id	sid	tid	token	lemma	pos
gojar-5993	1	1	global	global	ADJ
gojar-5993	1	2	online	online	PROPN
gojar-5993	1	3	journal	journal	PROPN
gojar-5993	1	4	of	of	ADP
gojar-5993	1	5	academic	academic	ADJ
gojar-5993	1	6	research	research	NOUN
gojar-5993	1	7	(	(	PUNCT
gojar-5993	1	8	gojar	gojar	NOUN
gojar-5993	1	9	)	)	PUNCT
gojar-5993	1	10	,	,	PUNCT
gojar-5993	1	11	vol	vol	NOUN
gojar-5993	1	12	.	.	PROPN
gojar-5993	1	13	4	4	NUM
gojar-5993	1	14	,	,	PUNCT
gojar-5993	1	15	no	no	INTJ
gojar-5993	1	16	.	.	NOUN
gojar-5993	1	17	1	1	NUM
gojar-5993	1	18	february	february	PROPN
gojar-5993	1	19	2025	2025	NUM
gojar-5993	1	20	global	global	ADJ
gojar-5993	1	21	online	online	ADJ
gojar-5993	1	22	journal	journal	PROPN
gojar-5993	1	23	of	of	ADP
gojar-5993	1	24	academic	academic	ADJ
gojar-5993	1	25	research	research	NOUN
gojar-5993	1	26	(	(	PUNCT
gojar-5993	1	27	gojar	gojar	NOUN
gojar-5993	1	28	)	)	PUNCT
gojar-5993	1	29	,	,	PUNCT
gojar-5993	1	30	vol	vol	NOUN
gojar-5993	1	31	.	.	PROPN
gojar-5993	1	32	4	4	NUM
gojar-5993	1	33	,	,	PUNCT
gojar-5993	1	34	no	no	INTJ
gojar-5993	1	35	.	.	NOUN
gojar-5993	1	36	1	1	NUM
gojar-5993	1	37	february	february	NOUN
gojar-5993	1	38	2025	2025	NUM
gojar-5993	1	39	60	60	NUM
gojar-5993	1	40	on	on	ADP
gojar-5993	1	41	boundedness	boundedness	NOUN
gojar-5993	1	42	and	and	CCONJ
gojar-5993	1	43	solution	solution	NOUN
gojar-5993	1	44	size	size	NOUN
gojar-5993	1	45	in	in	ADP
gojar-5993	1	46	rational	rational	ADJ
gojar-5993	1	47	linear	linear	ADJ
gojar-5993	1	48	programming	programming	NOUN
gojar-5993	1	49	and	and	CCONJ
gojar-5993	1	50	polyhedral	polyhedral	ADJ
gojar-5993	1	51	optimization	optimization	NOUN
gojar-5993	1	52	mark	mark	PROPN
gojar-5993	1	53	laisin	laisin	PROPN
gojar-5993	1	54	,	,	PUNCT
gojar-5993	1	55	collins	collins	PROPN
gojar-5993	1	56	edike	edike	PROPN
gojar-5993	1	57	&	&	CCONJ
gojar-5993	1	58	r.	r.	PROPN
gojar-5993	1	59	n.	n.	PROPN
gojar-5993	1	60	ujumadu	ujumadu	PROPN
gojar-5993	1	61	abstract	abstract	ADJ
gojar-5993	1	62	this	this	DET
gojar-5993	1	63	paper	paper	NOUN
gojar-5993	1	64	delves	delf	NOUN
gojar-5993	1	65	into	into	ADP
gojar-5993	1	66	the	the	DET
gojar-5993	1	67	theoretical	theoretical	ADJ
gojar-5993	1	68	and	and	CCONJ
gojar-5993	1	69	practical	practical	ADJ
gojar-5993	1	70	aspects	aspect	NOUN
gojar-5993	1	71	of	of	ADP
gojar-5993	1	72	boundedness	boundedness	NOUN
gojar-5993	1	73	and	and	CCONJ
gojar-5993	1	74	structural	structural	ADJ
gojar-5993	1	75	properties	property	NOUN
gojar-5993	1	76	in	in	ADP
gojar-5993	1	77	rational	rational	ADJ
gojar-5993	1	78	linear	linear	ADJ
gojar-5993	1	79	programming	programming	NOUN
gojar-5993	1	80	(	(	PUNCT
gojar-5993	1	81	lp	lp	NOUN
gojar-5993	1	82	)	)	PUNCT
gojar-5993	1	83	and	and	CCONJ
gojar-5993	1	84	polyhedral	polyhedral	ADJ
gojar-5993	1	85	optimization	optimization	NOUN
gojar-5993	1	86	.	.	PUNCT
gojar-5993	2	1	it	it	PRON
gojar-5993	2	2	provides	provide	VERB
gojar-5993	2	3	a	a	DET
gojar-5993	2	4	comprehensive	comprehensive	ADJ
gojar-5993	2	5	analysis	analysis	NOUN
gojar-5993	2	6	of	of	ADP
gojar-5993	2	7	conditions	condition	NOUN
gojar-5993	2	8	under	under	ADP
gojar-5993	2	9	which	which	PRON
gojar-5993	2	10	the	the	DET
gojar-5993	2	11	optimization	optimization	NOUN
gojar-5993	2	12	of	of	ADP
gojar-5993	2	13	linear	linear	PROPN
gojar-5993	2	14	functions	function	NOUN
gojar-5993	2	15	over	over	ADP
gojar-5993	2	16	rational	rational	ADJ
gojar-5993	2	17	polyhedra	polyhedra	NOUN
gojar-5993	2	18	remains	remain	VERB
gojar-5993	2	19	bounded	bound	VERB
gojar-5993	2	20	and	and	CCONJ
gojar-5993	2	21	establishes	establish	VERB
gojar-5993	2	22	explicit	explicit	ADJ
gojar-5993	2	23	constraints	constraint	NOUN
gojar-5993	2	24	on	on	ADP
gojar-5993	2	25	solution	solution	NOUN
gojar-5993	2	26	size	size	NOUN
gojar-5993	2	27	when	when	SCONJ
gojar-5993	2	28	optimal	optimal	ADJ
gojar-5993	2	29	solutions	solution	NOUN
gojar-5993	2	30	exist	exist	VERB
gojar-5993	2	31	.	.	PUNCT
gojar-5993	3	1	by	by	ADP
gojar-5993	3	2	exploring	explore	VERB
gojar-5993	3	3	the	the	DET
gojar-5993	3	4	interplay	interplay	NOUN
gojar-5993	3	5	between	between	ADP
gojar-5993	3	6	polyhedral	polyhedral	ADJ
gojar-5993	3	7	geometry	geometry	NOUN
gojar-5993	3	8	,	,	PUNCT
gojar-5993	3	9	integer	integer	NOUN
gojar-5993	3	10	hulls	hull	NOUN
gojar-5993	3	11	,	,	PUNCT
gojar-5993	3	12	and	and	CCONJ
gojar-5993	3	13	rational	rational	ADJ
gojar-5993	3	14	lp	lp	ADJ
gojar-5993	3	15	systems	system	NOUN
gojar-5993	3	16	,	,	PUNCT
gojar-5993	3	17	this	this	DET
gojar-5993	3	18	study	study	NOUN
gojar-5993	3	19	sheds	shed	VERB
gojar-5993	3	20	light	light	NOUN
gojar-5993	3	21	on	on	ADP
gojar-5993	3	22	fundamental	fundamental	ADJ
gojar-5993	3	23	principles	principle	NOUN
gojar-5993	3	24	that	that	PRON
gojar-5993	3	25	underlie	underlie	VERB
gojar-5993	3	26	modern	modern	ADJ
gojar-5993	3	27	optimization	optimization	NOUN
gojar-5993	3	28	techniques	technique	NOUN
gojar-5993	3	29	.	.	PUNCT
gojar-5993	4	1	key	key	ADJ
gojar-5993	4	2	findings	finding	NOUN
gojar-5993	4	3	include	include	VERB
gojar-5993	4	4	equivalence	equivalence	NOUN
gojar-5993	4	5	conditions	condition	NOUN
gojar-5993	4	6	for	for	ADP
gojar-5993	4	7	boundedness	boundedness	NOUN
gojar-5993	4	8	between	between	ADP
gojar-5993	4	9	rational	rational	ADJ
gojar-5993	4	10	polyhedra	polyhedra	NOUN
gojar-5993	4	11	and	and	CCONJ
gojar-5993	4	12	their	their	PRON
gojar-5993	4	13	integer	integer	NOUN
gojar-5993	4	14	hulls	hull	NOUN
gojar-5993	4	15	,	,	PUNCT
gojar-5993	4	16	as	as	ADV
gojar-5993	4	17	well	well	ADV
gojar-5993	4	18	as	as	ADP
gojar-5993	4	19	precise	precise	ADJ
gojar-5993	4	20	bounds	bound	NOUN
gojar-5993	4	21	on	on	ADP
gojar-5993	4	22	the	the	DET
gojar-5993	4	23	numerical	numerical	ADJ
gojar-5993	4	24	representation	representation	NOUN
gojar-5993	4	25	of	of	ADP
gojar-5993	4	26	optimal	optimal	ADJ
gojar-5993	4	27	solutions	solution	NOUN
gojar-5993	4	28	.	.	PUNCT
gojar-5993	5	1	these	these	DET
gojar-5993	5	2	results	result	VERB
gojar-5993	5	3	not	not	PART
gojar-5993	5	4	only	only	ADV
gojar-5993	5	5	enhance	enhance	VERB
gojar-5993	5	6	the	the	DET
gojar-5993	5	7	theoretical	theoretical	ADJ
gojar-5993	5	8	understanding	understanding	NOUN
gojar-5993	5	9	of	of	ADP
gojar-5993	5	10	lp	lp	ADJ
gojar-5993	5	11	and	and	CCONJ
gojar-5993	5	12	polyhedral	polyhedral	ADJ
gojar-5993	5	13	optimization	optimization	NOUN
gojar-5993	5	14	but	but	CCONJ
gojar-5993	5	15	also	also	ADV
gojar-5993	5	16	have	have	VERB
gojar-5993	5	17	significant	significant	ADJ
gojar-5993	5	18	implications	implication	NOUN
gojar-5993	5	19	for	for	ADP
gojar-5993	5	20	computational	computational	ADJ
gojar-5993	5	21	efficiency	efficiency	NOUN
gojar-5993	5	22	,	,	PUNCT
gojar-5993	5	23	algorithm	algorithm	NOUN
gojar-5993	5	24	design	design	NOUN
gojar-5993	5	25	,	,	PUNCT
gojar-5993	5	26	and	and	CCONJ
gojar-5993	5	27	numerical	numerical	ADJ
gojar-5993	5	28	stability	stability	NOUN
gojar-5993	5	29	in	in	ADP
gojar-5993	5	30	solving	solve	VERB
gojar-5993	5	31	real	real	ADJ
gojar-5993	5	32	-	-	PUNCT
gojar-5993	5	33	world	world	NOUN
gojar-5993	5	34	optimization	optimization	NOUN
gojar-5993	5	35	problems	problem	NOUN
gojar-5993	5	36	.	.	PUNCT
gojar-5993	6	1	the	the	DET
gojar-5993	6	2	discussion	discussion	NOUN
gojar-5993	6	3	is	be	AUX
gojar-5993	6	4	rooted	root	VERB
gojar-5993	6	5	in	in	ADP
gojar-5993	6	6	rigorous	rigorous	ADJ
gojar-5993	6	7	mathematical	mathematical	ADJ
gojar-5993	6	8	foundations	foundation	NOUN
gojar-5993	6	9	and	and	CCONJ
gojar-5993	6	10	extends	extend	VERB
gojar-5993	6	11	to	to	ADP
gojar-5993	6	12	practical	practical	ADJ
gojar-5993	6	13	applications	application	NOUN
gojar-5993	6	14	in	in	ADP
gojar-5993	6	15	areas	area	NOUN
gojar-5993	6	16	such	such	ADJ
gojar-5993	6	17	as	as	ADP
gojar-5993	6	18	mixed	mixed	ADJ
gojar-5993	6	19	-	-	PUNCT
gojar-5993	6	20	integer	integer	NOUN
gojar-5993	6	21	programming	programming	NOUN
gojar-5993	6	22	,	,	PUNCT
gojar-5993	6	23	computational	computational	ADJ
gojar-5993	6	24	geometry	geometry	NOUN
gojar-5993	6	25	,	,	PUNCT
gojar-5993	6	26	and	and	CCONJ
gojar-5993	6	27	combinatorial	combinatorial	ADJ
gojar-5993	6	28	optimization	optimization	NOUN
gojar-5993	6	29	.	.	PUNCT
gojar-5993	7	1	keywords	keyword	NOUN
gojar-5993	7	2	:	:	PUNCT
gojar-5993	7	3	rational	rational	ADJ
gojar-5993	7	4	linear	linear	ADJ
gojar-5993	7	5	programming	programming	NOUN
gojar-5993	7	6	,	,	PUNCT
gojar-5993	7	7	polyhedral	polyhedral	ADJ
gojar-5993	7	8	optimization	optimization	NOUN
gojar-5993	7	9	,	,	PUNCT
gojar-5993	7	10	boundedness	boundedness	NOUN
gojar-5993	7	11	conditions	condition	NOUN
gojar-5993	7	12	,	,	PUNCT
gojar-5993	7	13	integer	integer	NOUN
gojar-5993	7	14	hull	hull	NOUN
gojar-5993	7	15	,	,	PUNCT
gojar-5993	7	16	solution	solution	NOUN
gojar-5993	7	17	size	size	NOUN
gojar-5993	7	18	bounds	bound	NOUN
gojar-5993	7	19	,	,	PUNCT
gojar-5993	7	20	rational	rational	ADJ
gojar-5993	7	21	coefficients	coefficient	NOUN
gojar-5993	7	22	,	,	PUNCT
gojar-5993	7	23	computational	computational	ADJ
gojar-5993	7	24	geometry	geometry	NOUN
gojar-5993	7	25	,	,	PUNCT
gojar-5993	7	26	optimization	optimization	NOUN
gojar-5993	7	27	algorithms	algorithm	NOUN
gojar-5993	7	28	,	,	PUNCT
gojar-5993	7	29	numerical	numerical	ADJ
gojar-5993	7	30	stability	stability	NOUN
gojar-5993	7	31	.	.	PUNCT
gojar-5993	8	1	i.	i.	PROPN
gojar-5993	8	2	introduction	introduction	PROPN
gojar-5993	8	3	linear	linear	PROPN
gojar-5993	8	4	programming	programming	NOUN
gojar-5993	8	5	(	(	PUNCT
gojar-5993	8	6	lp	lp	NOUN
gojar-5993	8	7	)	)	PUNCT
gojar-5993	8	8	has	have	AUX
gojar-5993	8	9	had	have	VERB
gojar-5993	8	10	a	a	DET
gojar-5993	8	11	significant	significant	ADJ
gojar-5993	8	12	and	and	CCONJ
gojar-5993	8	13	enduring	enduring	ADJ
gojar-5993	8	14	influence	influence	NOUN
gojar-5993	8	15	,	,	PUNCT
gojar-5993	8	16	deeply	deeply	ADV
gojar-5993	8	17	connected	connect	VERB
gojar-5993	8	18	with	with	ADP
gojar-5993	8	19	the	the	DET
gojar-5993	8	20	evolution	evolution	NOUN
gojar-5993	8	21	of	of	ADP
gojar-5993	8	22	optimization	optimization	NOUN
gojar-5993	8	23	theory	theory	NOUN
gojar-5993	8	24	and	and	CCONJ
gojar-5993	8	25	computational	computational	ADJ
gojar-5993	8	26	methodologies	methodology	NOUN
gojar-5993	8	27	.	.	PUNCT
gojar-5993	9	1	scholars	scholar	NOUN
gojar-5993	9	2	have	have	AUX
gojar-5993	9	3	extensively	extensively	ADV
gojar-5993	9	4	traced	trace	VERB
gojar-5993	9	5	the	the	DET
gojar-5993	9	6	origins	origin	NOUN
gojar-5993	9	7	of	of	ADP
gojar-5993	9	8	lp	lp	NOUN
gojar-5993	9	9	back	back	ADV
gojar-5993	9	10	to	to	ADP
gojar-5993	9	11	the	the	DET
gojar-5993	9	12	1930s	1930	NOUN
gojar-5993	9	13	,	,	PUNCT
gojar-5993	9	14	highlighting	highlight	VERB
gojar-5993	9	15	leonid	leonid	PROPN
gojar-5993	9	16	kantorovich	kantorovich	PROPN
gojar-5993	9	17	’s	’s	PART
gojar-5993	9	18	groundbreaking	groundbreake	VERB
gojar-5993	9	19	work	work	NOUN
gojar-5993	9	20	in	in	ADP
gojar-5993	9	21	formulating	formulate	VERB
gojar-5993	9	22	optimization	optimization	NOUN
gojar-5993	9	23	problems	problem	NOUN
gojar-5993	9	24	to	to	PART
gojar-5993	9	25	address	address	VERB
gojar-5993	9	26	resource	resource	NOUN
gojar-5993	9	27	allocation	allocation	NOUN
gojar-5993	9	28	challenges	challenge	NOUN
gojar-5993	9	29	in	in	ADP
gojar-5993	9	30	economic	economic	ADJ
gojar-5993	9	31	planning	planning	NOUN
gojar-5993	9	32	(	(	PUNCT
gojar-5993	9	33	kantorovich	kantorovich	PROPN
