any DET ejpam-87 476 2 basis basis NOUN ejpam-87 476 3 that that PRON ejpam-87 476 4 has have VERB ejpam-87 476 5 the the DET ejpam-87 476 6 structure structure NOUN ejpam-87 476 7 presented present VERB ejpam-87 476 8 in in ADP ejpam-87 476 9 theorem theorem ADJ ejpam-87 476 10 2 2 NUM ejpam-87 476 11 is be AUX ejpam-87 476 12 suitable suitable ADJ ejpam-87 476 13 . . PUNCT ejpam-87 477 1 step step NOUN ejpam-87 477 2 1 1 NUM ejpam-87 477 3 . . PUNCT ejpam-87 477 4 check check VERB ejpam-87 477 5 for for ADP ejpam-87 477 6 primal primal ADJ ejpam-87 477 7 feasibility feasibility NOUN ejpam-87 477 8 . . PUNCT ejpam-87 478 1 if if SCONJ ejpam-87 478 2 b−1b b−1b X ejpam-87 478 3 ≥ ≥ NOUN ejpam-87 478 4 0 0 NUM ejpam-87 478 5 , , PUNCT ejpam-87 478 6 then then ADV ejpam-87 478 7 the the DET ejpam-87 478 8 solution solution NOUN ejpam-87 478 9 of of ADP ejpam-87 478 10 the the DET ejpam-87 478 11 relaxed relaxed ADJ ejpam-87 478 12 problem problem NOUN ejpam-87 478 13 terminates terminate VERB ejpam-87 478 14 . . PUNCT ejpam-87 479 1 go go VERB ejpam-87 479 2 to to PART ejpam-87 479 3 step step VERB ejpam-87 479 4 4 4 NUM ejpam-87 479 5 . . PUNCT ejpam-87 479 6 otherwise otherwise ADV ejpam-87 479 7 go go VERB ejpam-87 479 8 to to PART ejpam-87 479 9 step step VERB ejpam-87 479 10 2 2 NUM ejpam-87 479 11 . . PUNCT ejpam-87 480 1 step step NOUN ejpam-87 480 2 2 2 NUM ejpam-87 480 3 . . PUNCT ejpam-87 481 1 if if SCONJ ejpam-87 481 2 ( ( PUNCT ejpam-87 481 3 b−1b)j b−1b)j VERB ejpam-87 481 4 < < X ejpam-87 481 5 0 0 NUM ejpam-87 481 6 , , PUNCT ejpam-87 481 7 then then ADV ejpam-87 481 8 the the DET ejpam-87 481 9 jth jth PROPN ejpam-87 481 10 vector vector NOUN ejpam-87 481 11 in in ADP ejpam-87 481 12 b b PROPN ejpam-87 481 13 ( ( PUNCT ejpam-87 481 14 not not PART ejpam-87 481 15 necessarily necessarily ADV ejpam-87 481 16 equal equal ADJ ejpam-87 481 17 to to ADP ejpam-87 481 18 aj aj PROPN ejpam-87 481 19 ) ) PUNCT ejpam-87 481 20 is be AUX ejpam-87 481 21 a a DET ejpam-87 481 22 candidate candidate NOUN ejpam-87 481 23 to to PART ejpam-87 481 24 leave leave VERB ejpam-87 481 25 the the DET ejpam-87 481 26 basis basis NOUN ejpam-87 481 27 . . PUNCT ejpam-87 482 1 choose choose VERB ejpam-87 482 2 arbitrarily arbitrarily ADV ejpam-87 482 3 among among ADP ejpam-87 482 4 the the DET ejpam-87 482 5 candidates candidate NOUN ejpam-87 482 6 to to PART ejpam-87 482 7 leave leave VERB ejpam-87 482 8 the the DET ejpam-87 482 9 basis basis NOUN ejpam-87 482 10 . . PUNCT ejpam-87 483 1 go go VERB ejpam-87 483 2 to to PART ejpam-87 483 3 step step VERB ejpam-87 483 4 3 3 NUM ejpam-87 483 5 . . PUNCT ejpam-87 484 1 step step NOUN ejpam-87 484 2 3 3 NUM ejpam-87 484 3 . . PUNCT ejpam-87 484 4 include include VERB ejpam-87 484 5 the the DET ejpam-87 484 6 vector vector NOUN ejpam-87 484 7 al al PROPN ejpam-87 484 8 into into ADP ejpam-87 484 9 the the DET ejpam-87 484 10 basis basis NOUN ejpam-87 484 11 that that PRON ejpam-87 484 12 restores restore VERB ejpam-87 484 13 the the DET ejpam-87 484 14 dual dual ADJ ejpam-87 484 15 feasible feasible ADJ ejpam-87 484 16 basis basis NOUN ejpam-87 484 17 structure structure NOUN ejpam-87 484 18 . . PUNCT ejpam-87 485 1 go go VERB ejpam-87 485 2 to to PART ejpam-87 485 3 step step VERB ejpam-87 485 4 1 1 NUM ejpam-87 485 5 . . PUNCT ejpam-87 486 1 step step NOUN ejpam-87 486 2 4 4 NUM ejpam-87 486 3 . . PUNCT ejpam-87 487 1 if if SCONJ ejpam-87 487 2 the the DET ejpam-87 487 3 additional additional ADJ ejpam-87 487 4 constraint constraint NOUN ejpam-87 487 5 v0 v0 NOUN ejpam-87 487 6 + + CCONJ ejpam-87 487 7 ... ... PUNCT ejpam-87 488 1 + + CCONJ ejpam-87 488 2 vm vm PROPN ejpam-87 488 3 ≥ ≥ NOUN ejpam-87 488 4 vm+1 vm+1 PROPN ejpam-87 488 5 + + CCONJ ejpam-87 488 6 ... ... PUNCT ejpam-87 489 1 + + CCONJ ejpam-87 489 2 vn vn X ejpam-87 489 3 ( ( PUNCT ejpam-87 489 4 or or CCONJ ejpam-87 489 5 vm vm PROPN ejpam-87 489 6 + + CCONJ ejpam-87 489 7 ... ... PUNCT ejpam-87 490 1 + + CCONJ ejpam-87 490 2 vn vn X ejpam-87 490 3 ≥ ≥ PROPN ejpam-87 490 4 v0 v0 PROPN ejpam-87 490 5 + + CCONJ ejpam-87 490 6 ... ... PUNCT ejpam-87 491 1 + + CCONJ ejpam-87 491 2 vm−1 vm−1 NOUN ejpam-87 491 3 ) ) PUNCT ejpam-87 491 4 is be AUX ejpam-87 491 5 satisfied satisfied ADJ ejpam-87 491 6 , , PUNCT ejpam-87 491 7 then then ADV ejpam-87 491 8 the the DET ejpam-87 491 9 solution solution NOUN ejpam-87 491 10 of of ADP ejpam-87 491 11 problem problem NOUN ejpam-87 491 12 ( ( PUNCT ejpam-87 491 13 1.5 1.5 NUM ejpam-87 491 14 ) ) PUNCT ejpam-87 491 15 ( ( PUNCT ejpam-87 491 16 or or CCONJ ejpam-87 491 17 ( ( PUNCT ejpam-87 491 18 1.6 