gojar-5993	9	34	,	,	PUNCT
gojar-5993	9	35	1939	1939	NUM
gojar-5993	9	36	)	)	PUNCT
gojar-5993	9	37	.	.	PUNCT
gojar-5993	10	1	kantorovich	kantorovich	PROPN
gojar-5993	10	2	’s	’s	PART
gojar-5993	10	3	pioneering	pioneer	VERB
gojar-5993	10	4	contributions	contribution	NOUN
gojar-5993	10	5	established	establish	VERB
gojar-5993	10	6	the	the	DET
gojar-5993	10	7	foundation	foundation	NOUN
gojar-5993	10	8	of	of	ADP
gojar-5993	10	9	linear	linear	PROPN
gojar-5993	10	10	optimization	optimization	NOUN
gojar-5993	10	11	and	and	CCONJ
gojar-5993	10	12	global	global	ADJ
gojar-5993	10	13	online	online	ADJ
gojar-5993	10	14	journal	journal	PROPN
gojar-5993	10	15	of	of	ADP
gojar-5993	10	16	academic	academic	ADJ
gojar-5993	10	17	research	research	NOUN
gojar-5993	10	18	(	(	PUNCT
gojar-5993	10	19	gojar	gojar	NOUN
gojar-5993	10	20	)	)	PUNCT
gojar-5993	10	21	,	,	PUNCT
gojar-5993	10	22	vol	vol	NOUN
gojar-5993	10	23	.	.	PROPN
gojar-5993	10	24	4	4	NUM
gojar-5993	10	25	,	,	PUNCT
gojar-5993	10	26	no	no	INTJ
gojar-5993	10	27	.	.	NOUN
gojar-5993	10	28	1	1	NUM
gojar-5993	10	29	february	february	NOUN
gojar-5993	10	30	2025	2025	NUM
gojar-5993	10	31	61	61	NUM
gojar-5993	10	32	were	be	AUX
gojar-5993	10	33	later	later	ADV
gojar-5993	10	34	recognized	recognize	VERB
gojar-5993	10	35	with	with	ADP
gojar-5993	10	36	the	the	DET
gojar-5993	10	37	nobel	nobel	PROPN
gojar-5993	10	38	prize	prize	PROPN
gojar-5993	10	39	in	in	ADP
gojar-5993	10	40	economics	economic	NOUN
gojar-5993	10	41	,	,	PUNCT
gojar-5993	10	42	underscoring	underscore	VERB
gojar-5993	10	43	the	the	DET
gojar-5993	10	44	lasting	last	VERB
gojar-5993	10	45	impact	impact	NOUN
gojar-5993	10	46	of	of	ADP
gojar-5993	10	47	his	his	PRON
gojar-5993	10	48	work	work	NOUN
gojar-5993	10	49	.	.	PUNCT
gojar-5993	11	1	the	the	DET
gojar-5993	11	2	practical	practical	ADJ
gojar-5993	11	3	relevance	relevance	NOUN
gojar-5993	11	4	of	of	ADP
gojar-5993	11	5	lp	lp	NOUN
gojar-5993	11	6	has	have	AUX
gojar-5993	11	7	surged	surge	VERB
gojar-5993	11	8	during	during	ADP
gojar-5993	11	9	and	and	CCONJ
gojar-5993	11	10	after	after	ADP
gojar-5993	11	11	world	world	PROPN
gojar-5993	11	12	war	war	PROPN
gojar-5993	11	13	ii	ii	PROPN
gojar-5993	11	14	.	.	PUNCT
gojar-5993	12	1	researchers	researcher	NOUN
gojar-5993	12	2	,	,	PUNCT
gojar-5993	12	3	notably	notably	ADV
gojar-5993	12	4	george	george	PROPN
gojar-5993	12	5	dantzig	dantzig	PROPN
gojar-5993	12	6	,	,	PUNCT
gojar-5993	12	7	have	have	AUX
gojar-5993	12	8	developed	develop	VERB
gojar-5993	12	9	the	the	DET
gojar-5993	12	10	simplex	simplex	NOUN
gojar-5993	12	11	algorithm	algorithm	NOUN
gojar-5993	12	12	to	to	PART
gojar-5993	12	13	optimize	optimize	VERB
gojar-5993	12	14	military	military	ADJ
gojar-5993	12	15	logistics	logistic	NOUN
gojar-5993	12	16	and	and	CCONJ
gojar-5993	12	17	supply	supply	NOUN
gojar-5993	12	18	chains	chain	NOUN
gojar-5993	12	19	,	,	PUNCT
gojar-5993	12	20	a	a	DET
gojar-5993	12	21	milestone	milestone	NOUN
gojar-5993	12	22	in	in	ADP
gojar-5993	12	23	the	the	DET
gojar-5993	12	24	application	application	NOUN
gojar-5993	12	25	of	of	ADP
gojar-5993	12	26	mathematical	mathematical	ADJ
gojar-5993	12	27	optimization	optimization	NOUN
gojar-5993	12	28	(	(	PUNCT
gojar-5993	12	29	dantzig	dantzig	NOUN
gojar-5993	12	30	,	,	PUNCT
gojar-5993	12	31	1947	1947	NUM
gojar-5993	12	32	)	)	PUNCT
gojar-5993	12	33	.	.	PUNCT
gojar-5993	13	1	the	the	DET
gojar-5993	13	2	simplex	simplex	NOUN
gojar-5993	13	3	algorithm	algorithm	NOUN
gojar-5993	13	4	has	have	AUX
gojar-5993	13	5	remained	remain	VERB
gojar-5993	13	6	a	a	DET
gojar-5993	13	7	cornerstone	cornerstone	NOUN
gojar-5993	13	8	in	in	ADP
gojar-5993	13	9	solving	solve	VERB
gojar-5993	13	10	lp	lp	NOUN
gojar-5993	13	11	problems	problem	NOUN
gojar-5993	13	12	,	,	PUNCT
gojar-5993	13	13	celebrated	celebrate	VERB
gojar-5993	13	14	for	for	ADP
gojar-5993	13	15	its	its	PRON
gojar-5993	13	16	practical	practical	ADJ
gojar-5993	13	17	efficiency	efficiency	NOUN
gojar-5993	13	18	and	and	CCONJ
gojar-5993	13	19	ease	ease	NOUN
gojar-5993	13	20	of	of	ADP
gojar-5993	13	21	implementation	implementation	NOUN
gojar-5993	13	22	,	,	PUNCT
gojar-5993	13	23	despite	despite	SCONJ
gojar-5993	13	24	its	its	PRON
gojar-5993	13	25	potential	potential	ADJ
gojar-5993	13	26	exponential	exponential	ADJ
gojar-5993	13	27	time	time	NOUN
gojar-5993	13	28	complexity	complexity	NOUN
gojar-5993	13	29	in	in	ADP
gojar-5993	13	30	the	the	DET
gojar-5993	13	31	worst	bad	ADJ
gojar-5993	13	32	-	-	PUNCT
gojar-5993	13	33	case	case	NOUN
gojar-5993	13	34	scenarios	scenario	NOUN
gojar-5993	13	35	.	.	PUNCT
gojar-5993	14	1	the	the	DET
gojar-5993	14	2	study	study	NOUN
gojar-5993	14	3	of	of	ADP
gojar-5993	14	4	rational	rational	ADJ
gojar-5993	14	5	linear	linear	ADJ
gojar-5993	14	6	programming	programming	NOUN
gojar-5993	14	7	,	,	PUNCT
gojar-5993	14	8	characterized	characterize	VERB
gojar-5993	14	9	by	by	ADP
gojar-5993	14	10	constraints	constraint	NOUN
gojar-5993	14	11	and	and	CCONJ
gojar-5993	14	12	objectives	objective	NOUN
gojar-5993	14	13	expressed	express	VERB
gojar-5993	14	14	using	use	VERB
gojar-5993	14	15	rational	rational	ADJ
gojar-5993	14	16	coefficients	coefficient	NOUN
gojar-5993	14	17	,	,	PUNCT
gojar-5993	14	18	has	have	AUX
gojar-5993	14	19	gained	gain	VERB
gojar-5993	14	20	prominence	prominence	NOUN
gojar-5993	14	21	with	with	ADP
gojar-5993	14	22	advancements	advancement	NOUN
gojar-5993	14	23	in	in	ADP
gojar-5993	14	24	computational	computational	ADJ
gojar-5993	14	25	methodologies	methodology	NOUN
gojar-5993	14	26	.	.	PUNCT
gojar-5993	15	1	researchers	researcher	NOUN
gojar-5993	15	2	have	have	AUX
gojar-5993	15	3	extensively	extensively	ADV
gojar-5993	15	4	analyzed	analyze	VERB
gojar-5993	15	5	rational	rational	ADJ
gojar-5993	15	6	lp	lp	NOUN
gojar-5993	15	7	systems	system	NOUN
gojar-5993	15	8	,	,	PUNCT
gojar-5993	15	9	focusing	focus	VERB
gojar-5993	15	10	on	on	ADP
gojar-5993	15	11	their	their	PRON
gojar-5993	15	12	numerical	numerical	ADJ
gojar-5993	15	13	properties	property	NOUN
gojar-5993	15	14	,	,	PUNCT
gojar-5993	15	15	solution	solution	NOUN
gojar-5993	15	16	size	size	NOUN
gojar-5993	15	17	,	,	PUNCT
gojar-5993	15	18	and	and	CCONJ
gojar-5993	15	19	computational	computational	ADJ
gojar-5993	15	20	feasibility	feasibility	NOUN
gojar-5993	15	21	.	.	PUNCT
gojar-5993	16	1	during	during	ADP
gojar-5993	16	2	the	the	DET
gojar-5993	16	3	1980s	1980s	NUM
gojar-5993	16	4	,	,	PUNCT
gojar-5993	16	5	karmarkar	karmarkar	NOUN
gojar-5993	16	6	’s	’s	PART
gojar-5993	16	7	introduction	introduction	NOUN
gojar-5993	16	8	of	of	ADP
gojar-5993	16	9	the	the	DET
gojar-5993	16	10	polynomial	polynomial	ADJ
gojar-5993	16	11	-	-	PUNCT
gojar-5993	16	12	time	time	NOUN
gojar-5993	16	13	interior	interior	ADJ
gojar-5993	16	14	-	-	PUNCT
gojar-5993	16	15	point	point	NOUN
gojar-5993	16	16	method	method	NOUN
gojar-5993	16	17	has	have	AUX
gojar-5993	16	18	revolutionized	revolutionize	VERB
gojar-5993	16	19	the	the	DET
gojar-5993	16	20	field	field	NOUN
gojar-5993	16	21	,	,	PUNCT
gojar-5993	16	22	providing	provide	VERB
gojar-5993	16	23	an	an	DET
gojar-5993	16	24	alternative	alternative	NOUN
gojar-5993	16	25	to	to	ADP
gojar-5993	16	26	the	the	DET
gojar-5993	16	27	simplex	simplex	NOUN
gojar-5993	16	28	algorithm	algorithm	NOUN
gojar-5993	16	29	and	and	CCONJ
gojar-5993	16	30	emphasizing	emphasize	VERB
gojar-5993	16	31	the	the	DET
gojar-5993	16	32	significance	significance	NOUN
gojar-5993	16	33	of	of	ADP
gojar-5993	16	34	numerical	numerical	ADJ
gojar-5993	16	35	stability	stability	NOUN
gojar-5993	16	36	in	in	ADP
gojar-5993	16	37	optimization	optimization	NOUN
gojar-5993	16	38	(	(	PUNCT
gojar-5993	16	39	karmarkar	karmarkar	NOUN
gojar-5993	16	40	,	,	PUNCT
gojar-5993	16	41	1984	1984	NUM
gojar-5993	16	42	)	)	PUNCT
gojar-5993	16	43	.	.	PUNCT
gojar-5993	17	1	polyhedral	polyhedral	ADJ
gojar-5993	17	2	optimization	optimization	NOUN
gojar-5993	17	3	,	,	PUNCT
gojar-5993	17	4	a	a	DET
gojar-5993	17	5	core	core	NOUN
gojar-5993	17	6	area	area	NOUN
gojar-5993	17	7	of	of	ADP
gojar-5993	17	8	mathematical	mathematical	ADJ
gojar-5993	17	9	optimization	optimization	NOUN
gojar-5993	17	10	,	,	PUNCT
gojar-5993	17	11	has	have	AUX
gojar-5993	17	12	bridged	bridge	VERB
gojar-5993	17	13	critical	critical	ADJ
gojar-5993	17	14	concepts	concept	NOUN
gojar-5993	17	15	in	in	ADP
gojar-5993	17	16	combinatorics	combinatoric	NOUN
gojar-5993	17	17	,	,	PUNCT
gojar-5993	17	18	geometry	geometry	NOUN
gojar-5993	17	19	,	,	PUNCT
gojar-5993	17	20	and	and	CCONJ
gojar-5993	17	21	optimization	optimization	NOUN
gojar-5993	17	22	.	.	PUNCT
gojar-5993	18	1	scholars	scholar	NOUN
gojar-5993	18	2	have	have	AUX
gojar-5993	18	3	explored	explore	VERB
gojar-5993	18	4	the	the	DET
gojar-5993	18	5	geometric	geometric	ADJ
gojar-5993	18	6	properties	property	NOUN
gojar-5993	18	7	of	of	ADP
gojar-5993	18	8	feasible	feasible	ADJ
gojar-5993	18	9	regions	region	NOUN
gojar-5993	18	10	defined	define	VERB
gojar-5993	18	11	by	by	ADP
gojar-5993	18	12	linear	linear	PROPN
gojar-5993	18	13	inequalities	inequality	NOUN
gojar-5993	18	14	,	,	PUNCT
gojar-5993	18	15	offering	offer	VERB
gojar-5993	18	16	theoretical	theoretical	ADJ
gojar-5993	18	17	and	and	CCONJ
gojar-5993	18	18	practical	practical	ADJ
gojar-5993	18	19	insights	insight	NOUN
gojar-5993	18	20	for	for	ADP
gojar-5993	18	21	solving	solve	VERB
gojar-5993	18	22	complex	complex	ADJ
gojar-5993	18	23	problems	problem	NOUN
gojar-5993	18	24	.	.	PUNCT
gojar-5993	19	1	the	the	DET
gojar-5993	19	2	study	study	NOUN
gojar-5993	19	3	of	of	ADP
gojar-5993	19	4	polyhedra	polyhedra	PROPN
gojar-5993	19	5	has	have	AUX
gojar-5993	19	6	uncovered	uncover	VERB
gojar-5993	19	7	deep	deep	ADJ
gojar-5993	19	8	structural	structural	ADJ
gojar-5993	19	9	relationships	relationship	NOUN
gojar-5993	19	10	essential	essential	ADJ
gojar-5993	19	11	for	for	ADP
gojar-5993	19	12	various	various	ADJ
gojar-5993	19	13	optimization	optimization	NOUN
gojar-5993	19	14	tasks	task	NOUN
gojar-5993	19	15	,	,	PUNCT
gojar-5993	19	16	including	include	VERB
gojar-5993	19	17	vertex	vertex	NOUN
gojar-5993	19	18	enumeration	enumeration	NOUN
gojar-5993	19	19	and	and	CCONJ
gojar-5993	19	20	facet	facet	ADJ
gojar-5993	19	21	identification	identification	NOUN
gojar-5993	19	22	.	.	PUNCT
gojar-5993	20	1	among	among	ADP
gojar-5993	20	2	the	the	DET
gojar-5993	20	3	impactful	impactful	ADJ
gojar-5993	20	4	concepts	concept	NOUN
gojar-5993	20	5	in	in	ADP
gojar-5993	20	6	polyhedral	polyhedral	ADJ
gojar-5993	20	7	optimization	optimization	NOUN
gojar-5993	20	8	is	be	AUX
gojar-5993	20	9	the	the	DET
gojar-5993	20	10	integer	integer	NOUN
gojar-5993	20	11	hull	hull	NOUN
gojar-5993	20	12	,	,	PUNCT
gojar-5993	20	13	representing	represent	VERB
gojar-5993	20	14	the	the	DET
gojar-5993	20	15	convex	convex	ADJ
gojar-5993	20	16	hull	hull	NOUN
gojar-5993	20	17	of	of	ADP
gojar-5993	20	18	all	all	DET
gojar-5993	20	19	integer	integer	NOUN
gojar-5993	20	20	solutions	solution	NOUN
gojar-5993	20	21	within	within	ADP
gojar-5993	20	22	a	a	DET
gojar-5993	20	23	polyhedron	polyhedron	NOUN
gojar-5993	20	24	.	.	PUNCT
gojar-5993	21	1	this	this	DET
gojar-5993	21	2	concept	concept	NOUN
gojar-5993	21	3	has	have	AUX
gojar-5993	21	4	substantially	substantially	ADV
gojar-5993	21	5	advanced	advance	VERB
gojar-5993	21	6	the	the	DET
gojar-5993	21	7	theory	theory	NOUN
gojar-5993	21	8	and	and	CCONJ
gojar-5993	21	9	algorithms	algorithm	NOUN
gojar-5993	21	10	of	of	ADP
gojar-5993	21	11	integer	integer	NOUN
gojar-5993	21	12	programming	programming	NOUN
gojar-5993	21	13	and	and	CCONJ
gojar-5993	21	14	mixed	mixed	ADJ
gojar-5993	21	15	-	-	PUNCT
gojar-5993	21	16	integer	integer	NOUN
gojar-5993	21	17	programming	programming	NOUN
gojar-5993	21	18	,	,	PUNCT
gojar-5993	21	19	as	as	SCONJ
gojar-5993	21	20	emphasized	emphasize	VERB
gojar-5993	21	21	by	by	ADP
gojar-5993	21	22	nemhauser	nemhauser	NOUN
gojar-5993	21	23	and	and	CCONJ
gojar-5993	21	24	wolsey	wolsey	ADJ
gojar-5993	21	25	(	(	PUNCT
gojar-5993	21	26	1999	1999	NUM
gojar-5993	21	27	)	)	PUNCT
gojar-5993	21	28	.	.	PUNCT
gojar-5993	22	1	by	by	ADP
gojar-5993	22	2	enabling	enable	VERB
gojar-5993	22	3	the	the	DET
gojar-5993	22	4	transition	transition	NOUN
gojar-5993	22	5	from	from	ADP
gojar-5993	22	6	an	an	DET
gojar-5993	22	7	infinite	infinite	ADJ
gojar-5993	22	8	search	search	NOUN
gojar-5993	22	9	space	space	NOUN
gojar-5993	22	10	to	to	ADP
gojar-5993	22	11	a	a	DET
gojar-5993	22	12	finite	finite	NOUN
gojar-5993	22	13	and	and	CCONJ
gojar-5993	22	14	structured	structured	ADJ
gojar-5993	22	15	geometric	geometric	ADJ
gojar-5993	22	16	framework	framework	NOUN
gojar-5993	22	17	,	,	PUNCT
gojar-5993	22	18	the	the	DET
gojar-5993	22	19	integer	integer	NOUN
gojar-5993	22	20	hull	hull	NOUN
gojar-5993	22	21	has	have	AUX
gojar-5993	22	22	simplified	simplify	VERB
gojar-5993	22	23	the	the	DET
gojar-5993	22	24	analysis	analysis	NOUN
gojar-5993	22	25	of	of	ADP
gojar-5993	22	26	discrete	discrete	ADJ
gojar-5993	22	27	variable	variable	ADJ
gojar-5993	22	28	problems	problem	NOUN
gojar-5993	22	29	.	.	PUNCT
gojar-5993	23	1	recent	recent	ADJ
gojar-5993	23	2	advancements	advancement	NOUN
gojar-5993	23	3	in	in	ADP
gojar-5993	23	4	polyhedral	polyhedral	ADJ
gojar-5993	23	5	optimization	optimization	NOUN
gojar-5993	23	6	include	include	VERB
gojar-5993	23	7	the	the	DET
gojar-5993	23	8	construction	construction	NOUN
gojar-5993	23	9	and	and	CCONJ
gojar-5993	23	10	analysis	analysis	NOUN
gojar-5993	23	11	of	of	ADP
gojar-5993	23	12	rational	rational	ADJ
gojar-5993	23	13	polyhedra	polyhedra	NOUN
gojar-5993	23	14	on	on	ADP
gojar-5993	23	15	boards	board	NOUN
gojar-5993	23	16	.	.	PUNCT
gojar-5993	24	1	laisin	laisin	VERB
gojar-5993	24	2	et	et	PROPN
gojar-5993	24	3	al	al	PROPN
gojar-5993	24	4	.	.	PROPN
gojar-5993	25	1	(	(	PUNCT
gojar-5993	25	2	2024	2024	NUM
gojar-5993	25	3	)	)	PUNCT
gojar-5993	25	4	have	have	AUX
gojar-5993	25	5	demonstrated	demonstrate	VERB
gojar-5993	25	6	the	the	DET
gojar-5993	25	7	practical	practical	ADJ
gojar-5993	25	8	effectiveness	effectiveness	NOUN
gojar-5993	25	9	of	of	ADP
gojar-5993	25	10	polyhedral	polyhedral	ADJ
gojar-5993	25	11	techniques	technique	NOUN
gojar-5993	25	12	in	in	ADP
gojar-5993	25	13	modeling	modeling	NOUN
gojar-5993	25	14	and	and	CCONJ
gojar-5993	25	15	solving	solve	VERB
gojar-5993	25	16	problems	problem	NOUN
gojar-5993	25	17	involving	involve	VERB
gojar-5993	25	18	integral	integral	ADJ
gojar-5993	25	19	polyhedra	polyhedra	NOUN
gojar-5993	25	20	,	,	PUNCT
gojar-5993	25	21	offering	offer	VERB
gojar-5993	25	22	applications	application	NOUN
gojar-5993	25	23	in	in	ADP
gojar-5993	25	24	combinatorial	combinatorial	ADJ
gojar-5993	25	25	optimization	optimization	NOUN
gojar-5993	25	26	and	and	CCONJ
gojar-5993	25	27	computational	computational	ADJ
gojar-5993	25	28	geometry	geometry	NOUN
gojar-5993	25	29	.	.	PUNCT
gojar-5993	26	1	their	their	PRON
gojar-5993	26	2	work	work	NOUN
gojar-5993	26	3	has	have	AUX
gojar-5993	26	4	exemplified	exemplify	VERB
gojar-5993	26	5	how	how	SCONJ
gojar-5993	26	6	modern	modern	ADJ
gojar-5993	26	7	techniques	technique	NOUN
gojar-5993	26	8	can	can	AUX
gojar-5993	26	9	address	address	VERB
gojar-5993	26	10	both	both	DET
gojar-5993	26	11	theoretical	theoretical	ADJ
gojar-5993	26	12	challenges	challenge	NOUN
gojar-5993	26	13	and	and	CCONJ
gojar-5993	26	14	real	real	ADJ
gojar-5993	26	15	-	-	PUNCT
gojar-5993	26	16	world	world	NOUN
gojar-5993	26	17	applications	application	NOUN
gojar-5993	26	18	.	.	PUNCT
gojar-5993	27	1	this	this	DET
gojar-5993	27	2	paper	paper	NOUN
gojar-5993	27	3	examines	examine	VERB
gojar-5993	27	4	two	two	NUM
gojar-5993	27	5	fundamental	fundamental	ADJ
gojar-5993	27	6	aspects	aspect	NOUN
gojar-5993	27	7	of	of	ADP
gojar-5993	27	8	rational	rational	ADJ
gojar-5993	27	9	lp	lp	ADJ
gojar-5993	27	10	and	and	CCONJ
gojar-5993	27	11	polyhedral	polyhedral	ADJ
gojar-5993	27	12	global	global	ADJ
gojar-5993	27	13	online	online	PROPN
gojar-5993	27	14	journal	journal	PROPN
gojar-5993	27	15	of	of	ADP
gojar-5993	27	16	academic	academic	ADJ
gojar-5993	27	17	research	research	NOUN
gojar-5993	27	18	(	(	PUNCT
gojar-5993	27	19	gojar	gojar	NOUN
gojar-5993	27	20	)	)	PUNCT
gojar-5993	27	21	,	,	PUNCT
gojar-5993	27	22	vol	vol	NOUN
gojar-5993	27	23	.	.	PROPN
gojar-5993	27	24	4	4	NUM
gojar-5993	27	25	,	,	PUNCT
gojar-5993	27	26	no	no	INTJ
gojar-5993	27	27	.	.	NOUN
gojar-5993	27	28	1	1	NUM
gojar-5993	27	29	february	february	NOUN
gojar-5993	27	30	2025	2025	NUM
gojar-5993	27	31	62	62	NUM
gojar-5993	27	32	optimization	optimization	NOUN
gojar-5993	27	33	:	:	PUNCT
gojar-5993	27	34	i.	i.	NOUN
gojar-5993	27	35	conditions	condition	NOUN
gojar-5993	27	36	under	under	ADP
gojar-5993	27	37	which	which	PRON
gojar-5993	27	38	the	the	DET
gojar-5993	27	39	optimization	optimization	NOUN
gojar-5993	27	40	of	of	ADP
gojar-5993	27	41	a	a	DET
gojar-5993	27	42	linear	linear	ADJ
gojar-5993	27	43	function	function	NOUN
gojar-5993	27	44	over	over	ADP
gojar-5993	27	45	a	a	DET
gojar-5993	27	46	rational	rational	ADJ
gojar-5993	27	47	polyhedron	polyhedron	NOUN
gojar-5993	27	48	is	be	AUX
gojar-5993	27	49	bounded	bound	VERB
gojar-5993	27	50	.	.	PUNCT
gojar-5993	28	1	ii	ii	PROPN
gojar-5993	28	2	.	.	PROPN
gojar-5993	28	3	bounds	bound	VERB
gojar-5993	28	4	on	on	ADP
gojar-5993	28	5	the	the	DET
gojar-5993	28	6	size	size	NOUN
gojar-5993	28	7	of	of	ADP
gojar-5993	28	8	optimal	optimal	ADJ
gojar-5993	28	9	solutions	solution	NOUN
gojar-5993	28	10	.	.	PUNCT
gojar-5993	29	1	building	build	VERB
gojar-5993	29	2	on	on	ADP
gojar-5993	29	3	classical	classical	ADJ
gojar-5993	29	4	results	result	NOUN
gojar-5993	29	5	from	from	ADP
gojar-5993	29	6	schrijver	schrijver	PROPN
gojar-5993	29	7	(	(	PUNCT
gojar-5993	29	8	1998	1998	NUM
gojar-5993	29	9	)	)	PUNCT
gojar-5993	29	10	and	and	CCONJ
gojar-5993	29	11	others	other	NOUN
gojar-5993	29	12	,	,	PUNCT
gojar-5993	29	13	the	the	DET
gojar-5993	29	14	analysis	analysis	NOUN
gojar-5993	29	15	provides	provide	VERB
gojar-5993	29	16	refined	refined	ADJ
gojar-5993	29	17	bounds	bound	NOUN
gojar-5993	29	18	and	and	CCONJ
gojar-5993	29	19	structural	structural	ADJ
gojar-5993	29	20	insights	insight	NOUN
gojar-5993	29	21	critical	critical	ADJ
gojar-5993	29	22	for	for	ADP
gojar-5993	29	23	advancing	advance	VERB
gojar-5993	29	24	both	both	CCONJ
gojar-5993	29	25	theoretical	theoretical	ADJ
gojar-5993	29	26	understanding	understanding	NOUN
gojar-5993	29	27	and	and	CCONJ
gojar-5993	29	28	practical	practical	ADJ
gojar-5993	29	29	applications	application	NOUN
gojar-5993	29	30	in	in	ADP
gojar-5993	29	31	optimization	optimization	NOUN
gojar-5993	29	32	.	.	PUNCT
gojar-5993	30	1	ii	ii	PROPN
gojar-5993	30	2	.	.	PUNCT
gojar-5993	31	1	preliminaries	preliminary	NOUN
gojar-5993	31	2	and	and	CCONJ
gojar-5993	31	3	definitions	definition	NOUN
gojar-5993	31	4	definition	definition	NOUN
gojar-5993	31	5	2.1	2.1	NUM
gojar-5993	31	6	:	:	PUNCT
gojar-5993	31	7	sub	sub	ADJ
gojar-5993	31	8	-	-	ADJ
gojar-5993	31	9	determinant	determinant	ADJ
gojar-5993	31	10	let	let	VERB
gojar-5993	31	11	𝑨	𝑨	PRON
gojar-5993	31	12	be	be	AUX
gojar-5993	31	13	an	an	DET
gojar-5993	31	14	integral	integral	ADJ
gojar-5993	31	15	matrix	matrix	NOUN
gojar-5993	31	16	.	.	PUNCT
gojar-5993	32	1	a	a	DET
gojar-5993	32	2	sub	sub	ADJ
gojar-5993	32	3	-	-	ADJ
gojar-5993	32	4	determinant	determinant	ADJ
gojar-5993	32	5	of	of	ADP
gojar-5993	32	6	𝑨	𝑨	PROPN
gojar-5993	32	7	is	be	AUX
gojar-5993	32	8	|𝑩|	|𝑩|	PROPN
gojar-5993	32	9	for	for	ADP
gojar-5993	32	10	some	some	DET
gojar-5993	32	11	square	square	ADJ
gojar-5993	32	12	sub	sub	ADJ
gojar-5993	32	13	-	-	ADJ
gojar-5993	32	14	matrix	matrix	ADJ
gojar-5993	32	15	𝑩	𝑩	PROPN
gojar-5993	32	16	of	of	ADP
gojar-5993	32	17	𝑨	𝑨	PROPN
gojar-5993	32	18	(	(	PUNCT
gojar-5993	32	19	defined	define	VERB
gojar-5993	32	20	by	by	ADP
gojar-5993	32	21	arbitrary	arbitrary	ADJ
gojar-5993	32	22	row	row	NOUN
gojar-5993	32	23	and	and	CCONJ
gojar-5993	32	24	column	column	NOUN
gojar-5993	32	25	indices	index	NOUN
gojar-5993	32	26	)	)	PUNCT
gojar-5993	32	27	.	.	PUNCT
gojar-5993	33	1	we	we	PRON
gojar-5993	33	2	write𝚵(𝑨	write𝚵(𝑨	PROPN
gojar-5993	33	3	)	)	PUNCT
gojar-5993	33	4	for	for	ADP
gojar-5993	33	5	the	the	DET
gojar-5993	33	6	maximum	maximum	ADJ
gojar-5993	33	7	absolute	absolute	ADJ
gojar-5993	33	8	value	value	NOUN
gojar-5993	33	9	of	of	ADP
gojar-5993	33	10	the	the	DET
gojar-5993	33	11	sub	sub	NOUN
gojar-5993	33	12	-	-	NOUN
gojar-5993	33	13	determinants	determinant	NOUN
gojar-5993	33	14	of	of	ADP
gojar-5993	33	15	𝑨.	𝑨.	ADJ
gojar-5993	33	16	definition	definition	NOUN
gojar-5993	33	17	2.2	2.2	NUM
gojar-5993	33	18	:	:	PUNCT
gojar-5993	33	19	polyhedron	polyhedron	NOUN
gojar-5993	33	20	linear	linear	ADJ
gojar-5993	33	21	programming	programming	NOUN
gojar-5993	33	22	deals	deal	NOUN
gojar-5993	33	23	with	with	ADP
gojar-5993	33	24	optimizing	optimize	VERB
gojar-5993	33	25	a	a	DET
gojar-5993	33	26	linear	linear	ADJ
gojar-5993	33	27	objective	objective	ADJ
gojar-5993	33	28	function	function	NOUN
gojar-5993	33	29	of	of	ADP
gojar-5993	33	30	finitely	finitely	ADV
gojar-5993	33	31	many	many	ADJ
gojar-5993	33	32	variables	variable	NOUN
gojar-5993	33	33	subject	subject	ADJ
gojar-5993	33	34	to	to	ADP
gojar-5993	33	35	finitely	finitely	ADV
gojar-5993	33	36	many	many	ADJ
gojar-5993	33	37	linear	linear	ADJ
gojar-5993	33	38	inequalities	inequality	NOUN
gojar-5993	33	39	.	.	PUNCT
gojar-5993	34	1	so	so	ADV
gojar-5993	34	2	the	the	DET
gojar-5993	34	3	set	set	NOUN
gojar-5993	34	4	of	of	ADP
gojar-5993	34	5	feasible	feasible	ADJ
gojar-5993	34	6	solutions	solution	NOUN
gojar-5993	34	7	is	be	AUX
gojar-5993	34	8	the	the	DET
gojar-5993	34	9	intersection	intersection	NOUN
gojar-5993	34	10	of	of	ADP
gojar-5993	34	11	finitely	finitely	ADV
gojar-5993	34	12	many	many	ADJ
gojar-5993	34	13	half	half	ADJ
gojar-5993	34	14	spaces	space	NOUN
gojar-5993	34	15	.	.	PUNCT
gojar-5993	35	1	such	such	DET
gojar-5993	35	2	a	a	DET
gojar-5993	35	3	set	set	NOUN
gojar-5993	35	4	is	be	AUX
gojar-5993	35	5	called	call	VERB
gojar-5993	35	6	a	a	DET
gojar-5993	35	7	polyhedron	polyhedron	NOUN
gojar-5993	35	8	.	.	PUNCT
gojar-5993	36	1	definition	definition	NOUN
gojar-5993	36	2	2.3	2.3	NUM
gojar-5993	36	3	:	:	PUNCT
gojar-5993	36	4	polyhedron	polyhedron	NOUN
gojar-5993	36	5	in	in	ADP
gojar-5993	36	6	ℝ𝒏	ℝ𝒏	ADP
gojar-5993	36	7	it	it	PRON
gojar-5993	36	8	is	be	AUX
gojar-5993	36	9	a	a	DET
gojar-5993	36	10	set	set	NOUN
gojar-5993	36	11	of	of	ADP
gojar-5993	36	12	type	type	NOUN
gojar-5993	36	13	𝑷	𝑷	PROPN
gojar-5993	36	14	=	=	PUNCT
gojar-5993	37	1	{	{	PUNCT
gojar-5993	37	2	𝒙	𝒙	PROPN
gojar-5993	37	3	∈	∈	PROPN
gojar-5993	37	4	ℝ𝒏	ℝ𝒏	NOUN
gojar-5993	37	5	:	:	PUNCT
gojar-5993	37	6	𝑨𝒙	𝑨𝒙	PROPN
gojar-5993	37	7	≤	≤	NOUN
gojar-5993	37	8	𝒃	𝒃	VERB
gojar-5993	37	9	}	}	PUNCT
gojar-5993	37	10	for	for	ADP
gojar-5993	37	11	some	some	DET
gojar-5993	37	12	matrix	matrix	NOUN
gojar-5993	37	13	𝑨	𝑨	NOUN