1.6 NUM ejpam-87 491 19 ) ) PUNCT ejpam-87 491 20 ) ) PUNCT ejpam-87 491 21 terminates terminate VERB ejpam-87 491 22 . . PUNCT ejpam-87 492 1 otherwise otherwise ADV ejpam-87 492 2 go go VERB ejpam-87 492 3 to to PART ejpam-87 492 4 step step VERB ejpam-87 492 5 5 5 NUM ejpam-87 492 6 . . PUNCT ejpam-87 493 1 step step NOUN ejpam-87 493 2 5 5 NUM ejpam-87 493 3 . . PUNCT ejpam-87 494 1 reoptimize reoptimize VERB ejpam-87 494 2 the the DET ejpam-87 494 3 problem problem NOUN ejpam-87 494 4 with with ADP ejpam-87 494 5 the the DET ejpam-87 494 6 additional additional ADJ ejpam-87 494 7 constraint constraint NOUN ejpam-87 494 8 ( ( PUNCT ejpam-87 494 9 1.5a 1.5a NUM ejpam-87 494 10 ) ) PUNCT ejpam-87 494 11 or or CCONJ ejpam-87 494 12 ( ( PUNCT ejpam-87 494 13 1.6a 1.6a NUM ejpam-87 494 14 ): ): PUNCT ejpam-87 494 15 introduce introduce VERB ejpam-87 494 16 slack slack NOUN ejpam-87 494 17 variable variable ADJ ejpam-87 494 18 into into ADP ejpam-87 494 19 the the DET ejpam-87 494 20 additional additional ADJ ejpam-87 494 21 inequality inequality NOUN ejpam-87 494 22 constraint constraint NOUN ejpam-87 494 23 , , PUNCT ejpam-87 494 24 prescribe prescribe VERB ejpam-87 494 25 nonnegativity nonnegativity NOUN ejpam-87 494 26 relation relation NOUN ejpam-87 494 27 for for ADP ejpam-87 494 28 the the DET ejpam-87 494 29 slack slack NOUN ejpam-87 494 30 variable variable NOUN ejpam-87 494 31 , , PUNCT ejpam-87 494 32 set set VERB ejpam-87 494 33 up up ADP ejpam-87 494 34 the the DET ejpam-87 494 35 new new ADJ ejpam-87 494 36 dual dual ADJ ejpam-87 494 37 tableau tableau NOUN ejpam-87 494 38 and and CCONJ ejpam-87 494 39 carry carry VERB ejpam-87 494 40 out out ADP ejpam-87 494 41 the the DET ejpam-87 494 42 dual dual ADJ ejpam-87 494 43 method method NOUN ejpam-87 494 44 . . PUNCT ejpam-87 495 1 if if SCONJ ejpam-87 495 2 the the DET ejpam-87 495 3 sequence sequence NOUN ejpam-87 495 4 of of ADP ejpam-87 495 5 probabilities probability NOUN ejpam-87 495 6 p0 p0 NOUN ejpam-87 495 7 , , PUNCT ejpam-87 495 8 ... ... PUNCT ejpam-87 495 9 , , PUNCT ejpam-87 495 10 pn pn PROPN ejpam-87 495 11 is be AUX ejpam-87 495 12 increasing increase VERB ejpam-87 495 13 or or CCONJ ejpam-87 495 14 decreasing decrease VERB ejpam-87 495 15 , , PUNCT ejpam-87 495 16 i.e. i.e. X ejpam-87 495 17 , , PUNCT ejpam-87 495 18 if if SCONJ ejpam-87 495 19 m m NOUN ejpam-87 495 20 = = SYM ejpam-87 495 21 n n ADJ ejpam-87 495 22 or or CCONJ ejpam-87 495 23 m m PROPN ejpam-87 495 24 = = ADJ ejpam-87 495 25 0 0 NUM ejpam-87 495 26 , , PUNCT ejpam-87 495 27 then then ADV ejpam-87 495 28 the the DET ejpam-87 495 29 solution solution NOUN ejpam-87 495 30 of of ADP ejpam-87 495 31 problem problem NOUN ejpam-87 495 32 ( ( PUNCT ejpam-87 495 33 1.5 1.5 NUM ejpam-87 495 34 ) ) PUNCT ejpam-87 495 35 or or CCONJ ejpam-87 495 36 ( ( PUNCT ejpam-87 495 37 1.6 1.6 NUM ejpam-87 495 38 ) ) PUNCT ejpam-87 495 39 terminates terminate VERB ejpam-87 495 40 with with ADP ejpam-87 495 41 step step NOUN ejpam-87 495 42 3 3 NUM ejpam-87 495 43 . . PUNCT ejpam-87 496 1 no no DET ejpam-87 496 2 reoptimization reoptimization NOUN ejpam-87 496 3 is be AUX ejpam-87 496 4 needed need VERB ejpam-87 496 5 . . PUNCT ejpam-87 497 1 the the DET ejpam-87 497 2 relaxed relaxed ADJ ejpam-87 497 3 problem problem NOUN ejpam-87 497 4 is be AUX ejpam-87 497 5 equivalent equivalent ADJ ejpam-87 497 6 to to ADP ejpam-87 497 7 the the DET ejpam-87 497 8 original original ADJ ejpam-87 497 9 problem problem NOUN ejpam-87 497 10 ( ( PUNCT ejpam-87 497 11 1.5 1.5 NUM ejpam-87 497 12 ) ) PUNCT ejpam-87 497 13 or or CCONJ ejpam-87 497 14 ( ( PUNCT ejpam-87 497 15 1.6 1.6 NUM ejpam-87 497 16 ) ) PUNCT ejpam-87 497 17 . . PUNCT ejpam-87 498 1 7 7 X ejpam-87 498 2 . . X ejpam-87 498 3 application application NOUN ejpam-87 498 4 in in ADP ejpam-87 498 5 reliability reliability NOUN ejpam-87 498 6 let let VERB ejpam-87 498 7 a1 a1 NOUN ejpam-87 498 8 , , PUNCT ejpam-87 498 9 ... ... PUNCT ejpam-87 498 10 , , PUNCT ejpam-87 498 11 an an DET ejpam-87 498 12 be be AUX ejpam-87 498 13 independent independent ADJ ejpam-87 498 14 events event NOUN ejpam-87 498 15 and and CCONJ ejpam-87 498 16 define define VERB ejpam-87 498 17 the the DET ejpam-87 498 18 random random ADJ ejpam-87 498 19 variables variable NOUN ejpam-87 498 20 x1 x1 PROPN ejpam-87 498 21 , , PUNCT ejpam-87 498 22 ... ... PUNCT ejpam-87 498 23 , , PUNCT ejpam-87 498 24 xn xn PROPN ejpam-87 498 25 as as ADP ejpam-87 498 26 the the DET ejpam-87 498 27 characteristic characteristic ADJ ejpam-87 498 28 variables variable NOUN ejpam-87 498 29 corresponding correspond VERB ejpam-87 498 30 to to ADP ejpam-87 498 31 the the DET ejpam-87 498 32 above above ADJ ejpam-87 498 33 events event NOUN ejpam-87 498 34 , , PUNCT ejpam-87 498 35 respectively respectively ADV ejpam-87 498 36 , , PUNCT ejpam-87 498 37 i.e. i.e. X ejpam-87 498 38 , , PUNCT ejpam-87 498 39 xi xi X ejpam-87 498 40 = = SYM ejpam-87 498 41 { { PUNCT ejpam-87 498 42 1 1 NUM ejpam-87 498 43 if if SCONJ ejpam-87 498 44 ai ai VERB ejpam-87 498 45 occurs occur VERB ejpam-87 498 46 , , PUNCT ejpam-87 498 47 0 0 PUNCT ejpam-87 498 48 otherwise otherwise ADV ejpam-87 498 49 . . PUNCT ejpam-87 499 1 let let VERB ejpam-87 499 2 pi pi NOUN ejpam-87 499 3 = = PUNCT ejpam-87 499 4 p p X ejpam-87 499 5 ( ( PUNCT ejpam-87 499 6 xi xi X ejpam-87 499 7 = = SYM ejpam-87 499 8 1 1 NUM ejpam-87 499 9 ) ) PUNCT ejpam-87 499 10 , , PUNCT ejpam-87 499 11 i i PRON ejpam-87 499 12 = = NOUN ejpam-87 499 13 1 1 NUM ejpam-87 499 14 , , PUNCT ejpam-87 499 15 ... ... PUNCT ejpam-87 499 16 , , PUNCT ejpam-87 499 17 n. n. VERB ejpam-87 499 18 the the DET ejpam-87 499 19 random random ADJ ejpam-87 499 20 variables variable NOUN ejpam-87 499 21 x1 x1 PROPN ejpam-87 499 22 , , PUNCT ejpam-87 499 23 ... ... PUNCT ejpam-87 499 24 , , PUNCT ejpam-87 499 25 xn xn PROPN ejpam-87 499 26 have have AUX ejpam-87 499 27 logconcave logconcave VERB ejpam-87 499 28 discrete discrete ADJ ejpam-87 499 29 distributions distribution NOUN ejpam-87 499 30 . . PUNCT ejpam-87 500 1 since since SCONJ ejpam-87 500 2 the the DET ejpam-87 500 3 convolution convolution NOUN ejpam-87 500 4 of of ADP ejpam-87 500 5 discrete discrete ADJ ejpam-87 500 6 logconcave logconcave NOUN ejpam-87 500 7 sequences sequence NOUN ejpam-87 500 8 is be AUX ejpam-87 500 9 logconcave logconcave ADJ ejpam-87 500 10 ( ( PUNCT ejpam-87 500 11 see see VERB ejpam-87 500 12 , , PUNCT ejpam-87 500 13 e.g. e.g. ADV ejpam-87 500 14 , , PUNCT ejpam-87 500 15 prékopa prékopa VERB ejpam-87 501 1 [ [ X ejpam-87 501 2 11 11 NUM ejpam-87 501 3 ] ] NUM ejpam-87 501 4 ) ) PUNCT ejpam-87 501 5 , , PUNCT ejpam-87 501 6 it it PRON ejpam-87 501 7 follows follow VERB ejpam-87 501 8 that that SCONJ ejpam-87 501 9 the the DET ejpam-87 501 10 distribution distribution NOUN ejpam-87 501 11 of of ADP ejpam-87 501 12 x1 x1 PROPN ejpam-87 501 13 + + CCONJ ejpam-87 501 14 ... ... PUNCT ejpam-87 502 1 + + CCONJ ejpam-87 502 2 xn xn X ejpam-87 502 3 is be AUX ejpam-87 502 4 also also ADV ejpam-87 502 5 logconcave logconcave ADJ ejpam-87 502 6 . . PUNCT ejpam-87 503 1 in in ADP ejpam-87 503 2 many many ADJ ejpam-87 503 3 applications application NOUN ejpam-87 503 4 it it PRON ejpam-87 503 5 is be AUX ejpam-87 503 6 an an DET ejpam-87 503 7 important important ADJ ejpam-87 503 8 problem problem NOUN ejpam-87 503 9 to to PART ejpam-87 503 10 compute compute VERB ejpam-87 503 11 , , PUNCT ejpam-87 503 12 or or CCONJ ejpam-87 503 13 at at ADP ejpam-87 503 14 least least ADJ ejpam-87 503 15 approximate approximate ADJ ejpam-87 503 16 , , PUNCT ejpam-87 503 17 prékopa prékopa NOUN ejpam-87 503 18 , , PUNCT ejpam-87 503 19 m. m. NOUN ejpam-87 503 20 subasi subasi PROPN ejpam-87 503 21 , , PUNCT ejpam-87 503 22 e. e. PROPN ejpam-87 503 23 subasi subasi PROPN ejpam-87 503 24 / / SYM ejpam-87 503 25 eur eur PROPN ejpam-87 503 26 . . PUNCT ejpam-87 504 1 j. j. PROPN ejpam-87 504 2 pure pure PROPN ejpam-87 504 3 appl appl PROPN ejpam-87 504 4 . . PROPN ejpam-87 504 5 math math PROPN ejpam-87 504 6 , , PUNCT ejpam-87 504 7 1 1 NUM ejpam-87 504 8 ( ( PUNCT ejpam-87 504 9 2008 2008 NUM ejpam-87 504 10 ) ) PUNCT ejpam-87 504 11 , , PUNCT ejpam-87 504 12 ( ( PUNCT ejpam-87 504 13 60 60 NUM ejpam-87 504 14 - - SYM ejpam-87 504 15 81 81 NUM ejpam-87 504 16 ) ) PUNCT ejpam-87 504 17 76 76 NUM ejpam-87 504 18 e.g. e.g. ADV ejpam-87 504 19 , , PUNCT ejpam-87 504 20 by by ADP ejpam-87 504 21 the the DET ejpam-87 504 22 use use NOUN ejpam-87 504 23 of of ADP ejpam-87 504 24 bounds bound NOUN ejpam-87 504 25 , , PUNCT ejpam-87 504 26 the the DET ejpam-87 504 27 probability probability NOUN ejpam-87 504 28 p p X ejpam-87 504 29 ( ( PUNCT ejpam-87 504 30 x1 x1 PROPN ejpam-87 504 31 + + CCONJ ejpam-87 504 32 ... ... PUNCT ejpam-87 505 1 + + CCONJ ejpam-87 505 2 xn xn NUM ejpam-87 505 3 ≥ ≥ NOUN ejpam-87 505 4 1 1 NUM ejpam-87 505 5 ) ) PUNCT ejpam-87 505 6 . . PUNCT ejpam-87 506 1 ( ( PUNCT ejpam-87 506 2 7.1 7.1 NUM ejpam-87 506 3 ) ) PUNCT ejpam-87 506 4 if if SCONJ ejpam-87 506 5 i1 i1 PROPN ejpam-87 506 6 , , PUNCT ejpam-87 506 7 ... ... PUNCT ejpam-87 506 8 , , PUNCT ejpam-87 506 9 ic(n ic(n X ejpam-87 506 10 , , PUNCT ejpam-87 506 11 k k NOUN ejpam-87 506 12 ) ) PUNCT ejpam-87 