gojar-5993	37	14	∈	∈	PROPN
gojar-5993	37	15	ℝ𝒎×𝒏and	ℝ𝒎×𝒏and	PUNCT
gojar-5993	37	16	some	some	DET
gojar-5993	37	17	vector	vector	NOUN
gojar-5993	37	18	𝒃	𝒃	PROPN
gojar-5993	37	19	∈	∈	PROPN
gojar-5993	37	20	ℝ𝒎.	ℝ𝒎.	PROPN
gojar-5993	38	1	if	if	SCONJ
gojar-5993	38	2	a	a	PRON
gojar-5993	38	3	and	and	CCONJ
gojar-5993	38	4	b	b	NOUN
gojar-5993	38	5	are	be	AUX
gojar-5993	38	6	rational	rational	ADJ
gojar-5993	38	7	,	,	PUNCT
gojar-5993	38	8	then	then	ADV
gojar-5993	38	9	p	p	NOUN
gojar-5993	38	10	is	be	AUX
gojar-5993	38	11	a	a	DET
gojar-5993	38	12	rational	rational	ADJ
gojar-5993	38	13	polyhedron	polyhedron	NOUN
gojar-5993	38	14	.	.	PUNCT
gojar-5993	39	1	a	a	DET
gojar-5993	39	2	bounded	bounded	ADJ
gojar-5993	39	3	polyhedron	polyhedron	NOUN
gojar-5993	39	4	is	be	AUX
gojar-5993	39	5	also	also	ADV
gojar-5993	39	6	called	call	VERB
gojar-5993	39	7	a	a	DET
gojar-5993	39	8	polytope	polytope	NOUN
gojar-5993	39	9	.	.	PUNCT
gojar-5993	40	1	we	we	PRON
gojar-5993	40	2	denote	denote	VERB
gojar-5993	40	3	the	the	DET
gojar-5993	40	4	rank	rank	NOUN
gojar-5993	40	5	of	of	ADP
gojar-5993	40	6	a	a	DET
gojar-5993	40	7	matrix	matrix	NOUN
gojar-5993	40	8	a	a	PRON
gojar-5993	40	9	by	by	ADP
gojar-5993	40	10	𝒓𝒂𝒏𝒌(𝑨	𝒓𝒂𝒏𝒌(𝑨	PROPN
gojar-5993	40	11	)	)	PUNCT
gojar-5993	40	12	.	.	PUNCT
gojar-5993	41	1	the	the	DET
gojar-5993	41	2	dimension	dimension	NOUN
gojar-5993	41	3	dim	dim	NOUN
gojar-5993	41	4	x	x	PUNCT
gojar-5993	41	5	of	of	ADP
gojar-5993	41	6	a	a	DET
gojar-5993	41	7	nonempty	nonempty	ADJ
gojar-5993	41	8	set	set	NOUN
gojar-5993	41	9	:	:	PUNCT
gojar-5993	41	10	𝒙	𝒙	PROPN
gojar-5993	41	11	⊆	⊆	NUM
gojar-5993	41	12	ℝ𝒏	ℝ𝒏	NOUN
gojar-5993	41	13	is	be	AUX
gojar-5993	41	14	defined	define	VERB
gojar-5993	41	15	to	to	PART
gojar-5993	41	16	be	be	AUX
gojar-5993	41	17	𝒏	𝒏	PROPN
gojar-5993	41	18	−	−	PROPN
gojar-5993	41	19	𝐦𝐚𝐱	𝐦𝐚𝐱	PROPN
gojar-5993	41	20	𝒓𝒂𝒏𝒌(𝑨	𝒓𝒂𝒏𝒌(𝑨	PROPN
gojar-5993	41	21	)	)	PUNCT
gojar-5993	41	22	{	{	PUNCT
gojar-5993	41	23	𝒓𝒂𝒏𝒌(𝑨	𝒓𝒂𝒏𝒌(𝑨	NOUN
gojar-5993	41	24	):	):	PUNCT
gojar-5993	41	25	𝑨	𝑨	PROPN
gojar-5993	41	26	𝐢𝐬	𝐢𝐬	PROPN
gojar-5993	41	27	𝐚𝐧	𝐚𝐧	NOUN
gojar-5993	41	28	𝒏	𝒏	PROPN
gojar-5993	41	29	×	×	PROPN
gojar-5993	41	30	𝒏	𝒏	PROPN
gojar-5993	41	31	−	−	PROPN
gojar-5993	41	32	𝐦𝐚𝐭𝐫𝐢𝐱	𝐦𝐚𝐭𝐫𝐢𝐱	NOUN
gojar-5993	41	33	𝐰𝐢𝐭𝐡	𝐰𝐢𝐭𝐡	NOUN
gojar-5993	42	1	𝑨𝒙	𝑨𝒙	PROPN
gojar-5993	42	2	=	=	PUNCT
gojar-5993	42	3	𝑨𝒚	𝑨𝒚	NOUN
gojar-5993	42	4	𝐟𝐨𝐫	𝐟𝐨𝐫	NOUN
gojar-5993	42	5	𝐚𝐥𝐥	𝐚𝐥𝐥	VERB
gojar-5993	42	6	𝒙	𝒙	PROPN
gojar-5993	42	7	,	,	PUNCT
gojar-5993	42	8	𝒚	𝒚	PROPN
gojar-5993	42	9	∈	∈	PROPN
gojar-5993	42	10	𝑿	𝑿	PROPN
gojar-5993	42	11	}	}	PUNCT
gojar-5993	42	12	a	a	DET
gojar-5993	42	13	polyhedron	polyhedron	NOUN
gojar-5993	42	14	𝑷	𝑷	PROPN
gojar-5993	42	15	⊆	⊆	NUM
gojar-5993	42	16	ℝ𝒏	ℝ𝒏	PROPN
gojar-5993	42	17	is	be	AUX
gojar-5993	42	18	called	call	VERB
gojar-5993	42	19	full	full	ADJ
gojar-5993	42	20	-	-	PUNCT
gojar-5993	42	21	dimensional	dimensional	ADJ
gojar-5993	42	22	if	if	SCONJ
gojar-5993	42	23	𝐝𝐢𝐦	𝐝𝐢𝐦	NOUN
gojar-5993	42	24	𝑷	𝑷	NOUN
gojar-5993	42	25	=	=	SYM
gojar-5993	42	26	𝒏	𝒏	PROPN
gojar-5993	42	27	equivalently	equivalently	ADV
gojar-5993	42	28	,	,	PUNCT
gojar-5993	42	29	a	a	DET
gojar-5993	42	30	polyhedron	polyhedron	NOUN
gojar-5993	42	31	is	be	AUX
gojar-5993	42	32	full	full	ADV
gojar-5993	42	33	-	-	PUNCT
gojar-5993	42	34	dimensional	dimensional	ADJ
gojar-5993	42	35	if	if	SCONJ
gojar-5993	43	1	and	and	CCONJ
gojar-5993	43	2	only	only	ADV
gojar-5993	43	3	if	if	SCONJ
gojar-5993	43	4	there	there	PRON
gojar-5993	43	5	exist	exist	VERB
gojar-5993	43	6	a	a	DET
gojar-5993	43	7	point	point	NOUN
gojar-5993	43	8	𝒙∗	𝒙∗	NOUN
gojar-5993	43	9	in	in	ADP
gojar-5993	43	10	its	its	PRON
gojar-5993	43	11	interior	interior	NOUN
gojar-5993	43	12	.	.	PUNCT
gojar-5993	44	1	(	(	PUNCT
gojar-5993	44	2	genova	genova	PROPN
gojar-5993	44	3	and	and	CCONJ
gojar-5993	44	4	guliashki	guliashki	PROPN
gojar-5993	44	5	,	,	PUNCT
gojar-5993	44	6	2011	2011	NUM
gojar-5993	44	7	)	)	PUNCT
gojar-5993	44	8	.	.	PUNCT
gojar-5993	45	1	global	global	ADJ
gojar-5993	45	2	online	online	PROPN
gojar-5993	45	3	journal	journal	PROPN
gojar-5993	45	4	of	of	ADP
gojar-5993	45	5	academic	academic	ADJ
gojar-5993	45	6	research	research	NOUN
gojar-5993	45	7	(	(	PUNCT
gojar-5993	45	8	gojar	gojar	NOUN
gojar-5993	45	9	)	)	PUNCT
gojar-5993	45	10	,	,	PUNCT
gojar-5993	45	11	vol	vol	NOUN
gojar-5993	45	12	.	.	PROPN
gojar-5993	45	13	4	4	NUM
gojar-5993	45	14	,	,	PUNCT
gojar-5993	45	15	no	no	INTJ
gojar-5993	45	16	.	.	NOUN
gojar-5993	45	17	1	1	NUM
gojar-5993	45	18	february	february	PROPN
gojar-5993	45	19	2025	2025	NUM
gojar-5993	45	20	63	63	NUM
gojar-5993	45	21	proposition	proposition	NOUN
gojar-5993	45	22	2.1	2.1	NUM
gojar-5993	45	23	:	:	PUNCT
gojar-5993	45	24	nonempty	nonempty	ADJ
gojar-5993	45	25	polyhedron	polyhedron	NOUN
gojar-5993	45	26	:	:	PUNCT
gojar-5993	45	27	let	let	VERB
gojar-5993	45	28	𝑃	𝑃	VERB
gojar-5993	45	29	=	=	PRON
gojar-5993	45	30	{	{	PUNCT
gojar-5993	45	31	𝑥	𝑥	NOUN
gojar-5993	45	32	∶	∶	NOUN
gojar-5993	45	33	𝐴𝑥	𝐴𝑥	X
gojar-5993	45	34	≤	≤	ADJ
gojar-5993	45	35	𝑏	𝑏	NOUN
gojar-5993	45	36	}	}	PUNCT
gojar-5993	45	37	be	be	AUX
gojar-5993	45	38	a	a	DET
gojar-5993	45	39	nonempty	nonempty	ADJ
gojar-5993	45	40	polyhedron	polyhedron	NOUN
gojar-5993	45	41	.	.	PUNCT
gojar-5993	46	1	if	if	SCONJ
gojar-5993	46	2	c	c	PROPN
gojar-5993	46	3	is	be	AUX
gojar-5993	46	4	a	a	DET
gojar-5993	46	5	nonzero	nonzero	ADJ
gojar-5993	46	6	vector	vector	NOUN
gojar-5993	46	7	for	for	ADP
gojar-5993	46	8	which	which	PRON
gojar-5993	46	9	𝜹	𝜹	NUM
gojar-5993	46	10	∶=	∶=	NUM
gojar-5993	46	11	𝐦𝐚𝐱	𝐦𝐚𝐱	NOUN
gojar-5993	46	12	{	{	PUNCT
gojar-5993	46	13	𝒄𝒙	𝒄𝒙	PROPN
gojar-5993	46	14	∶	∶	VERB
gojar-5993	46	15	𝒙	𝒙	ADJ
gojar-5993	46	16	∈	∈	NOUN
gojar-5993	46	17	𝑷	𝑷	PROPN
gojar-5993	46	18	}	}	PUNCT
gojar-5993	46	19	is	be	AUX
gojar-5993	46	20	finite	finite	ADJ
gojar-5993	46	21	,	,	PUNCT
gojar-5993	46	22	then	then	ADV
gojar-5993	46	23	{	{	PUNCT
gojar-5993	46	24	𝒄𝒙	𝒄𝒙	PROPN
gojar-5993	46	25	∶	∶	VERB
gojar-5993	46	26	𝒙	𝒙	X
gojar-5993	47	1	=	=	SYM
gojar-5993	47	2	𝜹	𝜹	NOUN
gojar-5993	47	3	}	}	PUNCT
gojar-5993	47	4	is	be	AUX
gojar-5993	47	5	called	call	VERB
gojar-5993	47	6	a	a	DET
gojar-5993	47	7	supporting	support	VERB
gojar-5993	47	8	hyperplane	hyperplane	NOUN
gojar-5993	47	9	of	of	ADP
gojar-5993	47	10	p.	p.	PROPN
gojar-5993	47	11	a	a	DET
gojar-5993	47	12	face	face	NOUN
gojar-5993	47	13	of	of	ADP
gojar-5993	47	14	p	p	NOUN
gojar-5993	47	15	is	be	AUX
gojar-5993	47	16	p	p	PRON
gojar-5993	47	17	itself	itself	PRON
gojar-5993	47	18	or	or	CCONJ
gojar-5993	47	19	the	the	DET
gojar-5993	47	20	intersection	intersection	NOUN
gojar-5993	47	21	of	of	ADP
gojar-5993	47	22	p	p	NOUN
gojar-5993	47	23	with	with	ADP
gojar-5993	47	24	a	a	DET
gojar-5993	47	25	supporting	support	VERB
gojar-5993	47	26	hyperplane	hyperplane	NOUN
gojar-5993	47	27	of	of	ADP
gojar-5993	47	28	p.	p.	PROPN
gojar-5993	47	29	a	a	DET
gojar-5993	47	30	point	point	NOUN
gojar-5993	47	31	x	x	PUNCT
gojar-5993	47	32	for	for	ADP
gojar-5993	47	33	which	which	PRON
gojar-5993	47	34	{	{	PUNCT
gojar-5993	47	35	x	x	X
gojar-5993	47	36	}	}	PUNCT
gojar-5993	47	37	is	be	AUX
gojar-5993	47	38	a	a	DET
gojar-5993	47	39	face	face	NOUN
gojar-5993	47	40	is	be	AUX
gojar-5993	47	41	called	call	VERB
gojar-5993	47	42	a	a	DET
gojar-5993	47	43	vertex	vertex	NOUN
gojar-5993	47	44	of	of	ADP
gojar-5993	47	45	p	p	NOUN
gojar-5993	47	46	,	,	PUNCT
gojar-5993	47	47	and	and	CCONJ
gojar-5993	47	48	also	also	ADV
gojar-5993	47	49	a	a	DET
gojar-5993	47	50	basic	basic	ADJ
gojar-5993	47	51	solution	solution	NOUN
gojar-5993	47	52	of	of	ADP
gojar-5993	47	53	the	the	DET
gojar-5993	47	54	system	system	NOUN
gojar-5993	47	55	𝑨𝒙	𝑨𝒙	PROPN
gojar-5993	47	56	≤	≤	NUM
gojar-5993	47	57	𝒃	𝒃	PROPN
gojar-5993	47	58	(	(	PUNCT
gojar-5993	47	59	genova	genova	PROPN
gojar-5993	47	60	and	and	CCONJ
gojar-5993	47	61	guliashki	guliashki	PROPN
gojar-5993	47	62	,	,	PUNCT
gojar-5993	47	63	2011	2011	NUM
gojar-5993	47	64	)	)	PUNCT
gojar-5993	47	65	.	.	PUNCT
gojar-5993	48	1	proposition	proposition	NOUN
gojar-5993	48	2	2.2	2.2	NUM
gojar-5993	48	3	:	:	PUNCT
gojar-5993	48	4	let	let	VERB
gojar-5993	48	5	𝑃	𝑃	PROPN
gojar-5993	48	6	∶=	∶=	NUM
gojar-5993	48	7	{	{	PUNCT
gojar-5993	48	8	𝑥	𝑥	PRON
gojar-5993	48	9	∶	∶	NOUN
gojar-5993	49	1	𝐴𝑥	𝐴𝑥	X
gojar-5993	49	2	≤	≤	ADJ
gojar-5993	49	3	𝑏	𝑏	NOUN
gojar-5993	49	4	}	}	PUNCT
gojar-5993	49	5	be	be	AUX
gojar-5993	49	6	a	a	DET
gojar-5993	49	7	polyhedron	polyhedron	NOUN
gojar-5993	49	8	and	and	CCONJ
gojar-5993	49	9	𝑭	𝑭	NOUN
gojar-5993	49	10	⊆	⊆	PROPN
gojar-5993	49	11	𝑷.	𝑷.	PROPN
gojar-5993	49	12	then	then	ADV
gojar-5993	49	13	the	the	DET
gojar-5993	49	14	following	follow	VERB
gojar-5993	49	15	statements	statement	NOUN
gojar-5993	49	16	are	be	AUX
gojar-5993	49	17	equivalent	equivalent	ADJ
gojar-5993	49	18	:	:	PUNCT
gojar-5993	49	19	(	(	PUNCT
gojar-5993	49	20	a	a	X
gojar-5993	49	21	)	)	PUNCT
gojar-5993	49	22	f	f	NOUN
gojar-5993	49	23	is	be	AUX
gojar-5993	49	24	a	a	DET
gojar-5993	49	25	face	face	NOUN
gojar-5993	49	26	of	of	ADP
gojar-5993	49	27	p.	p.	NOUN
gojar-5993	49	28	(	(	PUNCT
gojar-5993	49	29	b	b	X
gojar-5993	49	30	)	)	PUNCT
gojar-5993	49	31	there	there	PRON
gojar-5993	49	32	exists	exist	VERB
gojar-5993	49	33	a	a	DET
gojar-5993	49	34	vector	vector	NOUN
gojar-5993	49	35	c	c	NOUN
gojar-5993	49	36	such	such	ADJ
gojar-5993	49	37	that	that	SCONJ
gojar-5993	49	38	𝜹	𝜹	NOUN
gojar-5993	49	39	∶=	∶=	NUM
gojar-5993	49	40	𝐦𝐚𝐱	𝐦𝐚𝐱	NOUN
gojar-5993	49	41	{	{	PUNCT
gojar-5993	49	42	𝒄𝒙	𝒄𝒙	PROPN
gojar-5993	49	43	∶	∶	VERB
gojar-5993	50	1	𝒙	𝒙	ADJ
gojar-5993	50	2	∈	∈	NOUN
gojar-5993	50	3	𝑷	𝑷	PROPN
gojar-5993	50	4	}	}	PUNCT
gojar-5993	50	5	is	be	AUX
gojar-5993	50	6	finite	finite	ADJ
gojar-5993	50	7	and	and	CCONJ
gojar-5993	50	8	𝑭	𝑭	NOUN
gojar-5993	50	9	=	=	PUNCT
gojar-5993	50	10	{	{	PUNCT
gojar-5993	50	11	𝒄𝒙	𝒄𝒙	NOUN
gojar-5993	50	12	=	=	PUNCT
gojar-5993	50	13	𝜹	𝜹	SYM
gojar-5993	50	14	∶	∶	VERB
gojar-5993	50	15	𝒙	𝒙	NOUN
gojar-5993	50	16	∈	∈	NOUN
gojar-5993	50	17	𝑷	𝑷	PROPN
gojar-5993	50	18	}	}	PUNCT
gojar-5993	50	19	(	(	PUNCT
gojar-5993	50	20	c	c	X
gojar-5993	50	21	)	)	PUNCT
gojar-5993	50	22	𝑭	𝑭	NOUN
gojar-5993	50	23	∶=	∶=	NUM
gojar-5993	50	24	{	{	PUNCT
gojar-5993	50	25	𝒙	𝒙	PROPN
gojar-5993	50	26	∈	∈	PROPN
gojar-5993	50	27	𝑷	𝑷	NOUN
gojar-5993	50	28	:	:	PUNCT
gojar-5993	50	29	𝑨′𝒙	𝑨′𝒙	X
gojar-5993	50	30	=	=	SYM
gojar-5993	50	31	𝒃′	𝒃′	PROPN
gojar-5993	50	32	}	}	PUNCT
gojar-5993	50	33	≠	≠	PROPN
gojar-5993	50	34	∅	∅	NOUN
gojar-5993	50	35	;	;	PUNCT
gojar-5993	50	36	for	for	ADP
gojar-5993	50	37	some	some	DET
gojar-5993	50	38	subsystem	subsystem	NOUN
gojar-5993	50	39	𝑨′𝒙	𝑨′𝒙	X
gojar-5993	50	40	≤	≤	NUM
gojar-5993	50	41	𝒃′	𝒃′	NOUN
gojar-5993	50	42	of	of	ADP
gojar-5993	50	43	𝑨𝒙	𝑨𝒙	PROPN
gojar-5993	50	44	≤	≤	PROPN
gojar-5993	50	45	𝒃	𝒃	PROPN
gojar-5993	50	46	(	(	PUNCT
gojar-5993	50	47	genova	genova	PROPN
gojar-5993	50	48	and	and	CCONJ
gojar-5993	50	49	guliashki	guliashki	PROPN
gojar-5993	50	50	,	,	PUNCT
gojar-5993	50	51	2011	2011	NUM
gojar-5993	50	52	)	)	PUNCT
gojar-5993	50	53	.	.	PUNCT
gojar-5993	51	1	corollary	corollary	ADJ
gojar-5993	51	2	2.1	2.1	NUM
gojar-5993	51	3	:	:	PUNCT
gojar-5993	51	4	let	let	VERB
gojar-5993	51	5	𝑃	𝑃	PRON
gojar-5993	51	6	be	be	AUX
gojar-5993	51	7	a	a	DET
gojar-5993	51	8	polyhedron	polyhedron	NOUN
gojar-5993	51	9	and	and	CCONJ
gojar-5993	51	10	𝐹	𝐹	PRON
gojar-5993	51	11	a	a	DET
gojar-5993	51	12	face	face	NOUN
gojar-5993	51	13	of	of	ADP
gojar-5993	51	14	𝑃.	𝑃.	PROPN
gojar-5993	51	15	then	then	ADV
gojar-5993	51	16	𝐹	𝐹	PROPN
gojar-5993	51	17	is	be	AUX
gojar-5993	51	18	again	again	ADV
gojar-5993	51	19	a	a	DET
gojar-5993	51	20	polyhedron	polyhedron	NOUN
gojar-5993	51	21	.	.	PUNCT
gojar-5993	52	1	furthermore	furthermore	ADV
gojar-5993	52	2	,	,	PUNCT
gojar-5993	52	3	a	a	DET
gojar-5993	52	4	set	set	NOUN
gojar-5993	52	5	𝐹′	𝐹′	SYM
gojar-5993	52	6	⊆	⊆	NUM
gojar-5993	52	7	𝐹	𝐹	PROPN
gojar-5993	52	8	is	be	AUX
gojar-5993	52	9	a	a	DET
gojar-5993	52	10	face	face	NOUN
gojar-5993	52	11	of	of	ADP
gojar-5993	52	12	𝑃	𝑃	NOUN
gojar-5993	52	13	if	if	NOUN
gojar-5993	53	1	and	and	CCONJ
gojar-5993	53	2	only	only	ADV
gojar-5993	53	3	if	if	SCONJ
gojar-5993	53	4	it	it	PRON
gojar-5993	53	5	is	be	AUX
gojar-5993	53	6	a	a	DET
gojar-5993	53	7	face	face	NOUN
gojar-5993	53	8	of	of	ADP
gojar-5993	53	9	𝐹	𝐹	PROPN
gojar-5993	53	10	(	(	PUNCT
gojar-5993	53	11	genova	genova	PROPN
gojar-5993	53	12	and	and	CCONJ
gojar-5993	53	13	guliashki	guliashki	PROPN
gojar-5993	53	14	,	,	PUNCT
gojar-5993	53	15	2011	2011	NUM
gojar-5993	53	16	)	)	PUNCT
gojar-5993	53	17	.	.	PUNCT
gojar-5993	54	1	proposition	proposition	NOUN
gojar-5993	54	2	2.3	2.3	NUM
gojar-5993	54	3	:	:	PUNCT
gojar-5993	54	4	let	let	VERB
gojar-5993	54	5	𝑷	𝑷	PROPN
gojar-5993	54	6	=	=	PRON
gojar-5993	54	7	{	{	PUNCT
gojar-5993	54	8	𝒙	𝒙	NOUN
gojar-5993	54	9	:	:	PUNCT
gojar-5993	54	10	𝑨𝒙	𝑨𝒙	PROPN
gojar-5993	54	11	≤	≤	NOUN
gojar-5993	54	12	𝒃	𝒃	AUX
gojar-5993	54	13	}	}	PUNCT
gojar-5993	54	14	be	be	AUX
gojar-5993	54	15	a	a	DET
gojar-5993	54	16	polyhedron	polyhedron	NOUN
gojar-5993	54	17	.	.	PUNCT
gojar-5993	55	1	a	a	DET
gojar-5993	55	2	nonempty	nonempty	NOUN
gojar-5993	55	3	subset	subset	VERB
gojar-5993	55	4	𝑭	𝑭	NOUN
gojar-5993	55	5	⊆	⊆	NUM
gojar-5993	55	6	𝑷	𝑷	PROPN
gojar-5993	55	7	is	be	AUX
gojar-5993	55	8	a	a	DET
gojar-5993	55	9	minimal	minimal	ADJ
gojar-5993	55	10	face	face	NOUN
gojar-5993	55	11	of	of	ADP
gojar-5993	55	12	𝑷	𝑷	PRON
gojar-5993	55	13	if	if	SCONJ
gojar-5993	56	1	and	and	CCONJ
gojar-5993	56	2	only	only	ADV
gojar-5993	56	3	if	if	SCONJ
gojar-5993	56	4	it	it	PRON
gojar-5993	56	5	is	be	AUX
gojar-5993	56	6	a	a	DET
gojar-5993	56	7	face	face	NOUN
gojar-5993	56	8	of	of	ADP
gojar-5993	56	9	;	;	PUNCT
gojar-5993	56	10	𝑭	𝑭	NOUN
gojar-5993	56	11	=	=	PUNCT
gojar-5993	56	12	{	{	PUNCT
gojar-5993	56	13	𝒙	𝒙	NUM
gojar-5993	56	14	:	:	PUNCT
gojar-5993	56	15	𝑨′𝒙	𝑨′𝒙	X
gojar-5993	56	16	=	=	SYM
gojar-5993	56	17	𝒃′	𝒃′	PROPN
gojar-5993	56	18	}	}	PUNCT
gojar-5993	56	19	for	for	ADP
gojar-5993	56	20	some	some	DET
gojar-5993	56	21	subsystem	subsystem	NOUN
gojar-5993	56	22	𝑨′𝒙	𝑨′𝒙	X
gojar-5993	56	23	≤	≤	NUM
gojar-5993	56	24	𝒃′	𝒃′	NOUN
gojar-5993	56	25	of	of	ADP
gojar-5993	56	26	𝑨𝒙	𝑨𝒙	PROPN
gojar-5993	56	27	≤	≤	PROPN
gojar-5993	56	28	𝒃	𝒃	NOUN
gojar-5993	56	29	(	(	PUNCT
gojar-5993	56	30	akif	akif	ADJ
gojar-5993	56	31	and	and	CCONJ
gojar-5993	56	32	cihan	cihan	NOUN
gojar-5993	56	33	,	,	PUNCT
gojar-5993	56	34	2008	2008	NUM
gojar-5993	56	35	)	)	PUNCT
gojar-5993	56	36	proposition	proposition	NOUN
gojar-5993	56	37	2.4	2.4	NUM
gojar-5993	56	38	:	:	PUNCT
gojar-5993	56	39	for	for	ADP
gojar-5993	56	40	any	any	DET
gojar-5993	56	41	rational	rational	ADJ
gojar-5993	56	42	square	square	ADJ
gojar-5993	56	43	matrix	matrix	NOUN
gojar-5993	56	44	a	a	PRON
gojar-5993	56	45	we	we	PRON
gojar-5993	56	46	have	have	VERB
gojar-5993	56	47	𝑠𝑖𝑧𝑒	𝑠𝑖𝑧𝑒	PROPN
gojar-5993	56	48	𝑑𝑒𝑡	𝑑𝑒𝑡	PROPN
gojar-5993	56	49	𝐴	𝐴	PROPN
gojar-5993	56	50	≤	≤	PROPN
gojar-5993	56	51	2𝑠𝑖𝑧𝑒(𝐴	2𝑠𝑖𝑧𝑒(𝐴	NUM
gojar-5993	56	52	)	)	PUNCT
gojar-5993	56	53	proposition	proposition	NOUN
gojar-5993	56	54	2.5	2.5	NUM
gojar-5993	56	55	:	:	PUNCT
gojar-5993	56	56	if	if	SCONJ
gojar-5993	56	57	𝒙	𝒙	X
gojar-5993	56	58	,	,	PUNCT
gojar-5993	56	59	𝒚	𝒚	PROPN
gojar-5993	56	60	∈	∈	PROPN
gojar-5993	56	61	ℚ𝒏	ℚ𝒏	PROPN
gojar-5993	56	62	are	be	AUX
gojar-5993	56	63	rational	rational	ADJ
gojar-5993	56	64	vectors	vector	NOUN
gojar-5993	56	65	,	,	PUNCT
gojar-5993	56	66	then	then	ADV
gojar-5993	56	67	𝒔𝒊𝒛𝒆(𝒙	𝒔𝒊𝒛𝒆(𝒙	X
gojar-5993	56	68	+	+	PROPN
gojar-5993	56	69	𝒚	𝒚	PROPN
gojar-5993	56	70	)	)	PUNCT
gojar-5993	56	71	≤	≤	NOUN
gojar-5993	56	72	𝟐(𝒔𝒊𝒛𝒆(𝒙	𝟐(𝒔𝒊𝒛𝒆(𝒙	NOUN
gojar-5993	56	73	)	)	PUNCT
gojar-5993	57	1	+	+	CCONJ
gojar-5993	57	2	𝒔𝒊𝒛𝒆(𝒚	𝒔𝒊𝒛𝒆(𝒚	NOUN
gojar-5993	57	3	)	)	PUNCT
gojar-5993	57	4	)	)	PUNCT
gojar-5993	57	5	𝒔𝒊𝒛𝒆(𝒙𝑻𝒚	𝒔𝒊𝒛𝒆(𝒙𝑻𝒚	NOUN
gojar-5993	57	6	)	)	PUNCT
gojar-5993	57	7	≤	≤	NOUN
gojar-5993	57	8	𝟐(𝒔𝒊𝒛𝒆(𝒙	𝟐(𝒔𝒊𝒛𝒆(𝒙	NOUN
gojar-5993	57	9	)	)	PUNCT
gojar-5993	58	1	+	+	CCONJ
gojar-5993	58	2	𝒔𝒊𝒛𝒆(𝒚))(𝐋𝐚𝐢𝐬𝐢𝐧	𝒔𝒊𝒛𝒆(𝒚))(𝐋𝐚𝐢𝐬𝐢𝐧	PROPN
gojar-5993	58	3	𝒆𝒕	𝒆𝒕	ADJ
gojar-5993	58	4	𝒂𝒍.	𝒂𝒍.	NOUN
gojar-5993	58	5	,	,	PUNCT
gojar-5993	58	6	𝟐𝟎𝟐𝟒	𝟐𝟎𝟐𝟒	NUM
gojar-5993	58	7	)	)	PUNCT
gojar-5993	58	8	.	.	PUNCT
gojar-5993	59	1	global	global	ADJ
gojar-5993	59	2	online	online	PROPN
gojar-5993	59	3	journal	journal	PROPN
gojar-5993	59	4	of	of	ADP
gojar-5993	59	5	academic	academic	ADJ
gojar-5993	59	6	research	research	NOUN
gojar-5993	59	7	(	(	PUNCT
gojar-5993	59	8	gojar	gojar	NOUN
gojar-5993	59	9	)	)	PUNCT
gojar-5993	59	10	,	,	PUNCT
gojar-5993	59	11	vol	vol	NOUN
gojar-5993	59	12	.	.	PROPN
gojar-5993	59	13	4	4	NUM
gojar-5993	59	14	,	,	PUNCT
gojar-5993	59	15	no	no	INTJ
gojar-5993	59	16	.	.	NOUN
gojar-5993	59	17	1	1	NUM
gojar-5993	59	18	february	february	NOUN
gojar-5993	59	19	2025	2025	NUM
gojar-5993	59	20	64	64	NUM
gojar-5993	59	21	definition	definition	NOUN
gojar-5993	59	22	2.4	2.4	NUM
gojar-5993	59	23	:	:	PUNCT
gojar-5993	59	24	integer	integer	NOUN
gojar-5993	59	25	programming	programming	NOUN
gojar-5993	59	26	problem	problem	NOUN
gojar-5993	59	27	(	(	PUNCT
gojar-5993	59	28	ipp	ipp	PROPN
gojar-5993	59	29	)	)	PUNCT
gojar-5993	59	30	the	the	DET
gojar-5993	59	31	ipp	ipp	PROPN
gojar-5993	59	32	is	be	AUX
gojar-5993	59	33	a	a	DET
gojar-5993	59	34	special	special	ADJ
gojar-5993	59	35	class	class	NOUN
gojar-5993	59	36	of	of	ADP
gojar-5993	59	37	linear	linear	PROPN
gojar-5993	59	38	programming	programming	NOUN
gojar-5993	59	39	problem	problem	NOUN
gojar-5993	59	40	(	(	PUNCT
gojar-5993	59	41	lpp	lpp	PROPN
gojar-5993	59	42	)	)	PUNCT
gojar-5993	59	43	where	where	SCONJ
gojar-5993	59	44	all	all	PRON
gojar-5993	59	45	or	or	CCONJ
gojar-5993	59	46	some	some	PRON
gojar-5993	59	47	of	of	ADP
gojar-5993	59	48	the	the	DET
gojar-5993	59	49	variables	variable	NOUN
gojar-5993	59	50	in	in	ADP
gojar-5993	59	51	the	the	DET
gojar-5993	59	52	optimal	optimal	ADJ
gojar-5993	59	53	solution	solution	NOUN
gojar-5993	59	54	are	be	AUX
gojar-5993	59	55	restricted	restrict	VERB
gojar-5993	59	56	to	to	PART
gojar-5993	59	57	assume	assume	VERB
gojar-5993	59	58	nonnegative	nonnegative	ADJ
gojar-5993	59	59	-	-	PUNCT
gojar-5993	59	60	integer	integer	NOUN
gojar-5993	59	61	values	value	NOUN
gojar-5993	59	62	.	.	PUNCT
gojar-5993	60	1	thus	thus	ADV
gojar-5993	60	2	the	the	DET
gojar-5993	60	3	general	general	ADJ
gojar-5993	60	4	ipp	ipp	PROPN
gojar-5993	60	5	can	can	AUX
gojar-5993	60	6	be	be	AUX
gojar-5993	60	7	stated	state	VERB
gojar-5993	60	8	as	as	SCONJ
gojar-5993	60	9	follows	follow	VERB
gojar-5993	60	10	:	:	PUNCT
gojar-5993	60	11	optimize	optimize	VERB
gojar-5993	60	12	the	the	DET
gojar-5993	60	13	linear	linear	ADJ
gojar-5993	60	14	function	function	NOUN
gojar-5993	60	15	𝐎𝐩𝐭𝐢𝐦𝐢𝐳𝐞	𝐎𝐩𝐭𝐢𝐦𝐢𝐳𝐞	PROPN
gojar-5993	60	16	𝒁	𝒁	PROPN
gojar-5993	60	17	=	=	PUNCT
gojar-5993	60	18	∑	∑	PUNCT
gojar-5993	60	19	𝒄𝒊𝒙𝒊	𝒄𝒊𝒙𝒊	VERB
gojar-5993	60	20	𝒏	𝒏	PROPN
gojar-5993	60	21	𝒊=𝟏	𝒊=𝟏	NOUN
gojar-5993	60	22	…	…	PUNCT
gojar-5993	60	23	(	(	PUNCT
gojar-5993	60	24	𝟏	𝟏	X
gojar-5993	60	25	)	)	PUNCT
gojar-5993	60	26	subject	subject	NOUN
gojar-5993	60	27	to	to	ADP
gojar-5993	60	28	the	the	DET
gojar-5993	60	29	constraints	constraint	NOUN
gojar-5993	60	30	.	.	PUNCT
gojar-5993	61	1	∑	∑	ADV
gojar-5993	61	2	𝒂𝒊𝒋𝒙𝒊	𝒂𝒊𝒋𝒙𝒊	PROPN
gojar-5993	61	3	𝒏	𝒏	PROPN
gojar-5993	61	4	𝒊=𝟏	𝒊=𝟏	NOUN
gojar-5993	61	5	≤	≤	ADV
gojar-5993	61	6	𝒃𝒊	𝒃𝒊	ADP
gojar-5993	61	7	,	,	PUNCT
gojar-5993	61	8	𝒋	𝒋	X
gojar-5993	61	9	=	=	SYM
gojar-5993	61	10	𝟏	𝟏	PROPN
gojar-5993	61	11	,	,	PUNCT
gojar-5993	61	12	𝟐	𝟐	NUM
gojar-5993	61	13	,	,	PUNCT
gojar-5993	61	14	…	…	PUNCT
gojar-5993	61	15	,	,	PUNCT
gojar-5993	61	16	𝒎	𝒎	X
gojar-5993	61	17	…	…	PUNCT
gojar-5993	61	18	(	(	PUNCT
gojar-5993	61	19	𝟐	𝟐	X
gojar-5993	61	20	)	)	PUNCT
gojar-5993	61	21	𝒙𝒊	𝒙𝒊	PROPN
gojar-5993	61	22	≥	≥	NUM
gojar-5993	61	23	𝟎	𝟎	NUM
gojar-5993	61	24	and	and	CCONJ
gojar-5993	61	25	some	some	DET
gojar-5993	61	26	𝒙𝒊	𝒙𝒊	PROPN
gojar-5993	61	27	are	be	AUX
gojar-5993	61	28	integers	integer	NOUN
gojar-5993	61	29	.	.	PUNCT
gojar-5993	62	1	there	there	PRON
gojar-5993	62	2	are	be	VERB
gojar-5993	62	3	two	two	NUM
gojar-5993	62	4	types	type	NOUN
gojar-5993	62	5	of	of	ADP
gojar-5993	62	6	the	the	DET
gojar-5993	62	7	integer	integer	NOUN
gojar-5993	62	8	programming	programming	NOUN
gojar-5993	62	9	problems	problem	NOUN
gojar-5993	62	10	(	(	PUNCT
gojar-5993	62	11	elmuti	elmuti	PROPN
gojar-5993	62	12	,	,	PUNCT
gojar-5993	62	13	2003	2003	NUM
gojar-5993	62	14	;	;	PUNCT
gojar-5993	62	15	genova	genova	PROPN
gojar-5993	62	16	and	and	CCONJ
gojar-5993	62	17	guliashki	guliashki	PROPN
gojar-5993	62	18	,	,	PUNCT
gojar-5993	62	19	2011	2011	NUM
gojar-5993	62	20	)	)	PUNCT
gojar-5993	62	21	.	.	PUNCT
gojar-5993	63	1	definition	definition	NOUN
gojar-5993	63	2	2.5	2.5	NUM
gojar-5993	63	3	:	:	PUNCT
gojar-5993	63	4	all	all	DET
gojar-5993	63	5	integer	integer	NOUN
gojar-5993	63	6	programming	programming	NOUN
gojar-5993	63	7	problem	problem	NOUN
gojar-5993	63	8	an	an	DET
gojar-5993	63	9	ipp	ipp	PROPN
gojar-5993	63	10	.	.	PUNCT
gojar-5993	63	11	is	be	AUX
gojar-5993	63	12	termed	term	VERB
gojar-5993	63	13	as	as	ADP
gojar-5993	63	14	all	all	DET
gojar-5993	63	15	ipp	ipp	PROPN
gojar-5993	63	16	or	or	CCONJ
gojar-5993	63	17	pure	pure	ADJ
gojar-5993	63	18	ipp	ipp	PROPN
gojar-5993	63	19	if	if	SCONJ
gojar-5993	63	20	all	all	DET
gojar-5993	63	21	the	the	DET
gojar-5993	63	22	variables	variable	NOUN
gojar-5993	63	23	in	in	ADP
gojar-5993	63	24	the	the	DET
gojar-5993	63	25	optimal	optimal	ADJ
gojar-5993	63	26	solution	solution	NOUN
gojar-5993	63	27	are	be	AUX
gojar-5993	63	28	restricted	restrict	VERB
gojar-5993	63	29	to	to	PART
gojar-5993	63	30	assume	assume	VERB
gojar-5993	63	31	non	non	ADJ
gojar-5993	63	32	-	-	ADJ
gojar-5993	63	33	negative	negative	ADJ
gojar-5993	63	34	integer	integer	NOUN
gojar-5993	63	35	values	value	NOUN
gojar-5993	63	36	.	.	PUNCT
gojar-5993	64	1	definition	definition	NOUN
gojar-5993	64	2	2.6	2.6	NUM
gojar-5993	64	3	:	:	PUNCT
gojar-5993	64	4	mixed	mixed	ADJ
gojar-5993	64	5	integer	integer	NOUN
gojar-5993	64	6	programming	programming	NOUN
gojar-5993	64	7	problem	problem	NOUN
gojar-5993	64	8	(	(	PUNCT
gojar-5993	64	9	mipp	mipp	PROPN
gojar-5993	64	10	)	)	PUNCT
gojar-5993	64	11	an	an	DET
gojar-5993	64	12	ipp	ipp	NOUN
gojar-5993	64	13	is	be	AUX
gojar-5993	64	14	termed	term	VERB
gojar-5993	64	15	as	as	ADP
gojar-5993	64	16	mixed	mixed	ADJ
gojar-5993	64	17	mipp	mipp	NOUN
gojar-5993	64	18	if	if	SCONJ
gojar-5993	64	19	only	only	ADV
gojar-5993	64	20	some	some	DET
gojar-5993	64	21	variables	variable	NOUN
gojar-5993	64	22	in	in	ADP
gojar-5993	64	23	the	the	DET
gojar-5993	64	24	optimal	optimal	ADJ
gojar-5993	64	25	solution	solution	NOUN
gojar-5993	64	26	are	be	AUX
gojar-5993	64	27	restricted	restrict	VERB
gojar-5993	64	28	to	to	PART
gojar-5993	64	29	assume	assume	VERB
gojar-5993	64	30	non	non	ADJ
gojar-5993	64	31	-	-	ADJ
gojar-5993	64	32	negative	negative	ADJ
gojar-5993	64	33	integer	integer	NOUN
gojar-5993	64	34	values	value	NOUN