506 13 designate designate VERB ejpam-87 506 14 the the DET ejpam-87 506 15 k k ADJ ejpam-87 506 16 - - ADJ ejpam-87 506 17 element element ADJ ejpam-87 506 18 subsets subset NOUN ejpam-87 506 19 of of ADP ejpam-87 506 20 the the DET ejpam-87 506 21 set set NOUN ejpam-87 506 22 { { PUNCT ejpam-87 506 23 1 1 NUM ejpam-87 506 24 , , PUNCT ejpam-87 506 25 ... ... PUNCT ejpam-87 506 26 , , PUNCT ejpam-87 506 27 n n CCONJ ejpam-87 506 28 } } PUNCT ejpam-87 506 29 and and CCONJ ejpam-87 506 30 jl jl NOUN ejpam-87 506 31 = = PUNCT ejpam-87 506 32 { { PUNCT ejpam-87 506 33 1 1 NUM ejpam-87 506 34 , , PUNCT ejpam-87 506 35 ... ... PUNCT ejpam-87 506 36 , , PUNCT ejpam-87 506 37 n}\il n}\il NOUN ejpam-87 506 38 , , PUNCT ejpam-87 506 39 l l NOUN ejpam-87 506 40 = = SYM ejpam-87 506 41 1 1 NUM ejpam-87 506 42 , , PUNCT ejpam-87 506 43 ... ... PUNCT ejpam-87 506 44 , , PUNCT ejpam-87 506 45 c(n c(n PROPN ejpam-87 506 46 , , PUNCT ejpam-87 506 47 k k PROPN ejpam-87 506 48 ) ) PUNCT ejpam-87 506 49 , , PUNCT ejpam-87 506 50 then then ADV ejpam-87 506 51 we we PRON ejpam-87 506 52 have have VERB ejpam-87 506 53 the the DET ejpam-87 506 54 equation equation NOUN ejpam-87 506 55 p p NOUN ejpam-87 506 56 ( ( PUNCT ejpam-87 506 57 x1 x1 PROPN ejpam-87 506 58 + + CCONJ ejpam-87 506 59 ... ... PUNCT ejpam-87 507 1 + + CCONJ ejpam-87 507 2 xn xn NUM ejpam-87 507 3 ≥ ≥ NOUN ejpam-87 507 4 1 1 NUM ejpam-87 507 5 ) ) PUNCT ejpam-87 507 6 = = SYM ejpam-87 507 7 n∑ n∑ NOUN ejpam-87 507 8 k=1 k=1 PUNCT ejpam-87 508 1 c(n c(n PROPN ejpam-87 508 2 , , PUNCT ejpam-87 508 3 k)∑ k)∑ ADJ ejpam-87 508 4 l=1 l=1 PROPN ejpam-87 508 5 ∏ ∏ PROPN ejpam-87 508 6 i∈il i∈il PROPN ejpam-87 508 7 pi pi PROPN ejpam-87 508 8 ∏ ∏ PROPN ejpam-87 508 9 j∈jl j∈jl NOUN ejpam-87 508 10 ( ( PUNCT ejpam-87 508 11 1− 1− NUM ejpam-87 508 12 pj pj PROPN ejpam-87 508 13 ) ) PUNCT ejpam-87 508 14 , , PUNCT ejpam-87 508 15 ( ( PUNCT ejpam-87 508 16 7.2 7.2 NUM ejpam-87 508 17 ) ) PUNCT ejpam-87 509 1 where where SCONJ ejpam-87 509 2 c(n c(n PROPN ejpam-87 509 3 , , PUNCT ejpam-87 509 4 k k NOUN ejpam-87 509 5 ) ) PUNCT ejpam-87 509 6 = = SYM ejpam-87 509 7 ( ( PUNCT ejpam-87 509 8 n n X ejpam-87 509 9 k k NOUN ejpam-87 509 10 ) ) PUNCT ejpam-87 509 11 . . PUNCT ejpam-87 510 1 if if SCONJ ejpam-87 510 2 n n PRON ejpam-87 510 3 is be AUX ejpam-87 510 4 large large ADJ ejpam-87 510 5 , , PUNCT ejpam-87 510 6 then then ADV ejpam-87 510 7 the the DET ejpam-87 510 8 calculation calculation NOUN ejpam-87 510 9 of of ADP ejpam-87 510 10 the the DET ejpam-87 510 11 probabilities probability NOUN ejpam-87 510 12 on on ADP ejpam-87 510 13 the the DET ejpam-87 510 14 right right ADJ ejpam-87 510 15 hand hand NOUN ejpam-87 510 16 side side NOUN ejpam-87 510 17 of of ADP ejpam-87 510 18 ( ( PUNCT ejpam-87 510 19 7.2 7.2 NUM ejpam-87 510 20 ) ) PUNCT ejpam-87 510 21 may may AUX ejpam-87 510 22 be be AUX ejpam-87 510 23 hard hard ADJ ejpam-87 510 24 , , PUNCT ejpam-87 510 25 even even ADV ejpam-87 510 26 impossible impossible ADJ ejpam-87 510 27 . . PUNCT ejpam-87 511 1 however however ADV ejpam-87 511 2 , , PUNCT ejpam-87 511 3 we we PRON ejpam-87 511 4 can can AUX ejpam-87 511 5 calculate calculate VERB ejpam-87 511 6 lower low ADJ ejpam-87 511 7 and and CCONJ ejpam-87 511 8 upper upper ADJ ejpam-87 511 9 bounds bound NOUN ejpam-87 511 10 for for ADP ejpam-87 511 11 the the DET ejpam-87 511 12 probability probability NOUN ejpam-87 511 13 on on ADP ejpam-87 511 14 the the DET ejpam-87 511 15 left left ADJ ejpam-87 511 16 hand hand NOUN ejpam-87 511 17 side side NOUN ejpam-87 511 18 of of ADP ejpam-87 511 19 ( ( PUNCT ejpam-87 511 20 7.2 7.2 NUM ejpam-87 511 21 ) ) PUNCT ejpam-87 511 22 by by ADP ejpam-87 511 23 the the DET ejpam-87 511 24 use use NOUN ejpam-87 511 25 of of ADP ejpam-87 511 26 the the DET ejpam-87 511 27 sums sum NOUN ejpam-87 511 28 : : PUNCT ejpam-87 511 29 sk sk PROPN ejpam-87 511 30 = = PUNCT ejpam-87 511 31 ∑ ∑ NOUN ejpam-87 511 32 1≤i1< 1≤i1< NUM ejpam-87 511 33 ... ... PUNCT ejpam-87 511 34 > X ejpam-87 553 14 0 0 PUNCT ejpam-87 553 15 . . PUNCT ejpam-87 554 1 thus thus ADV ejpam-87 554 2 b b X ejpam-87 554 3 is be AUX ejpam-87 554 4 optimal optimal ADJ ejpam-87 554 5 and and CCONJ ejpam-87 554 6 the the DET ejpam-87 554 7 optimum optimum ADJ ejpam-87 554 8 value value NOUN ejpam-87 554 9 of of ADP ejpam-87 554 10 the the DET ejpam-87 554 11 relaxed relaxed ADJ ejpam-87 554 12 problem problem NOUN ejpam-87 554 13 ( ( PUNCT ejpam-87 554 14 1.6 1.6 NUM ejpam-87 554 15 ) ) PUNCT ejpam-87 554 16 is be AUX ejpam-87 554 17 0.931460733 0.931460733 NUM ejpam-87 554 18 . . PUNCT ejpam-87 555 1 the the DET ejpam-87 555 2 solution solution NOUN ejpam-87 555 3 of of ADP ejpam-87 555 4 the the DET ejpam-87 555 5 relaxed relaxed ADJ ejpam-87 555 6 problem problem NOUN ejpam-87 555 7 terminates terminate VERB ejpam-87 555 8 . . PUNCT ejpam-87 556 1 step step NOUN ejpam-87 556 2 4 4 NUM ejpam-87 556 3 . . PUNCT ejpam-87 557 1 the the DET ejpam-87 557 2 additional additional ADJ ejpam-87 557 3 constraint constraint NOUN ejpam-87 557 4 ( ( PUNCT ejpam-87 557 5 1.6a 1.6a NUM ejpam-87 557 6 ) ) PUNCT ejpam-87 557 7 is be AUX ejpam-87 557 8 equivalent equivalent ADJ ejpam-87 557 9 to to PART ejpam-87 557 10 v5 v5 VERB ejpam-87 557 11 + + CCONJ ejpam-87 557 12 ... ... PUNCT ejpam-87 558 1 + + CCONJ ejpam-87 558 2 v10 v10 NOUN ejpam-87 558 3 − − PROPN ejpam-87 558 4 v0 v0 NOUN ejpam-87 558 5 − − PROPN ejpam-87 558 6 ... ... PUNCT ejpam-87 558 7 − − PROPN ejpam-87 558 8 v4 v4 NOUN ejpam-87 558 9 = = PUNCT ejpam-87 558 10 −0.02938134 −0.02938134 X ejpam-87 558 11 < < X ejpam-87 558 12 0 0 NUM ejpam-87 558 13 . . PUNCT ejpam-87 559 1 the the DET ejpam-87 559 2 optimal optimal ADJ ejpam-87 559 3 solution solution NOUN ejpam-87 559 4 to to ADP ejpam-87 559 5 the the DET ejpam-87 559 6 relaxed relaxed ADJ ejpam-87 559 7 problem problem NOUN ejpam-87 559 8 does do AUX ejpam-87 559 9 not not PART ejpam-87 559 10 satisfy satisfy VERB ejpam-87 559 11 constraint constraint NOUN ejpam-87 559 12 ( ( PUNCT ejpam-87 559 13 1.6a 1.6a NUM ejpam-87 559 14 ) ) PUNCT ejpam-87 559 15 . . PUNCT ejpam-87 560 1 step step NOUN ejpam-87 560 2 5 5 NUM ejpam-87 560 3 . . PUNCT ejpam-87 561 1 in in ADP ejpam-87 561 2 order order NOUN ejpam-87 561 3 to to PART ejpam-87 561 4 ensure ensure VERB ejpam-87 561 5 the the DET ejpam-87 561 6 mode mode NOUN ejpam-87 561 7 of of ADP ejpam-87 561 8 the the DET ejpam-87 561 9 distribution distribution NOUN ejpam-87 561 10 is be AUX ejpam-87 561 11 5 5 NUM ejpam-87 561 12 we we PRON ejpam-87 561 13 prescribe prescribe VERB ejpam-87 561 14 ( ( PUNCT ejpam-87 561 15 1.6a 1.6a NUM ejpam-87 561 16 ) ) PUNCT ejpam-87 561 17 as as ADP ejpam-87 561 18 an an DET ejpam-87 561 19 additional additional ADJ ejpam-87 561 20 constraint constraint NOUN ejpam-87 561 21 : : PUNCT ejpam-87 561 22 v5 v5 PROPN ejpam-87 561 23 + + CCONJ ejpam-87 561 24 ... ... PUNCT ejpam-87 562 1 + + CCONJ ejpam-87 562 2 v10 v10 NOUN ejpam-87 562 3 − − PROPN ejpam-87 562 4 v0 v0 NOUN ejpam-87 562 5 − − PROPN ejpam-87 562 6 ... ... PUNCT ejpam-87 562 7 − − PROPN ejpam-87 563 1 v4 v4 PROPN ejpam-87 563 2 ≥ ≥ NOUN ejpam-87 563 3 0 0 PUNCT ejpam-87 563 4 . . PUNCT ejpam-87 564 1 let let VERB ejpam-87 564 2 us we PRON ejpam-87 564 3 rewrite rewrite VERB ejpam-87 564 4 the the DET ejpam-87 564 5 constraint constraint NOUN ejpam-87 564 6 in in ADP ejpam-87 564 7 the the DET ejpam-87 564 8 form form NOUN ejpam-87 564 9 v5 v5 PROPN ejpam-87 564 10 + + CCONJ ejpam-87 564 11 ... ... PUNCT ejpam-87 565 1 + + CCONJ ejpam-87 565 2 v10 v10 NOUN ejpam-87 565 3 − − PROPN ejpam-87 565 4 v0 v0 NOUN ejpam-87 565 5 − − PROPN ejpam-87 565 6 ... ... PUNCT ejpam-87 565 7 − − PROPN ejpam-87 566 1 v4 v4 NOUN ejpam-87 566 2 − − NOUN ejpam-87 566 3 v11 v11 NOUN ejpam-87 566 4 = = NOUN ejpam-87 566 5 0 0 PROPN ejpam-87 566 6 , , PUNCT ejpam-87 566 7 where where SCONJ ejpam-87 566 8 v11 v11 NOUN ejpam-87 566 9 ≥ ≥ NOUN ejpam-87 566 10 0 0 NUM ejpam-87 566 11 is be AUX ejpam-87 566 12 slack slack NOUN ejpam-87 566 13 variable variable ADJ ejpam-87 566 14 . . PUNCT ejpam-87 567 1 we we PRON ejpam-87 567 2 use use VERB ejpam-87 567 3 the the DET ejpam-87 567 4 dual dual ADJ ejpam-87 567 5 method method NOUN ejpam-87 567 6 to to PART ejpam-87 567 7 reoptimize reoptimize VERB ejpam-87 567 8 the the DET ejpam-87 567 9 problem problem NOUN ejpam-87 567 10 ( ( PUNCT ejpam-87 567 11 see see VERB ejpam-87 567 12 , , PUNCT ejpam-87 567 13 e.g. e.g. ADV ejpam-87 567 14 , , PUNCT ejpam-87 567 15 [ [ X ejpam-87 567 16 12 12 NUM ejpam-87 567 17 ] ] PUNCT ejpam-87 567 18 ) ) PUNCT ejpam-87 567 19 after after ADP ejpam-87 567 20 applying apply VERB ejpam-87 567 21 the the DET ejpam-87 567 22 dual dual ADJ ejpam-87 567 23 method method NOUN ejpam-87 567 24 to to ADP ejpam-87 567 25 the the DET ejpam-87 567 26 new new ADJ ejpam-87 567 27 problem problem NOUN ejpam-87 567 28 , , PUNCT ejpam-87 567 29 we we PRON ejpam-87 567 30 obtain obtain VERB ejpam-87 567 31 the the DET ejpam-87 567 32 optimal optimal ADJ ejpam-87 567 33 basis basis NOUN ejpam-87 567 34 and and CCONJ ejpam-87 567 35 the the DET ejpam-87 567 36 optimum optimum ADJ ejpam-87 567 37 value value NOUN ejpam-87 567 38 of of ADP ejpam-87 567 39 