gojar-5993	64	35	while	while	SCONJ
gojar-5993	64	36	the	the	DET
gojar-5993	64	37	remaining	remain	VERB
gojar-5993	64	38	variables	variable	NOUN
gojar-5993	64	39	are	be	AUX
gojar-5993	64	40	free	free	ADJ
gojar-5993	64	41	to	to	PART
gojar-5993	64	42	take	take	VERB
gojar-5993	64	43	any	any	DET
gojar-5993	64	44	non	non	ADJ
gojar-5993	64	45	-	-	ADJ
gojar-5993	64	46	negative	negative	ADJ
gojar-5993	64	47	values	value	NOUN
gojar-5993	64	48	(	(	PUNCT
gojar-5993	64	49	gupta	gupta	PROPN
gojar-5993	64	50	et	et	PROPN
gojar-5993	64	51	al	al	PROPN
gojar-5993	64	52	.	.	PROPN
gojar-5993	64	53	,	,	PUNCT
gojar-5993	64	54	2014	2014	NUM
gojar-5993	64	55	)	)	PUNCT
gojar-5993	64	56	.	.	PUNCT
gojar-5993	65	1	importance	importance	NOUN
gojar-5993	65	2	of	of	ADP
gojar-5993	65	3	ipp	ipp	PROPN
gojar-5993	65	4	quite	quite	ADV
gojar-5993	65	5	often	often	ADV
gojar-5993	65	6	,	,	PUNCT
gojar-5993	65	7	in	in	ADP
gojar-5993	65	8	business	business	NOUN
gojar-5993	65	9	and	and	CCONJ
gojar-5993	65	10	industry	industry	NOUN
gojar-5993	66	1	,	,	PUNCT
gojar-5993	66	2	we	we	PRON
gojar-5993	66	3	require	require	VERB
gojar-5993	66	4	the	the	DET
gojar-5993	66	5	discrete	discrete	ADJ
gojar-5993	66	6	nature	nature	NOUN
gojar-5993	66	7	or	or	CCONJ
gojar-5993	66	8	values	value	NOUN
gojar-5993	66	9	of	of	ADP
gojar-5993	66	10	the	the	DET
gojar-5993	66	11	variables	variable	NOUN
gojar-5993	66	12	involved	involve	VERB
gojar-5993	66	13	in	in	ADP
gojar-5993	66	14	many	many	ADJ
gojar-5993	66	15	decision	decision	NOUN
gojar-5993	66	16	making	make	VERB
gojar-5993	66	17	situations	situation	NOUN
gojar-5993	66	18	.	.	PUNCT
gojar-5993	67	1	for	for	ADP
gojar-5993	67	2	example	example	NOUN
gojar-5993	67	3	,	,	PUNCT
gojar-5993	67	4	in	in	ADP
gojar-5993	67	5	a	a	DET
gojar-5993	67	6	factory	factory	NOUN
gojar-5993	67	7	manufacturing	manufacture	VERB
gojar-5993	67	8	trucks	truck	NOUN
gojar-5993	67	9	or	or	CCONJ
gojar-5993	67	10	cars	car	NOUN
gojar-5993	67	11	etc	etc	X
gojar-5993	67	12	.	.	X
gojar-5993	68	1	the	the	DET
gojar-5993	68	2	quantity	quantity	NOUN
gojar-5993	68	3	or	or	CCONJ
gojar-5993	68	4	number	number	NOUN
gojar-5993	68	5	manufactured	manufacture	VERB
gojar-5993	68	6	can	can	AUX
gojar-5993	68	7	be	be	AUX
gojar-5993	68	8	a	a	DET
gojar-5993	68	9	whole	whole	ADJ
gojar-5993	68	10	discrete	discrete	ADJ
gojar-5993	68	11	number	number	NOUN
gojar-5993	68	12	only	only	ADV
gojar-5993	68	13	as	as	ADP
gojar-5993	68	14	a	a	DET
gojar-5993	68	15	fraction	fraction	NOUN
gojar-5993	68	16	of	of	ADP
gojar-5993	68	17	truck	truck	NOUN
gojar-5993	68	18	or	or	CCONJ
gojar-5993	68	19	car	car	NOUN
gojar-5993	68	20	is	be	AUX
gojar-5993	68	21	not	not	PART
gojar-5993	68	22	required	require	VERB
gojar-5993	68	23	.	.	PUNCT
gojar-5993	69	1	in	in	ADP
gojar-5993	69	2	assignment	assignment	NOUN
gojar-5993	69	3	problems	problem	NOUN
gojar-5993	69	4	and	and	CCONJ
gojar-5993	69	5	travelling	travel	VERB
gojar-5993	69	6	salesman	salesman	NOUN
gojar-5993	69	7	problems	problem	NOUN
gojar-5993	69	8	etc	etc	X
gojar-5993	69	9	.	.	PUNCT
gojar-5993	70	1	the	the	DET
gojar-5993	70	2	variables	variable	NOUN
gojar-5993	70	3	involved	involve	VERB
gojar-5993	70	4	can	can	AUX
gojar-5993	70	5	assume	assume	VERB
gojar-5993	70	6	integer	integer	NOUN
gojar-5993	70	7	values	value	NOUN
gojar-5993	70	8	only	only	ADV
gojar-5993	70	9	.	.	PUNCT
gojar-5993	71	1	in	in	ADP
gojar-5993	71	2	allocation	allocation	NOUN
gojar-5993	71	3	of	of	ADP
gojar-5993	71	4	goods	good	NOUN
gojar-5993	71	5	,	,	PUNCT
gojar-5993	71	6	a	a	DET
gojar-5993	71	7	shipment	shipment	NOUN
gojar-5993	71	8	must	must	AUX
gojar-5993	71	9	involve	involve	VERB
gojar-5993	71	10	a	a	DET
gojar-5993	71	11	discrete	discrete	ADJ
gojar-5993	71	12	number	number	NOUN
gojar-5993	71	13	of	of	ADP
gojar-5993	71	14	trucks	truck	NOUN
gojar-5993	71	15	etc	etc	X
gojar-5993	71	16	.	.	X
gojar-5993	71	17	in	in	ADP
gojar-5993	71	18	sequencing	sequence	VERB
gojar-5993	71	19	and	and	CCONJ
gojar-5993	71	20	routing	routing	NOUN
gojar-5993	71	21	decisions	decision	NOUN
gojar-5993	71	22	we	we	PRON
gojar-5993	71	23	require	require	VERB
gojar-5993	71	24	the	the	DET
gojar-5993	71	25	discrete	discrete	ADJ
gojar-5993	71	26	values	value	NOUN
gojar-5993	71	27	of	of	ADP
gojar-5993	71	28	variables	variable	NOUN
gojar-5993	71	29	.	.	PUNCT
gojar-5993	72	1	thus	thus	ADV
gojar-5993	72	2	we	we	PRON
gojar-5993	72	3	come	come	VERB
gojar-5993	72	4	across	across	ADP
gojar-5993	72	5	many	many	ADJ
gojar-5993	72	6	integer	integer	NOUN
gojar-5993	72	7	programming	programming	NOUN
gojar-5993	72	8	problems	problem	NOUN
gojar-5993	72	9	and	and	CCONJ
gojar-5993	72	10	hence	hence	ADV
gojar-5993	72	11	need	need	VERB
gojar-5993	72	12	some	some	DET
gojar-5993	72	13	systematic	systematic	ADJ
gojar-5993	72	14	procedure	procedure	NOUN
gojar-5993	72	15	for	for	ADP
gojar-5993	72	16	obtaining	obtain	VERB
gojar-5993	72	17	the	the	DET
gojar-5993	72	18	exact	exact	ADJ
gojar-5993	72	19	optimal	optimal	ADJ
gojar-5993	72	20	integer	integer	NOUN
gojar-5993	72	21	solution	solution	NOUN
gojar-5993	72	22	to	to	ADP
gojar-5993	72	23	such	such	ADJ
gojar-5993	72	24	problems	problem	NOUN
gojar-5993	72	25	(	(	PUNCT
gojar-5993	72	26	elmuti	elmuti	PROPN
gojar-5993	72	27	,	,	PUNCT
gojar-5993	72	28	2003	2003	NUM
gojar-5993	72	29	;	;	PUNCT
gojar-5993	72	30	genova	genova	PROPN
gojar-5993	72	31	and	and	CCONJ
gojar-5993	72	32	guliashki	guliashki	PROPN
gojar-5993	72	33	,	,	PUNCT
gojar-5993	72	34	2011	2011	NUM
gojar-5993	72	35	)	)	PUNCT
gojar-5993	72	36	.	.	PUNCT
gojar-5993	73	1	global	global	ADJ
gojar-5993	73	2	online	online	PROPN
gojar-5993	73	3	journal	journal	PROPN
gojar-5993	73	4	of	of	ADP
gojar-5993	73	5	academic	academic	ADJ
gojar-5993	73	6	research	research	NOUN
gojar-5993	73	7	(	(	PUNCT
gojar-5993	73	8	gojar	gojar	NOUN
gojar-5993	73	9	)	)	PUNCT
gojar-5993	73	10	,	,	PUNCT
gojar-5993	73	11	vol	vol	NOUN
gojar-5993	73	12	.	.	PROPN
gojar-5993	73	13	4	4	NUM
gojar-5993	73	14	,	,	PUNCT
gojar-5993	73	15	no	no	INTJ
gojar-5993	73	16	.	.	NOUN
gojar-5993	73	17	1	1	NUM
gojar-5993	73	18	february	february	NOUN
gojar-5993	73	19	2025	2025	NUM
gojar-5993	73	20	65	65	NUM
gojar-5993	73	21	iii	iii	NOUN
gojar-5993	73	22	.	.	PUNCT
gojar-5993	73	23	main	main	ADJ
gojar-5993	73	24	results	result	NOUN
gojar-5993	73	25	lemma	lemma	PROPN
gojar-5993	73	26	:	:	PUNCT
gojar-5993	73	27	(	(	PUNCT
gojar-5993	73	28	boundedness	boundedness	NOUN
gojar-5993	73	29	equivalence	equivalence	NOUN
gojar-5993	73	30	)	)	PUNCT
gojar-5993	73	31	let	let	VERB
gojar-5993	73	32	𝑷	𝑷	PROPN
gojar-5993	73	33	=	=	PRON
gojar-5993	73	34	{	{	PUNCT
gojar-5993	73	35	𝒙	𝒙	NOUN
gojar-5993	73	36	:	:	PUNCT
gojar-5993	73	37	𝑨𝒙	𝑨𝒙	PROPN
gojar-5993	73	38	≤	≤	NOUN
gojar-5993	73	39	𝒃	𝒃	AUX
gojar-5993	73	40	}	}	PUNCT
gojar-5993	73	41	be	be	AUX
gojar-5993	73	42	some	some	DET
gojar-5993	73	43	rational	rational	ADJ
gojar-5993	73	44	polyhedron	polyhedron	NOUN
gojar-5993	73	45	whose	whose	DET
gojar-5993	73	46	integer	integer	NOUN
gojar-5993	73	47	hull	hull	NOUN
gojar-5993	73	48	is	be	AUX
gojar-5993	73	49	nonempty	nonempty	ADJ
gojar-5993	73	50	,	,	PUNCT
gojar-5993	73	51	and	and	CCONJ
gojar-5993	73	52	let	let	VERB
gojar-5993	73	53	𝒄	𝒄	NOUN
gojar-5993	73	54	be	be	AUX
gojar-5993	73	55	some	some	DET
gojar-5993	73	56	vector	vector	NOUN
gojar-5993	73	57	(	(	PUNCT
gojar-5993	73	58	not	not	PART
gojar-5993	73	59	necessarily	necessarily	ADV
gojar-5993	73	60	rational	rational	ADJ
gojar-5993	73	61	)	)	PUNCT
gojar-5993	73	62	.	.	PUNCT
gojar-5993	74	1	then	then	ADV
gojar-5993	74	2	𝐦𝐚𝐱	𝐦𝐚𝐱	PROPN
gojar-5993	74	3	{	{	PUNCT
gojar-5993	74	4	𝒄𝒙	𝒄𝒙	PROPN
gojar-5993	74	5	:	:	PUNCT
gojar-5993	74	6	𝒙	𝒙	PROPN
gojar-5993	74	7	∈	∈	PROPN
gojar-5993	74	8	𝑷	𝑷	PROPN
gojar-5993	74	9	}	}	PUNCT
gojar-5993	74	10	is	be	AUX
gojar-5993	74	11	bounded	bound	VERB
gojar-5993	74	12	if	if	SCONJ
gojar-5993	74	13	and	and	CCONJ
gojar-5993	74	14	only	only	ADV
gojar-5993	74	15	if	if	SCONJ
gojar-5993	74	16	𝐦𝐚𝐱	𝐦𝐚𝐱	PROPN
gojar-5993	74	17	{	{	PUNCT
gojar-5993	74	18	𝒄𝒙	𝒄𝒙	PROPN
gojar-5993	74	19	:	:	PUNCT
gojar-5993	74	20	𝒙	𝒙	PROPN
gojar-5993	74	21	∈	∈	PROPN
gojar-5993	74	22	𝑷𝟏	𝑷𝟏	NOUN
gojar-5993	74	23	}	}	PUNCT
gojar-5993	74	24	is	be	AUX
gojar-5993	74	25	bounded	bound	VERB
gojar-5993	74	26	.	.	PUNCT
gojar-5993	75	1	proof	proof	NOUN
gojar-5993	75	2	:	:	PUNCT
gojar-5993	75	3	suppose	suppose	VERB
gojar-5993	75	4	𝐦𝐚𝐱	𝐦𝐚𝐱	PROPN
gojar-5993	75	5	{	{	PUNCT
gojar-5993	75	6	𝒄𝒙	𝒄𝒙	PROPN
gojar-5993	75	7	:	:	PUNCT
gojar-5993	75	8	𝒙	𝒙	PROPN
gojar-5993	75	9	∈	∈	PROPN
gojar-5993	75	10	𝑷}is	𝑷}is	PROPN
gojar-5993	75	11	unbounded	unbounde	VERB
gojar-5993	75	12	.	.	PUNCT
gojar-5993	76	1	then	then	ADV
gojar-5993	76	2	corollary	corollary	ADJ
gojar-5993	76	3	3.2.8	3.2.8	NUM
gojar-5993	76	4	says	say	VERB
gojar-5993	76	5	that	that	SCONJ
gojar-5993	76	6	the	the	DET
gojar-5993	76	7	system	system	NOUN
gojar-5993	76	8	𝒚𝑨	𝒚𝑨	NOUN
gojar-5993	76	9	=	=	PROPN
gojar-5993	76	10	𝒄.	𝒄.	PROPN
gojar-5993	76	11	𝒚	𝒚	PROPN
gojar-5993	76	12	≥	≥	X
gojar-5993	76	13	𝟎	𝟎	NUM
gojar-5993	76	14	has	have	VERB
gojar-5993	76	15	no	no	DET
gojar-5993	76	16	solution	solution	NOUN
gojar-5993	76	17	.	.	PUNCT
gojar-5993	77	1	by	by	ADP
gojar-5993	77	2	corollary	corollary	ADJ
gojar-5993	77	3	3.26	3.26	NUM
gojar-5993	77	4	there	there	PRON
gojar-5993	77	5	is	be	VERB
gojar-5993	77	6	a	a	DET
gojar-5993	77	7	vector	vector	NOUN
gojar-5993	77	8	𝒛.	𝒛.	NOUN
gojar-5993	77	9	with	with	ADP
gojar-5993	77	10	𝒆𝒛	𝒆𝒛	PROPN
gojar-5993	77	11	<	<	X
gojar-5993	77	12	0	0	PROPN
gojar-5993	77	13	and	and	CCONJ
gojar-5993	77	14	𝑨𝒛	𝑨𝒛	PROPN
gojar-5993	77	15	≥	≥	NOUN
gojar-5993	77	16	𝟎.	𝟎.	PUNCT
gojar-5993	77	17	then	then	ADV
gojar-5993	77	18	the	the	DET
gojar-5993	77	19	𝑳𝑷	𝑳𝑷	PROPN
gojar-5993	77	20	𝐦𝐢𝐧	𝐦𝐢𝐧	NOUN
gojar-5993	77	21	{	{	PUNCT
gojar-5993	77	22	𝒄𝒛	𝒄𝒛	ADP
gojar-5993	77	23	:	:	PUNCT
gojar-5993	77	24	𝑨𝒛	𝑨𝒛	PROPN
gojar-5993	77	25	≥	≥	NOUN
gojar-5993	77	26	𝟎	𝟎	NUM
gojar-5993	77	27	,	,	PUNCT
gojar-5993	77	28	−∥≤	−∥≤	PROPN
gojar-5993	77	29	𝒛	𝒛	NUM
gojar-5993	77	30	≤∥	≤∥	PUNCT
gojar-5993	77	31	}	}	PUNCT
gojar-5993	77	32	is	be	AUX
gojar-5993	77	33	feasible	feasible	ADJ
gojar-5993	77	34	.	.	PUNCT
gojar-5993	78	1	let	let	VERB
gojar-5993	78	2	𝒛∗	𝒛∗	PROPN
gojar-5993	78	3	be	be	AUX
gojar-5993	78	4	an	an	DET
gojar-5993	78	5	optimum	optimum	ADJ
gojar-5993	78	6	basic	basic	ADJ
gojar-5993	78	7	solution	solution	NOUN
gojar-5993	78	8	of	of	ADP
gojar-5993	78	9	this	this	DET
gojar-5993	78	10	𝑳𝑷.	𝑳𝑷.	NOUN
gojar-5993	78	11	𝒛∗	𝒛∗	PROPN
gojar-5993	78	12	is	be	AUX
gojar-5993	78	13	rational	rational	ADJ
gojar-5993	78	14	as	as	SCONJ
gojar-5993	78	15	it	it	PRON
gojar-5993	78	16	is	be	AUX
gojar-5993	78	17	a	a	DET
gojar-5993	78	18	vertex	vertex	NOUN
gojar-5993	78	19	of	of	ADP
gojar-5993	78	20	a	a	DET
gojar-5993	78	21	rational	rational	ADJ
gojar-5993	78	22	polytope	polytope	NOUN
gojar-5993	78	23	.	.	PUNCT
gojar-5993	79	1	multiply	multiply	VERB
gojar-5993	79	2	𝒛∗	𝒛∗	PROPN
gojar-5993	79	3	by	by	ADP
gojar-5993	79	4	a	a	DET
gojar-5993	79	5	suitable	suitable	ADJ
gojar-5993	79	6	natural	natural	ADJ
gojar-5993	79	7	number	number	NOUN
gojar-5993	79	8	to	to	PART
gojar-5993	79	9	obtain	obtain	VERB
gojar-5993	79	10	an	an	DET
gojar-5993	79	11	integral	integral	ADJ
gojar-5993	79	12	vector	vector	NOUN
gojar-5993	79	13	𝝎	𝝎	NOUN
gojar-5993	79	14	with	with	ADP
gojar-5993	79	15	𝑨𝝎	𝑨𝝎	PROPN
gojar-5993	79	16	≥	≥	NOUN
gojar-5993	79	17	𝟎	𝟎	NUM
gojar-5993	79	18	and	and	CCONJ
gojar-5993	79	19	𝒄𝝎	𝒄𝝎	ADV