problem problem NOUN ejpam-87 567 40 ( ( PUNCT ejpam-87 567 41 1.6 1.6 NUM ejpam-87 567 42 ) ) PUNCT ejpam-87 567 43 , , PUNCT ejpam-87 567 44 i.e. i.e. X ejpam-87 567 45 , , PUNCT ejpam-87 567 46 the the DET ejpam-87 567 47 lower lower ADV ejpam-87 567 48 bound bind VERB ejpam-87 567 49 for for ADP ejpam-87 567 50 the the DET ejpam-87 567 51 probability probability NOUN ejpam-87 567 52 of of ADP ejpam-87 567 53 the the DET ejpam-87 567 54 union union NOUN ejpam-87 567 55 as as SCONJ ejpam-87 567 56 given give VERB ejpam-87 567 57 below below ADV ejpam-87 567 58 : : PUNCT ejpam-87 567 59   PROPN ejpam-87 567 60   PROPN ejpam-87 567 61 v0 v0 PROPN ejpam-87 567 62 v3 v3 PROPN ejpam-87 567 63 v4 v4 PROPN ejpam-87 567 64 v5 v5 PROPN ejpam-87 567 65 v10 v10 PROPN ejpam-87 567 66   PUNCT ejpam-87 568 1   NUM ejpam-87 568 2 = = SYM ejpam-87 568 3   X ejpam-87 568 4   PROPN ejpam-87 568 5 0.0685393 0.0685393 NUM ejpam-87 568 6 0.0117919 0.0117919 NUM ejpam-87 568 7 0.0468601 0.0468601 NUM ejpam-87 568 8 0.0097938 0.0097938 NUM ejpam-87 568 9 0.09780996 0.09780996 NUM ejpam-87 568 10   PUNCT ejpam-87 569 1   NUM ejpam-87 569 2 and and CCONJ ejpam-87 569 3 0.931905905 0.931905905 NUM ejpam-87 569 4 ≤ ≤ NUM ejpam-87 569 5 p p NOUN ejpam-87 569 6 ( ( PUNCT ejpam-87 569 7 ν ν X ejpam-87 569 8 ≥ ≥ NOUN ejpam-87 569 9 1 1 NUM ejpam-87 569 10 ) ) PUNCT ejpam-87 569 11 . . PUNCT ejpam-87 570 1 references reference NOUN ejpam-87 570 2 81 81 NUM ejpam-87 570 3 references reference NOUN ejpam-87 570 4 [ [ X ejpam-87 570 5 1 1 NUM ejpam-87 570 6 ] ] X ejpam-87 570 7 e. e. PROPN ejpam-87 570 8 boros boros PROPN ejpam-87 570 9 , , PUNCT ejpam-87 570 10 a. a. NOUN ejpam-87 570 11 prékopa prékopa PROPN ejpam-87 570 12 , , PUNCT ejpam-87 570 13 closed close VERB ejpam-87 570 14 form form NOUN ejpam-87 570 15 two two NUM ejpam-87 570 16 - - PUNCT ejpam-87 570 17 sided sided ADJ ejpam-87 570 18 bounds bound NOUN ejpam-87 570 19 for for ADP ejpam-87 570 20 probabilities probability NOUN ejpam-87 570 21 that that PRON ejpam-87 570 22 exactly exactly ADV ejpam-87 570 23 r r NOUN ejpam-87 570 24 and and CCONJ ejpam-87 570 25 at at ADP ejpam-87 570 26 least least ADJ ejpam-87 570 27 r r NOUN ejpam-87 570 28 out out ADP ejpam-87 570 29 of of ADP ejpam-87 570 30 n n PRON ejpam-87 570 31 events event NOUN ejpam-87 570 32 occur occur VERB ejpam-87 570 33 . . PUNCT ejpam-87 571 1 math math NOUN ejpam-87 571 2 . . PUNCT ejpam-87 572 1 oper oper PROPN ejpam-87 572 2 . . PUNCT ejpam-87 572 3 res res PROPN ejpam-87 572 4 . . PROPN ejpam-87 572 5 , , PUNCT ejpam-87 572 6 14 14 NUM ejpam-87 572 7 : : PUNCT ejpam-87 573 1 317–347 317–347 NUM ejpam-87 573 2 ( ( PUNCT ejpam-87 573 3 1989 1989 NUM ejpam-87 573 4 ) ) PUNCT ejpam-87 573 5 . . PUNCT ejpam-87 574 1 [ [ X ejpam-87 574 2 2 2 X ejpam-87 574 3 ] ] X ejpam-87 574 4 j. j. PROPN ejpam-87 574 5 bukszár bukszár PROPN ejpam-87 574 6 , , PUNCT ejpam-87 574 7 a. a. NOUN ejpam-87 574 8 prékopa prékopa PROPN ejpam-87 574 9 , , PUNCT ejpam-87 574 10 probability probability NOUN ejpam-87 574 11 bounds bound VERB ejpam-87 574 12 with with ADP ejpam-87 574 13 cherry cherry NOUN ejpam-87 574 14 trees tree NOUN ejpam-87 574 15 . . PUNCT ejpam-87 575 1 math math NOUN ejpam-87 575 2 . . PUNCT ejpam-87 576 1 oper oper PROPN ejpam-87 576 2 . . PUNCT ejpam-87 576 3 res res PROPN ejpam-87 576 4 . . PROPN ejpam-87 576 5 , , PUNCT ejpam-87 576 6 26 26 NUM ejpam-87 576 7 : : PUNCT ejpam-87 576 8 174–192 174–192 NUM ejpam-87 576 9 ( ( PUNCT ejpam-87 576 10 2001 2001 NUM ejpam-87 576 11 ) ) PUNCT ejpam-87 576 12 . . PUNCT ejpam-87 577 1 [ [ X ejpam-87 577 2 3 3 X ejpam-87 577 3 ] ] X ejpam-87 577 4 j. j. PROPN ejpam-87 577 5 bukszár bukszár PROPN ejpam-87 577 6 , , PUNCT ejpam-87 577 7 hypermultitrees hypermultitree NOUN ejpam-87 577 8 and and CCONJ ejpam-87 577 9 bonferroni bonferroni PROPN ejpam-87 577 10 inequalities inequality NOUN ejpam-87 577 11 . . PUNCT ejpam-87 578 1 mathemtical mathemtical ADJ ejpam-87 578 2 inequalities inequality NOUN ejpam-87 578 3 and and CCONJ ejpam-87 578 4 applications application NOUN ejpam-87 578 5 , , PUNCT ejpam-87 578 6 6 6 NUM ejpam-87 578 7 : : PUNCT ejpam-87 578 8 727–745 727–745 NUM ejpam-87 578 9 ( ( PUNCT ejpam-87 578 10 2003 2003 NUM ejpam-87 578 11 ) ) PUNCT ejpam-87 578 12 . . PUNCT ejpam-87 579 1 [ [ X ejpam-87 579 2 4 4 NUM ejpam-87 579 3 ] ] X ejpam-87 579 4 d.a d.a PROPN ejpam-87 579 5 . . PROPN ejpam-87 579 6 dawson dawson PROPN ejpam-87 579 7 , , PUNCT ejpam-87 579 8 a. a. PROPN ejpam-87 579 9 sankoff sankoff PROPN ejpam-87 579 10 , , PUNCT ejpam-87 579 11 an an DET ejpam-87 579 12 inequality inequality NOUN ejpam-87 579 13 for for ADP ejpam-87 579 14 probabilities probability NOUN ejpam-87 579 15 . . PUNCT ejpam-87 580 1 proceedings proceeding NOUN ejpam-87 580 2 of of ADP ejpam-87 580 3 the the DET ejpam-87 580 4 american american PROPN ejpam-87 580 5 mathematical mathematical PROPN ejpam-87 580 6 society society NOUN ejpam-87 580 7 , , PUNCT ejpam-87 580 8 18 18 NUM ejpam-87 580 9 : : PUNCT ejpam-87 580 10 504–507 504–507 NUM ejpam-87 580 11 ( ( PUNCT ejpam-87 580 12 1967 1967 NUM ejpam-87 580 13 ) ) PUNCT ejpam-87 580 14 . . PUNCT ejpam-87 581 1 [ [ X ejpam-87 581 2 5 5 X ejpam-87 581 3 ] ] PUNCT ejpam-87 581 4 j. j. PROPN ejpam-87 581 5 gessel gessel PROPN ejpam-87 581 6 , , PUNCT ejpam-87 581 7 g. g. PROPN ejpam-87 581 8 viennot viennot PROPN ejpam-87 581 9 , , PUNCT ejpam-87 581 10 binomial binomial ADJ ejpam-87 581 11 determinants determinant NOUN ejpam-87 581 12 , , PUNCT ejpam-87 581 13 paths path NOUN ejpam-87 581 14 , , PUNCT ejpam-87 581 15 and and CCONJ ejpam-87 581 16 hook hook NOUN ejpam-87 581 17 length length NOUN ejpam-87 581 18 formulae formulae NOUN ejpam-87 581 19 . . PUNCT ejpam-87 582 1 advences advence NOUN ejpam-87 582 2 in in ADP ejpam-87 582 3 mathematics mathematics PROPN ejpam-87 582 4 58 58 NUM ejpam-87 582 5 : : PUNCT ejpam-87 582 6 300–321 300–321 NUM ejpam-87 582 7 ( ( PUNCT ejpam-87 582 8 1985 1985 NUM ejpam-87 582 9 ) ) PUNCT ejpam-87 582 10 . . PUNCT ejpam-87 583 1 [ [ X ejpam-87 583 2 6 6 NUM ejpam-87 583 3 ] ] SYM ejpam-87 583 4 s.m s.m PROPN ejpam-87 583 5 . . PROPN ejpam-87 583 6 kwerel kwerel PROPN ejpam-87 583 7 , , PUNCT ejpam-87 583 8 most most ADV ejpam-87 583 9 stringent stringent ADJ ejpam-87 583 10 bounds bound NOUN ejpam-87 583 11 on on ADP ejpam-87 583 12 aggregated aggregate VERB ejpam-87 583 13 probabilities probability NOUN ejpam-87 583 14 of of ADP ejpam-87 583 15 partially partially ADV ejpam-87 583 16 specified specify VERB ejpam-87 583 17 dependent dependent ADJ ejpam-87 583 18 probability probability NOUN ejpam-87 583 19 systems system NOUN ejpam-87 583 20 . . PUNCT ejpam-87 584 1 j. j. PROPN ejpam-87 584 2 amer amer PROPN ejpam-87 584 3 . . PUNCT ejpam-87 585 1 stat stat PROPN ejpam-87 585 2 . . PUNCT ejpam-87 586 1 assoc assoc PROPN ejpam-87 586 2 . . PROPN ejpam-87 586 3 , , PUNCT ejpam-87 587 1 70 70 NUM ejpam-87 587 2 : : PUNCT ejpam-87 587 3 472–479 472–479 NUM ejpam-87 587 4 ( ( PUNCT ejpam-87 587 5 1975 1975 NUM ejpam-87 587 6 ) ) PUNCT ejpam-87 587 7 . . PUNCT ejpam-87 588 1 [ [ X ejpam-87 588 2 7 7 X ejpam-87 588 3 ] ] PUNCT ejpam-87 588 4 a. a. NOUN ejpam-87 588 5 prékopa prékopa PROPN ejpam-87 588 6 , , PUNCT ejpam-87 588 7 boole boole PROPN ejpam-87 588 8 - - PUNCT ejpam-87 588 9 bonferroni bonferroni PROPN ejpam-87 588 10 inequalities inequality NOUN ejpam-87 588 11 and and CCONJ ejpam-87 588 12 linear linear PROPN ejpam-87 588 13 programming programming NOUN ejpam-87 588 14 . . PUNCT ejpam-87 589 1 operations operation NOUN ejpam-87 589 2 research research VERB ejpam-87 589 3 36 36 NUM ejpam-87 589 4 : : PUNCT ejpam-87 589 5 145–162 145–162 NUM ejpam-87 589 6 ( ( PUNCT ejpam-87 589 7 1988 1988 NUM ejpam-87 589 8 ) ) PUNCT ejpam-87 589 9 . . PUNCT ejpam-87 590 1 [ [ X ejpam-87 590 2 8 8 NUM ejpam-87 590 3 ] ] PUNCT ejpam-87 590 4 a. a. NOUN ejpam-87 590 5 prékopa prékopa PROPN ejpam-87 590 6 , , PUNCT ejpam-87 590 7 totally totally ADV ejpam-87 590 8 positive positive ADJ ejpam-87 590 9 linear linear ADJ ejpam-87 590 10 programming programming NOUN ejpam-87 590 11 problems problem NOUN ejpam-87 590 12 , , PUNCT ejpam-87 590 13 in in ADP ejpam-87 590 14 l.v l.v PROPN ejpam-87 590 15 . . PROPN ejpam-87 590 16 kantorovich kantorovich PROPN ejpam-87 590 17 memorial memorial ADJ ejpam-87 590 18 volume volume NOUN ejpam-87 590 19 , , PUNCT ejpam-87 590 20 oxford oxford PROPN ejpam-87 590 21 univ univ PROPN ejpam-87 590 22 . . PUNCT ejpam-87 591 1 press press PROPN ejpam-87 591 2 , , PUNCT ejpam-87 591 3 new new PROPN ejpam-87 591 4 york york PROPN ejpam-87 591 5 , , PUNCT ejpam-87 591 6 197–207 197–207 NUM ejpam-87 591 7 , , PUNCT ejpam-87 591 8 1989 1989 NUM ejpam-87 591 9 . . PUNCT ejpam-87 592 1 [ [ X ejpam-87 592 2 9 9 NUM ejpam-87 592 3 ] ] PUNCT ejpam-87 592 4 a. a. NOUN ejpam-87 592 5 prékopa prékopa PROPN ejpam-87 592 6 , , PUNCT ejpam-87 592 7 sharp sharp ADJ ejpam-87 592 8 bounds bound NOUN ejpam-87 592 9 on on ADP ejpam-87 592 10 probabilities probability NOUN ejpam-87 592 11 using use VERB ejpam-87 592 12 linear linear ADJ ejpam-87 592 13 programming programming NOUN ejpam-87 592 14 . . PUNCT ejpam-87 593 1 operations operation NOUN ejpam-87 593 2 