gojar-5993	79	20	<	<	X
gojar-5993	79	21	0	0	X
gojar-5993	79	22	.	.	PUNCT
gojar-5993	80	1	let	let	VERB
gojar-5993	80	2	𝒗	𝒗	PROPN
gojar-5993	80	3	∈	∈	PROPN
gojar-5993	80	4	𝑷𝟏	𝑷𝟏	NOUN
gojar-5993	80	5	be	be	AUX
gojar-5993	80	6	some	some	DET
gojar-5993	80	7	integer	integer	NOUN
gojar-5993	80	8	vector	vector	NOUN
gojar-5993	80	9	.	.	PUNCT
gojar-5993	81	1	then	then	ADV
gojar-5993	81	2	𝒗	𝒗	PROPN
gojar-5993	81	3	−	−	NOUN
gojar-5993	81	4	𝒌𝝎	𝒌𝝎	ADP
gojar-5993	81	5	∈	∈	PROPN
gojar-5993	81	6	𝑷𝟏	𝑷𝟏	NOUN
gojar-5993	81	7	for	for	ADP
gojar-5993	81	8	all	all	DET
gojar-5993	81	9	𝒌	𝒌	PROPN
gojar-5993	81	10	∈	∈	PROPN
gojar-5993	81	11	ℕ	ℕ	PROPN
gojar-5993	81	12	,	,	PUNCT
gojar-5993	81	13	and	and	CCONJ
gojar-5993	81	14	thus	thus	ADV
gojar-5993	81	15	𝐦𝐚𝐱	𝐦𝐚𝐱	NOUN
gojar-5993	81	16	{	{	PUNCT
gojar-5993	81	17	𝒄𝒙	𝒄𝒙	PROPN
gojar-5993	81	18	:	:	PUNCT
gojar-5993	81	19	𝒙	𝒙	PROPN
gojar-5993	81	20	∈	∈	PROPN
gojar-5993	81	21	𝑷𝟏	𝑷𝟏	NOUN
gojar-5993	81	22	}	}	PUNCT
gojar-5993	81	23	is	be	AUX
gojar-5993	81	24	unbounded	unbounde	VERB
gojar-5993	81	25	.	.	PUNCT
gojar-5993	82	1	the	the	DET
gojar-5993	82	2	other	other	ADJ
gojar-5993	82	3	direction	direction	NOUN
gojar-5993	82	4	is	be	AUX
gojar-5993	82	5	trivial	trivial	ADJ
gojar-5993	82	6	.	.	PUNCT
gojar-5993	83	1	theorem	theorem	ADJ
gojar-5993	83	2	(	(	PUNCT
gojar-5993	83	3	rational	rational	ADJ
gojar-5993	83	4	matrices	matrix	NOUN
gojar-5993	83	5	and	and	CCONJ
gojar-5993	83	6	vertices	vertex	NOUN
gojar-5993	83	7	of	of	ADP
gojar-5993	83	8	polytopes	polytope	NOUN
gojar-5993	83	9	)	)	PUNCT
gojar-5993	83	10	consider	consider	VERB
gojar-5993	83	11	the	the	DET
gojar-5993	83	12	rational	rational	ADJ
gojar-5993	83	13	linear	linear	NOUN
gojar-5993	83	14	programming	programming	NOUN
gojar-5993	83	15	(	(	PUNCT
gojar-5993	83	16	lp	lp	NOUN
gojar-5993	83	17	)	)	PUNCT
gojar-5993	83	18	problem	problem	NOUN
gojar-5993	83	19	:	:	PUNCT
gojar-5993	83	20	𝑳𝑷	𝑳𝑷	PROPN
gojar-5993	83	21	:	:	PUNCT
gojar-5993	83	22	𝒎𝒂𝒙	𝒎𝒂𝒙	NOUN
gojar-5993	83	23	{	{	PUNCT
gojar-5993	83	24	𝒄𝑻𝒙	𝒄𝑻𝒙	NOUN
gojar-5993	83	25	:	:	PUNCT
gojar-5993	83	26	𝑨𝒙	𝑨𝒙	PROPN
gojar-5993	83	27	≤	≤	NOUN
gojar-5993	83	28	𝒃	𝒃	ADP
gojar-5993	83	29	}	}	PUNCT
gojar-5993	83	30	where	where	SCONJ
gojar-5993	83	31	a	a	PRON
gojar-5993	83	32	and	and	CCONJ
gojar-5993	83	33	b	b	NOUN
gojar-5993	83	34	are	be	AUX
gojar-5993	83	35	rational	rational	ADJ
gojar-5993	83	36	.	.	PUNCT
gojar-5993	84	1	suppose	suppose	VERB
gojar-5993	84	2	this	this	DET
gojar-5993	84	3	lp	lp	NOUN
gojar-5993	84	4	has	have	VERB
gojar-5993	84	5	an	an	DET
gojar-5993	84	6	optimum	optimum	ADJ
gojar-5993	84	7	solution	solution	NOUN
gojar-5993	84	8	.	.	PUNCT
gojar-5993	85	1	then	then	ADV
gojar-5993	85	2	the	the	DET
gojar-5993	85	3	following	follow	VERB
gojar-5993	85	4	hold	hold	NOUN
gojar-5993	85	5	:	:	PUNCT
gojar-5993	85	6	(	(	PUNCT
gojar-5993	85	7	i	i	NOUN
gojar-5993	85	8	)	)	PUNCT
gojar-5993	85	9	bounded	bound	VERB
gojar-5993	85	10	size	size	NOUN
gojar-5993	85	11	solution	solution	NOUN
gojar-5993	85	12	:	:	PUNCT
gojar-5993	85	13	there	there	PRON
gojar-5993	85	14	exists	exist	VERB
gojar-5993	85	15	an	an	DET
gojar-5993	85	16	optimum	optimum	ADJ
gojar-5993	85	17	solution	solution	NOUN
gojar-5993	85	18	x	x	PUNCT
gojar-5993	85	19	such	such	ADJ
gojar-5993	85	20	that	that	SCONJ
gojar-5993	85	21	:	:	PUNCT
gojar-5993	85	22	𝒔𝒊𝒛𝒆(𝒙	𝒔𝒊𝒛𝒆(𝒙	X
gojar-5993	85	23	)	)	PUNCT
gojar-5993	85	24	≤	≤	NUM
gojar-5993	85	25	𝟒𝒏(𝒔𝒊𝒛𝒆(𝑨	𝟒𝒏(𝒔𝒊𝒛𝒆(𝑨	NOUN
gojar-5993	85	26	)	)	PUNCT
gojar-5993	85	27	+	+	CCONJ
gojar-5993	86	1	𝒔𝒊𝒛𝒆(𝒃	𝒔𝒊𝒛𝒆(𝒃	NOUN
gojar-5993	86	2	)	)	PUNCT
gojar-5993	86	3	)	)	PUNCT
gojar-5993	86	4	(	(	PUNCT
gojar-5993	86	5	ii	ii	NOUN
gojar-5993	86	6	)	)	PUNCT
gojar-5993	86	7	special	special	ADJ
gojar-5993	86	8	case	case	NOUN
gojar-5993	86	9	(	(	PUNCT
gojar-5993	86	10	unit	unit	NOUN
gojar-5993	86	11	vector	vector	NOUN
gojar-5993	86	12	b	b	PROPN
gojar-5993	86	13	):	):	PUNCT
gojar-5993	86	14	if	if	SCONJ
gojar-5993	86	15	𝒃	𝒃	NOUN
gojar-5993	86	16	=	=	SYM
gojar-5993	86	17	𝒆𝒊	𝒆𝒊	NOUN
gojar-5993	86	18	𝒐𝒓	𝒐𝒓	NOUN
gojar-5993	86	19	𝒃	𝒃	NOUN
gojar-5993	86	20	=	=	VERB
gojar-5993	86	21	−𝒆𝒊	−𝒆𝒊	NOUN
gojar-5993	86	22	for	for	ADP
gojar-5993	86	23	some	some	DET
gojar-5993	86	24	unit	unit	NOUN
gojar-5993	86	25	vector	vector	NOUN
gojar-5993	86	26	𝒆𝒊	𝒆𝒊	PROPN
gojar-5993	86	27	there	there	ADV
gojar-5993	86	28	exists	exist	VERB
gojar-5993	86	29	a	a	DET
gojar-5993	86	30	nonsingular	nonsingular	ADJ
gojar-5993	86	31	submatrix	submatrix	NOUN
gojar-5993	86	32	𝑨′	𝑨′	NOUN
gojar-5993	86	33	of	of	ADP
gojar-5993	86	34	a	a	PRON
gojar-5993	86	35	and	and	CCONJ
gojar-5993	86	36	an	an	DET
gojar-5993	86	37	optimum	optimum	ADJ
gojar-5993	86	38	solution	solution	NOUN
gojar-5993	86	39	x	x	PUNCT
gojar-5993	86	40	such	such	ADJ
gojar-5993	86	41	that	that	SCONJ
gojar-5993	86	42	:	:	PUNCT
gojar-5993	86	43	𝒔𝒊𝒛𝒆(𝒙	𝒔𝒊𝒛𝒆(𝒙	X
gojar-5993	86	44	)	)	PUNCT
gojar-5993	86	45	≤	≤	NUM
gojar-5993	86	46	𝟒𝒏(𝒔𝒊𝒛𝒆(𝑨	𝟒𝒏(𝒔𝒊𝒛𝒆(𝑨	NOUN
gojar-5993	86	47	)	)	PUNCT
gojar-5993	86	48	+	+	CCONJ
gojar-5993	87	1	𝒔𝒊𝒛𝒆(𝒃	𝒔𝒊𝒛𝒆(𝒃	NOUN
gojar-5993	87	2	)	)	PUNCT
gojar-5993	87	3	)	)	PUNCT
gojar-5993	87	4	with	with	ADP
gojar-5993	87	5	each	each	DET
gojar-5993	87	6	component	component	NOUN
gojar-5993	87	7	of	of	ADP
gojar-5993	87	8	x	x	SYM
gojar-5993	87	9	satisfying	satisfying	NOUN
gojar-5993	87	10	:	:	PUNCT
gojar-5993	87	11	𝒔𝒊𝒛𝒆(𝒄𝒐𝒎𝒑𝒐𝒏𝒆𝒏𝒕	𝒔𝒊𝒛𝒆(𝒄𝒐𝒎𝒑𝒐𝒏𝒆𝒏𝒕	PROPN
gojar-5993	87	12	𝒐𝒇	𝒐𝒇	PUNCT
gojar-5993	87	13	𝒙	𝒙	X
gojar-5993	87	14	)	)	PUNCT
gojar-5993	87	15	≤	≤	NOUN
gojar-5993	87	16	𝟒(𝒔𝒊𝒛𝒆(𝑨	𝟒(𝒔𝒊𝒛𝒆(𝑨	NOUN
gojar-5993	87	17	)	)	PUNCT
gojar-5993	87	18	+	+	NUM
gojar-5993	88	1	𝒔𝒊𝒛𝒆(𝒃	𝒔𝒊𝒛𝒆(𝒃	NOUN
gojar-5993	88	2	)	)	PUNCT
gojar-5993	88	3	)	)	PUNCT
gojar-5993	88	4	global	global	ADJ
gojar-5993	88	5	online	online	PROPN
gojar-5993	88	6	journal	journal	PROPN
gojar-5993	88	7	of	of	ADP
gojar-5993	88	8	academic	academic	ADJ
gojar-5993	88	9	research	research	NOUN
gojar-5993	88	10	(	(	PUNCT
gojar-5993	88	11	gojar	gojar	NOUN
gojar-5993	88	12	)	)	PUNCT
gojar-5993	88	13	,	,	PUNCT
gojar-5993	88	14	vol	vol	NOUN
gojar-5993	88	15	.	.	PROPN
gojar-5993	89	1	4	4	NUM
gojar-5993	89	2	,	,	PUNCT
gojar-5993	89	3	no	no	INTJ
gojar-5993	89	4	.	.	NOUN
gojar-5993	89	5	1	1	NUM
gojar-5993	89	6	february	february	NOUN
gojar-5993	89	7	2025	2025	NUM
gojar-5993	89	8	66	66	NUM
gojar-5993	89	9	(	(	PUNCT
gojar-5993	89	10	iii	iii	NOUN
gojar-5993	89	11	)	)	PUNCT
gojar-5993	89	12	reduced	reduce	VERB
gojar-5993	89	13	submatrix	submatrix	NOUN
gojar-5993	89	14	case	case	NOUN
gojar-5993	89	15	:	:	PUNCT
gojar-5993	89	16	if	if	SCONJ
gojar-5993	89	17	𝒃	𝒃	NOUN
gojar-5993	89	18	=	=	SYM
gojar-5993	89	19	𝒆𝒊	𝒆𝒊	NOUN
gojar-5993	89	20	𝒐𝒓	𝒐𝒓	NOUN
gojar-5993	89	21	𝒃	𝒃	NOUN
gojar-5993	89	22	=	=	VERB
gojar-5993	89	23	−𝒆𝒊	−𝒆𝒊	NOUN
gojar-5993	89	24	for	for	ADP
gojar-5993	89	25	some	some	DET
gojar-5993	89	26	unit	unit	NOUN
gojar-5993	89	27	vector	vector	NOUN
gojar-5993	89	28	𝒆𝒊	𝒆𝒊	PROPN
gojar-5993	89	29	,	,	PUNCT
gojar-5993	89	30	then	then	ADV
gojar-5993	89	31	there	there	PRON
gojar-5993	89	32	exists	exist	VERB
gojar-5993	89	33	a	a	DET
gojar-5993	89	34	non	non	ADJ
gojar-5993	89	35	-	-	ADJ
gojar-5993	89	36	singular	singular	ADJ
gojar-5993	89	37	submatrix	submatrix	NOUN
gojar-5993	89	38	𝑨′	𝑨′	NOUN
gojar-5993	89	39	of	of	ADP
gojar-5993	89	40	a	a	PRON
gojar-5993	89	41	and	and	CCONJ
gojar-5993	89	42	an	an	DET
gojar-5993	89	43	optimum	optimum	ADJ
gojar-5993	89	44	solution	solution	NOUN
gojar-5993	89	45	x	x	PUNCT
gojar-5993	89	46	such	such	ADJ
gojar-5993	89	47	that	that	SCONJ
gojar-5993	89	48	:	:	PUNCT
gojar-5993	89	49	𝒔𝒊𝒛𝒆(𝒙	𝒔𝒊𝒛𝒆(𝒙	X
gojar-5993	89	50	)	)	PUNCT
gojar-5993	89	51	≤	≤	NOUN
gojar-5993	89	52	𝟒𝒏	𝟒𝒏	ADJ
gojar-5993	89	53	⋅	⋅	PROPN
gojar-5993	89	54	𝒔𝒊𝒛𝒆(𝑨′	𝒔𝒊𝒛𝒆(𝑨′	NUM
gojar-5993	89	55	)	)	PUNCT
gojar-5993	89	56	proof	proof	NOUN
gojar-5993	89	57	the	the	DET
gojar-5993	89	58	proof	proof	NOUN
gojar-5993	89	59	of	of	ADP
gojar-5993	89	60	theorem	theorem	ADJ
gojar-5993	89	61	4.3	4.3	NUM
gojar-5993	89	62	relies	relie	NOUN
gojar-5993	89	63	on	on	ADP
gojar-5993	89	64	these	these	DET
gojar-5993	89	65	definitions	definition	NOUN
gojar-5993	89	66	2.1	2.1	NUM
gojar-5993	89	67	,	,	PUNCT
gojar-5993	89	68	2.2	2.2	NUM
gojar-5993	89	69	,	,	PUNCT
gojar-5993	89	70	2.3	2.3	NUM
gojar-5993	89	71	,	,	PUNCT
gojar-5993	89	72	2.4	2.4	NUM
gojar-5993	89	73	and	and	CCONJ
gojar-5993	89	74	2.5	2.5	NUM
gojar-5993	89	75	respectively	respectively	ADV
gojar-5993	89	76	,	,	PUNCT
gojar-5993	89	77	to	to	PART
gojar-5993	89	78	analyse	analyse	VERB
gojar-5993	89	79	the	the	DET
gojar-5993	89	80	structure	structure	NOUN
gojar-5993	89	81	and	and	CCONJ
gojar-5993	89	82	properties	property	NOUN
gojar-5993	89	83	of	of	ADP
gojar-5993	89	84	the	the	DET
gojar-5993	89	85	lp	lp	PROPN
gojar-5993	89	86	problem	problem	NOUN
gojar-5993	89	87	.	.	PUNCT
gojar-5993	90	1	task	task	NOUN
gojar-5993	90	2	1	1	NUM
gojar-5993	90	3	:	:	PUNCT
gojar-5993	90	4	to	to	PART
gojar-5993	90	5	show	show	VERB
gojar-5993	90	6	that	that	SCONJ
gojar-5993	90	7	,	,	PUNCT
gojar-5993	90	8	for	for	ADP
gojar-5993	90	9	a	a	DET
gojar-5993	90	10	given	give	VERB
gojar-5993	90	11	lp	lp	NOUN
gojar-5993	90	12	,	,	PUNCT
gojar-5993	90	13	there	there	PRON
gojar-5993	90	14	exists	exist	VERB
gojar-5993	90	15	a	a	DET
gojar-5993	90	16	solution	solution	NOUN
gojar-5993	90	17	with	with	ADP
gojar-5993	90	18	bounded	bounded	ADJ
gojar-5993	90	19	size	size	NOUN
gojar-5993	90	20	.	.	PUNCT
gojar-5993	91	1	by	by	ADP
gojar-5993	91	2	the	the	DET
gojar-5993	91	3	fundamental	fundamental	ADJ
gojar-5993	91	4	theorem	theorem	NOUN
gojar-5993	91	5	of	of	ADP
gojar-5993	91	6	linear	linear	PROPN
gojar-5993	91	7	programming	programming	NOUN
gojar-5993	91	8	,	,	PUNCT
gojar-5993	91	9	there	there	PRON
gojar-5993	91	10	exists	exist	VERB
gojar-5993	91	11	an	an	DET
gojar-5993	91	12	optimum	optimum	ADJ
gojar-5993	91	13	solution	solution	NOUN
gojar-5993	91	14	𝒙∗	𝒙∗	NOUN
gojar-5993	91	15	at	at	ADP
gojar-5993	91	16	a	a	DET
gojar-5993	91	17	vertex	vertex	NOUN
gojar-5993	91	18	of	of	ADP
gojar-5993	91	19	the	the	DET
gojar-5993	91	20	feasible	feasible	ADJ
gojar-5993	91	21	polytope	polytope	NOUN
gojar-5993	91	22	{	{	PUNCT
gojar-5993	91	23	𝒙	𝒙	X
gojar-5993	91	24	:	:	PUNCT
gojar-5993	91	25	𝑨𝒙	𝑨𝒙	PROPN
gojar-5993	91	26	≤	≤	NOUN
gojar-5993	91	27	𝒃	𝒃	ADP
gojar-5993	91	28	}	}	PUNCT
gojar-5993	91	29	.	.	PUNCT
gojar-5993	92	1	o	o	NOUN
gojar-5993	92	2	vertex	vertex	NOUN
gojar-5993	92	3	characterization	characterization	NOUN
gojar-5993	92	4	:	:	PUNCT
gojar-5993	92	5	a	a	DET
gojar-5993	92	6	vertex	vertex	NOUN
gojar-5993	92	7	𝒙∗	𝒙∗	NOUN
gojar-5993	92	8	corresponds	correspond	VERB
gojar-5993	92	9	to	to	ADP
gojar-5993	92	10	a	a	DET
gojar-5993	92	11	subset	subset	NOUN
gojar-5993	92	12	of	of	ADP
gojar-5993	92	13	constraints	constraint	NOUN
gojar-5993	92	14	in	in	ADP
gojar-5993	92	15	𝑨𝒙	𝑨𝒙	PROPN
gojar-5993	92	16	≤	≤	NOUN
gojar-5993	92	17	𝒃	𝒃	NOUN
gojar-5993	92	18	that	that	PRON
gojar-5993	92	19	are	be	AUX
gojar-5993	92	20	active	active	ADJ
gojar-5993	92	21	(	(	PUNCT
gojar-5993	92	22	i.e.	i.e.	X
gojar-5993	92	23	,	,	PUNCT
gojar-5993	92	24	satisfied	satisfied	ADJ
gojar-5993	92	25	as	as	ADP
gojar-5993	92	26	equalities	equality	NOUN
gojar-5993	92	27	)	)	PUNCT
gojar-5993	92	28	.	.	PUNCT
gojar-5993	93	1	by	by	ADP
gojar-5993	93	2	corollary	corollary	ADJ
gojar-5993	93	3	2.1	2.1	NUM
gojar-5993	93	4	.	.	PUNCT
gojar-5993	94	1	the	the	DET
gojar-5993	94	2	maximum	maximum	NOUN
gojar-5993	94	3	is	be	AUX
gojar-5993	94	4	attained	attain	VERB