research research NOUN ejpam-87 593 3 , , PUNCT ejpam-87 593 4 38 38 NUM ejpam-87 593 5 : : PUNCT ejpam-87 593 6 227–239 227–239 NUM ejpam-87 593 7 ( ( PUNCT ejpam-87 593 8 1990 1990 NUM ejpam-87 593 9 ) ) PUNCT ejpam-87 593 10 . . PUNCT ejpam-87 594 1 [ [ X ejpam-87 594 2 10 10 NUM ejpam-87 594 3 ] ] PUNCT ejpam-87 594 4 a. a. NOUN ejpam-87 594 5 prékopa prékopa PROPN ejpam-87 594 6 , , PUNCT ejpam-87 594 7 the the DET ejpam-87 594 8 discrete discrete ADJ ejpam-87 594 9 moment moment NOUN ejpam-87 594 10 problem problem NOUN ejpam-87 594 11 and and CCONJ ejpam-87 594 12 linear linear PROPN ejpam-87 594 13 programming programming NOUN ejpam-87 594 14 . . PUNCT ejpam-87 595 1 discrete discrete ADJ ejpam-87 595 2 applied apply VERB ejpam-87 595 3 mathematics mathematic NOUN ejpam-87 595 4 27 27 NUM ejpam-87 595 5 : : PUNCT ejpam-87 595 6 235–254 235–254 NUM ejpam-87 595 7 ( ( PUNCT ejpam-87 595 8 1990 1990 NUM ejpam-87 595 9 ) ) PUNCT ejpam-87 595 10 . . PUNCT ejpam-87 596 1 [ [ X ejpam-87 596 2 11 11 NUM ejpam-87 596 3 ] ] PUNCT ejpam-87 596 4 a. a. NOUN ejpam-87 596 5 prékopa prékopa PROPN ejpam-87 596 6 , , PUNCT ejpam-87 596 7 stochastic stochastic ADJ ejpam-87 596 8 programming programming NOUN ejpam-87 596 9 , , PUNCT ejpam-87 596 10 kluwer kluwer NOUN ejpam-87 596 11 academic academic ADJ ejpam-87 596 12 publishers publisher NOUN ejpam-87 596 13 , , PUNCT ejpam-87 596 14 dordtecht dordtecht NOUN ejpam-87 596 15 , , PUNCT ejpam-87 596 16 boston boston PROPN ejpam-87 596 17 , , PUNCT ejpam-87 596 18 1995 1995 NUM ejpam-87 596 19 . . PUNCT ejpam-87 597 1 [ [ X ejpam-87 597 2 12 12 NUM ejpam-87 597 3 ] ] PUNCT ejpam-87 597 4 a. a. NOUN ejpam-87 597 5 prékopa prékopa PROPN ejpam-87 597 6 , , PUNCT ejpam-87 597 7 a a DET ejpam-87 597 8 brief brief ADJ ejpam-87 597 9 introduction introduction NOUN ejpam-87 597 10 to to ADP ejpam-87 597 11 linear linear PROPN ejpam-87 597 12 programming programming NOUN ejpam-87 597 13 . . PUNCT ejpam-87 598 1 math math NOUN ejpam-87 598 2 . . PUNCT ejpam-87 599 1 scientist scientist NOUN ejpam-87 599 2 21 21 NUM ejpam-87 599 3 : : PUNCT ejpam-87 599 4 85–111 85–111 NUM ejpam-87 599 5 ( ( PUNCT ejpam-87 599 6 1996 1996 NUM ejpam-87 599 7 ) ) PUNCT ejpam-87 599 8 . . PUNCT ejpam-87 600 1 [ [ X ejpam-87 600 2 13 13 NUM ejpam-87 600 3 ] ] PUNCT ejpam-87 600 4 a. a. NOUN ejpam-87 600 5 prékopa prékopa PROPN ejpam-87 600 6 , , PUNCT ejpam-87 600 7 l. l. PROPN ejpam-87 600 8 gao gao PROPN ejpam-87 600 9 , , PUNCT ejpam-87 600 10 bounding bound VERB ejpam-87 600 11 the the DET ejpam-87 600 12 probability probability NOUN ejpam-87 600 13 of of ADP ejpam-87 600 14 the the DET ejpam-87 600 15 union union NOUN ejpam-87 600 16 of of ADP ejpam-87 600 17 events event NOUN ejpam-87 600 18 by by ADP ejpam-87 600 19 the the DET ejpam-87 600 20 use use NOUN ejpam-87 600 21 of of ADP ejpam-87 600 22 aggregation aggregation NOUN ejpam-87 600 23 and and CCONJ ejpam-87 600 24 disaggregation disaggregation NOUN ejpam-87 600 25 in in ADP ejpam-87 600 26 linear linear PROPN ejpam-87 600 27 programs program NOUN ejpam-87 600 28 . . PUNCT ejpam-87 601 1 discrete discrete VERB ejpam-87 601 2 applied apply VERB ejpam-87 601 3 mathematics mathematic NOUN ejpam-87 601 4 145 145 NUM ejpam-87 601 5 : : PUNCT ejpam-87 602 1 444–454 444–454 NUM ejpam-87 602 2 ( ( PUNCT ejpam-87 602 3 2005 2005 NUM ejpam-87 602 4 ) ) PUNCT ejpam-87 602 5 . . PUNCT ejpam-87 603 1 [ [ X ejpam-87 603 2 14 14 NUM ejpam-87 603 3 ] ] X ejpam-87 603 4 e. e. PROPN ejpam-87 603 5 subasi subasi PROPN ejpam-87 603 6 , , PUNCT ejpam-87 603 7 m. m. NOUN ejpam-87 603 8 subasi subasi PROPN ejpam-87 603 9 , , PUNCT ejpam-87 603 10 a. a. NOUN ejpam-87 603 11 prékopa prékopa PROPN ejpam-87 603 12 , , PUNCT ejpam-87 603 13 discrete discrete ADJ ejpam-87 603 14 moment moment NOUN ejpam-87 603 15 problems problem NOUN ejpam-87 603 16 with with ADP ejpam-87 603 17 distributions distribution NOUN ejpam-87 603 18 known know VERB ejpam-87 603 19 to to PART ejpam-87 603 20 be be AUX ejpam-87 603 21 unimodal unimodal ADJ ejpam-87 603 22 . . PUNCT ejpam-87 604 1 mathematical mathematical ADJ ejpam-87 604 2 inequalities inequality NOUN ejpam-87 604 3 and and CCONJ ejpam-87 604 4 applications application NOUN ejpam-87 604 5 , , PUNCT ejpam-87 604 6 accepted accept VERB ejpam-87 604 7 . . PUNCT ejpam-87 605 1 available available ADJ ejpam-87 605 2 as as ADP ejpam-87 605 3 rutcor rutcor ADJ ejpam-87 605 4 research research NOUN ejpam-87 605 5 report report NOUN ejpam-87 605 6 rrr rrr PROPN ejpam-87 605 7 15 15 NUM ejpam-87 605 8 - - SYM ejpam-87 605 9 2007 2007 NUM ejpam-87 605 10 . . PUNCT