gojar-5993	94	5	in	in	ADP
gojar-5993	94	6	a	a	DET
gojar-5993	94	7	face	face	NOUN
gojar-5993	94	8	𝑭	𝑭	NOUN
gojar-5993	94	9	of	of	ADP
gojar-5993	94	10	{	{	PUNCT
gojar-5993	94	11	𝒙	𝒙	PROPN
gojar-5993	94	12	∶	∶	NOUN
gojar-5993	94	13	𝑨𝒙	𝑨𝒙	PROPN
gojar-5993	94	14	≤	≤	NOUN
gojar-5993	94	15	𝒃	𝒃	ADP
gojar-5993	94	16	}	}	PUNCT
gojar-5993	94	17	.	.	PUNCT
gojar-5993	95	1	let	let	VERB
gojar-5993	95	2	𝑰	𝑰	PROPN
gojar-5993	95	3	⊆	⊆	NUM
gojar-5993	95	4	{	{	PUNCT
gojar-5993	95	5	𝟏	𝟏	NUM
gojar-5993	95	6	,	,	PUNCT
gojar-5993	95	7	𝟐	𝟐	NUM
gojar-5993	95	8	,	,	PUNCT
gojar-5993	95	9	…	…	PUNCT
gojar-5993	95	10	,	,	PUNCT
gojar-5993	95	11	𝒎	𝒎	X
gojar-5993	95	12	}	}	PUNCT
gojar-5993	95	13	denote	denote	VERB
gojar-5993	95	14	the	the	DET
gojar-5993	95	15	indices	index	NOUN
gojar-5993	95	16	of	of	ADP
gojar-5993	95	17	active	active	ADJ
gojar-5993	95	18	constraints	constraint	NOUN
gojar-5993	95	19	,	,	PUNCT
gojar-5993	95	20	and	and	CCONJ
gojar-5993	95	21	let	let	VERB
gojar-5993	95	22	𝑨𝑰	𝑨𝑰	PROPN
gojar-5993	95	23	denote	denote	VERB
gojar-5993	95	24	the	the	DET
gojar-5993	95	25	submatrix	submatrix	NOUN
gojar-5993	95	26	of	of	ADP
gojar-5993	95	27	a	a	DET
gojar-5993	95	28	corresponding	corresponding	NOUN
gojar-5993	95	29	to	to	ADP
gojar-5993	95	30	these	these	DET
gojar-5993	95	31	constraints	constraint	NOUN
gojar-5993	95	32	.	.	PUNCT
gojar-5993	96	1	at	at	ADP
gojar-5993	96	2	a	a	DET
gojar-5993	96	3	vertex	vertex	NOUN
gojar-5993	96	4	,	,	PUNCT
gojar-5993	96	5	the	the	DET
gojar-5993	96	6	system	system	NOUN
gojar-5993	96	7	can	can	AUX
gojar-5993	96	8	be	be	AUX
gojar-5993	96	9	written	write	VERB
gojar-5993	96	10	as	as	ADP
gojar-5993	96	11	:	:	PUNCT
gojar-5993	96	12	𝑨𝑰𝒙	𝑨𝑰𝒙	X
gojar-5993	96	13	=	=	PUNCT
gojar-5993	96	14	𝒃𝑰	𝒃𝑰	ADJ
gojar-5993	96	15	where	where	SCONJ
gojar-5993	96	16	𝒃𝑰is	𝒃𝑰is	PROPN
gojar-5993	96	17	the	the	DET
gojar-5993	96	18	corresponding	correspond	VERB
gojar-5993	96	19	sub	sub	NOUN
gojar-5993	96	20	vector	vector	NOUN
gojar-5993	96	21	of	of	ADP
gojar-5993	96	22	b.	b.	PROPN
gojar-5993	96	23	o	o	PROPN
gojar-5993	96	24	non	non	ADJ
gojar-5993	96	25	-	-	NOUN
gojar-5993	96	26	singularity	singularity	NOUN
gojar-5993	96	27	of	of	ADP
gojar-5993	96	28	𝑨𝑰	𝑨𝑰	PROPN
gojar-5993	96	29	:	:	PUNCT
gojar-5993	96	30	for	for	SCONJ
gojar-5993	96	31	𝒙∗	𝒙∗	NOUN
gojar-5993	96	32	to	to	PART
gojar-5993	96	33	be	be	AUX
gojar-5993	96	34	a	a	DET
gojar-5993	96	35	vertex	vertex	NOUN
gojar-5993	96	36	,	,	PUNCT
gojar-5993	96	37	the	the	DET
gojar-5993	96	38	matrix	matrix	NOUN
gojar-5993	96	39	𝑨𝑰	𝑨𝑰	PROPN
gojar-5993	96	40	must	must	AUX
gojar-5993	96	41	be	be	AUX
gojar-5993	96	42	non	non	ADJ
gojar-5993	96	43	-	-	ADJ
gojar-5993	96	44	singular	singular	ADJ
gojar-5993	96	45	(	(	PUNCT
gojar-5993	96	46	invertible	invertible	ADJ
gojar-5993	96	47	)	)	PUNCT
gojar-5993	96	48	,	,	PUNCT
gojar-5993	96	49	and	and	CCONJ
gojar-5993	96	50	∣	∣	ADJ
gojar-5993	96	51	𝑰	𝑰	PROPN
gojar-5993	96	52	∣=	∣=	PROPN
gojar-5993	96	53	𝒏.	𝒏.	ADJ
gojar-5993	96	54	o	o	NOUN
gojar-5993	96	55	size	size	NOUN
gojar-5993	96	56	of	of	ADP
gojar-5993	96	57	solution	solution	NOUN
gojar-5993	96	58	:	:	PUNCT
gojar-5993	96	59	solving	solve	VERB
gojar-5993	96	60	𝑨𝑰𝒙	𝑨𝑰𝒙	NOUN
gojar-5993	96	61	=	=	PUNCT
gojar-5993	96	62	𝒃𝑰	𝒃𝑰	ADJ
gojar-5993	96	63	𝒙	𝒙	NOUN
gojar-5993	96	64	=	=	SYM
gojar-5993	96	65	𝑨𝑰	𝑨𝑰	PROPN
gojar-5993	96	66	−𝟏𝒃𝑰	−𝟏𝒃𝑰	NOUN
gojar-5993	96	67	using	use	VERB
gojar-5993	96	68	bounds	bound	NOUN
gojar-5993	96	69	on	on	ADP
gojar-5993	96	70	the	the	DET
gojar-5993	96	71	size	size	NOUN
gojar-5993	96	72	of	of	ADP
gojar-5993	96	73	𝑨𝑰	𝑨𝑰	PROPN
gojar-5993	96	74	and	and	CCONJ
gojar-5993	96	75	𝒃𝑰	𝒃𝑰	PROPN
gojar-5993	96	76	,	,	PUNCT
gojar-5993	96	77	and	and	CCONJ
gojar-5993	96	78	the	the	DET
gojar-5993	96	79	fact	fact	NOUN
gojar-5993	96	80	that	that	SCONJ
gojar-5993	96	81	𝑨𝑰	𝑨𝑰	PROPN
gojar-5993	96	82	is	be	AUX
gojar-5993	96	83	rational	rational	ADJ
gojar-5993	96	84	,	,	PUNCT
gojar-5993	96	85	the	the	DET
gojar-5993	96	86	entries	entry	NOUN
gojar-5993	96	87	of	of	ADP
gojar-5993	96	88	are	be	AUX
gojar-5993	96	89	bounded	bound	VERB
gojar-5993	96	90	in	in	ADP
gojar-5993	96	91	terms	term	NOUN
gojar-5993	96	92	of	of	ADP
gojar-5993	96	93	𝒔𝒊𝒛𝒆(𝑨𝑰	𝒔𝒊𝒛𝒆(𝑨𝑰	NOUN
gojar-5993	96	94	)	)	PUNCT
gojar-5993	96	95	.	.	PUNCT
gojar-5993	97	1	specifically	specifically	ADV
gojar-5993	97	2	,	,	PUNCT
gojar-5993	97	3	the	the	DET
gojar-5993	97	4	size	size	NOUN
gojar-5993	97	5	of	of	ADP
gojar-5993	97	6	x	x	PROPN
gojar-5993	97	7	is	be	AUX
gojar-5993	97	8	bounded	bound	VERB
gojar-5993	97	9	by	by	ADP
gojar-5993	97	10	:	:	PUNCT
gojar-5993	97	11	𝒔𝒊𝒛𝒆(𝒙	𝒔𝒊𝒛𝒆(𝒙	NOUN
gojar-5993	97	12	)	)	PUNCT
gojar-5993	97	13	≤	≤	NUM
gojar-5993	97	14	𝟒𝒏(𝒔𝒊𝒛𝒆(𝑨	𝟒𝒏(𝒔𝒊𝒛𝒆(𝑨	NOUN
gojar-5993	97	15	)	)	PUNCT
gojar-5993	97	16	+	+	CCONJ
gojar-5993	97	17	𝒔𝒊𝒛𝒆(𝒃	𝒔𝒊𝒛𝒆(𝒃	NOUN
gojar-5993	97	18	)	)	PUNCT
gojar-5993	97	19	)	)	PUNCT
gojar-5993	97	20	.	.	PUNCT
gojar-5993	98	1	global	global	ADJ
gojar-5993	98	2	online	online	PROPN
gojar-5993	98	3	journal	journal	PROPN
gojar-5993	98	4	of	of	ADP
gojar-5993	98	5	academic	academic	ADJ
gojar-5993	98	6	research	research	NOUN
gojar-5993	98	7	(	(	PUNCT
gojar-5993	98	8	gojar	gojar	NOUN
gojar-5993	98	9	)	)	PUNCT
gojar-5993	98	10	,	,	PUNCT
gojar-5993	98	11	vol	vol	NOUN
gojar-5993	98	12	.	.	PROPN
gojar-5993	98	13	4	4	NUM
gojar-5993	98	14	,	,	PUNCT
gojar-5993	98	15	no	no	INTJ
gojar-5993	98	16	.	.	NOUN
gojar-5993	98	17	1	1	NUM
gojar-5993	98	18	february	february	PROPN
gojar-5993	98	19	2025	2025	NUM
gojar-5993	98	20	67	67	NUM
gojar-5993	98	21	task	task	NOUN
gojar-5993	98	22	2	2	NUM
gojar-5993	98	23	:	:	PUNCT
gojar-5993	98	24	to	to	PART
gojar-5993	98	25	show	show	VERB
gojar-5993	98	26	that	that	SCONJ
gojar-5993	98	27	for	for	ADP
gojar-5993	98	28	a	a	DET
gojar-5993	98	29	give	give	NOUN
gojar-5993	98	30	lp	lp	NOUN
gojar-5993	98	31	has	have	VERB
gojar-5993	98	32	a	a	DET
gojar-5993	98	33	special	special	ADJ
gojar-5993	98	34	case	case	NOUN
gojar-5993	98	35	(	(	PUNCT
gojar-5993	98	36	𝒃	𝒃	PROPN
gojar-5993	98	37	=	=	SYM
gojar-5993	98	38	𝒆𝒊	𝒆𝒊	NOUN
gojar-5993	98	39	𝒐𝒓	𝒐𝒓	NOUN
gojar-5993	98	40	𝒃	𝒃	NOUN
gojar-5993	98	41	=	=	SYM
gojar-5993	98	42	−𝒆𝒊	−𝒆𝒊	NOUN
gojar-5993	98	43	)	)	PUNCT
gojar-5993	98	44	if	if	SCONJ
gojar-5993	98	45	𝒃	𝒃	NOUN
gojar-5993	98	46	=	=	SYM
gojar-5993	98	47	𝒆𝒊	𝒆𝒊	NOUN
gojar-5993	98	48	𝒐𝒓	𝒐𝒓	NOUN
gojar-5993	98	49	𝒃	𝒃	NOUN
gojar-5993	98	50	=	=	SYM
gojar-5993	98	51	−𝒆𝒊	−𝒆𝒊	NOUN
gojar-5993	98	52	,	,	PUNCT
gojar-5993	98	53	where𝒆𝒊	where𝒆𝒊	PROPN
gojar-5993	98	54	is	be	AUX
gojar-5993	98	55	a	a	DET
gojar-5993	98	56	unit	unit	NOUN
gojar-5993	98	57	vector	vector	NOUN
gojar-5993	98	58	,	,	PUNCT
gojar-5993	98	59	the	the	DET
gojar-5993	98	60	lp	lp	NOUN
gojar-5993	98	61	corresponds	correspond	VERB
gojar-5993	98	62	to	to	ADP
gojar-5993	98	63	finding	find	VERB
gojar-5993	98	64	the	the	DET
gojar-5993	98	65	maximum	maximum	ADJ
gojar-5993	98	66	value	value	NOUN
gojar-5993	98	67	of	of	ADP
gojar-5993	98	68	𝒄𝑻𝒙	𝒄𝑻𝒙	NOUN
gojar-5993	98	69	along	along	ADP
gojar-5993	98	70	a	a	DET
gojar-5993	98	71	specific	specific	ADJ
gojar-5993	98	72	axis	axis	NOUN
gojar-5993	98	73	defined	define	VERB
gojar-5993	98	74	by	by	ADP
gojar-5993	98	75	𝒆𝒊.	𝒆𝒊.	NOUN
gojar-5993	98	76	o	o	PROPN
gojar-5993	98	77	existence	existence	NOUN
gojar-5993	98	78	of	of	ADP
gojar-5993	98	79	a	a	DET
gojar-5993	98	80	non	non	ADJ
gojar-5993	98	81	-	-	ADJ
gojar-5993	98	82	singular	singular	ADJ
gojar-5993	98	83	submatrix	submatrix	NOUN
gojar-5993	98	84	:	:	PUNCT
gojar-5993	98	85	as	as	ADP
gojar-5993	98	86	in	in	ADP
gojar-5993	98	87	task	task	NOUN
gojar-5993	98	88	1	1	NUM
gojar-5993	98	89	,	,	PUNCT
gojar-5993	98	90	there	there	PRON
gojar-5993	98	91	exists	exist	VERB
gojar-5993	98	92	a	a	DET
gojar-5993	98	93	vertex	vertex	NOUN
gojar-5993	98	94	solution	solution	NOUN
gojar-5993	98	95	𝒙∗	𝒙∗	NOUN
gojar-5993	98	96	,	,	PUNCT
gojar-5993	98	97	and	and	CCONJ
gojar-5993	98	98	the	the	DET
gojar-5993	98	99	active	active	ADJ
gojar-5993	98	100	constraints	constraint	NOUN
gojar-5993	98	101	correspond	correspond	VERB
gojar-5993	98	102	to	to	ADP
gojar-5993	98	103	a	a	DET
gojar-5993	98	104	nonsingular	nonsingular	ADJ
gojar-5993	98	105	submatrix	submatrix	NOUN
gojar-5993	98	106	𝑨′	𝑨′	NOUN
gojar-5993	98	107	of	of	ADP
gojar-5993	98	108	a.	a.	NOUN
gojar-5993	98	109	o	o	NOUN
gojar-5993	98	110	bound	bind	VERB
gojar-5993	98	111	on	on	ADP
gojar-5993	98	112	solution	solution	NOUN
gojar-5993	98	113	size	size	NOUN
gojar-5993	98	114	:	:	PUNCT
gojar-5993	98	115	similar	similar	ADJ
gojar-5993	98	116	to	to	ADP
gojar-5993	98	117	the	the	DET
gojar-5993	98	118	general	general	ADJ
gojar-5993	98	119	case	case	NOUN
gojar-5993	98	120	,	,	PUNCT
gojar-5993	98	121	the	the	DET
gojar-5993	98	122	size	size	NOUN
gojar-5993	98	123	of	of	ADP
gojar-5993	98	124	x	x	PROPN
gojar-5993	98	125	is	be	AUX
gojar-5993	98	126	bounded	bound	VERB
gojar-5993	98	127	by	by	ADP
gojar-5993	98	128	:	:	PUNCT
gojar-5993	98	129	𝒔𝒊𝒛𝒆(𝒙	𝒔𝒊𝒛𝒆(𝒙	NOUN
gojar-5993	98	130	)	)	PUNCT
gojar-5993	98	131	≤	≤	NUM
gojar-5993	98	132	𝟒𝒏(𝒔𝒊𝒛𝒆(𝑨	𝟒𝒏(𝒔𝒊𝒛𝒆(𝑨	NOUN
gojar-5993	98	133	)	)	PUNCT
gojar-5993	99	1	+	+	CCONJ
gojar-5993	100	1	𝒔𝒊𝒛𝒆(𝒃	𝒔𝒊𝒛𝒆(𝒃	NOUN
gojar-5993	100	2	)	)	PUNCT
gojar-5993	100	3	)	)	PUNCT
gojar-5993	101	1	,	,	PUNCT
gojar-5993	101	2	with	with	ADP
gojar-5993	101	3	the	the	DET
gojar-5993	101	4	size	size	NOUN
gojar-5993	101	5	of	of	ADP
gojar-5993	101	6	each	each	DET
gojar-5993	101	7	component	component	NOUN
gojar-5993	101	8	of	of	ADP
gojar-5993	101	9	x	x	PART
gojar-5993	101	10	further	far	ADV
gojar-5993	101	11	bounded	bound	VERB
gojar-5993	101	12	by	by	ADP
gojar-5993	101	13	:	:	PUNCT
gojar-5993	101	14	𝒔𝒊𝒛𝒆(𝒄𝒐𝒎𝒑𝒐𝒏𝒆𝒏𝒕	𝒔𝒊𝒛𝒆(𝒄𝒐𝒎𝒑𝒐𝒏𝒆𝒏𝒕	PROPN
gojar-5993	101	15	𝒐𝒇	𝒐𝒇	PUNCT
gojar-5993	101	16	𝒙	𝒙	X
gojar-5993	101	17	)	)	PUNCT
gojar-5993	101	18	≤	≤	NOUN
gojar-5993	101	19	𝟒(𝒔𝒊𝒛𝒆(𝑨	𝟒(𝒔𝒊𝒛𝒆(𝑨	NOUN
gojar-5993	101	20	)	)	PUNCT
gojar-5993	101	21	+	+	NUM
gojar-5993	101	22	𝒔𝒊𝒛𝒆(𝒃	𝒔𝒊𝒛𝒆(𝒃	NOUN
gojar-5993	101	23	)	)	PUNCT
gojar-5993	101	24	)	)	PUNCT
gojar-5993	101	25	.	.	PUNCT
gojar-5993	102	1	task	task	NOUN
gojar-5993	102	2	3	3	NUM
gojar-5993	102	3	:	:	PUNCT
gojar-5993	102	4	to	to	PART
gojar-5993	102	5	show	show	VERB
gojar-5993	102	6	that	that	SCONJ
gojar-5993	102	7	for	for	ADP
gojar-5993	102	8	a	a	DET
gojar-5993	102	9	give	give	NOUN
gojar-5993	102	10	lp	lp	NOUN
gojar-5993	102	11	has	have	AUX
gojar-5993	102	12	reduced	reduce	VERB
gojar-5993	102	13	submatrix	submatrix	NOUN
gojar-5993	102	14	:	:	PUNCT
gojar-5993	102	15	let	let	VERB
gojar-5993	102	16	𝑭′	𝑭′	PROPN
gojar-5993	102	17	⊆	⊆	NUM
gojar-5993	102	18	𝑭	𝑭	PROPN
gojar-5993	102	19	be	be	AUX
gojar-5993	102	20	a	a	DET
gojar-5993	102	21	minimal	minimal	ADJ
gojar-5993	102	22	face	face	NOUN
gojar-5993	102	23	.	.	PUNCT
gojar-5993	103	1	by	by	ADP
gojar-5993	103	2	corollary	corollary	ADJ
gojar-5993	103	3	2.1𝑭′	2.1𝑭′	PROPN
gojar-5993	103	4	=	=	SYM
gojar-5993	103	5	{	{	PUNCT
gojar-5993	103	6	𝒙	𝒙	NUM
gojar-5993	103	7	∶	∶	NOUN
gojar-5993	103	8	𝑨′𝒙	𝑨′𝒙	X
gojar-5993	103	9	=	=	SYM
gojar-5993	103	10	𝒃′	𝒃′	PROPN
gojar-5993	103	11	}	}	PUNCT
gojar-5993	103	12	for	for	ADP
gojar-5993	103	13	some	some	DET
gojar-5993	103	14	subsystem	subsystem	NOUN
gojar-5993	103	15	𝑨′𝒙	𝑨′𝒙	X
gojar-5993	103	16	≤	≤	NUM
gojar-5993	103	17	𝒃′	𝒃′	NOUN
gojar-5993	103	18	𝐨𝐟	𝐨𝐟	ADP
gojar-5993	103	19	𝑨𝒙	𝑨𝒙	PROPN
gojar-5993	103	20	≤	≤	PUNCT
gojar-5993	103	21	𝒃.	𝒃.	NOUN
gojar-5993	103	22	then	then	ADV
gojar-5993	103	23	,	,	PUNCT
gojar-5993	103	24	in	in	ADP
gojar-5993	103	25	the	the	DET
gojar-5993	103	26	special	special	ADJ
gojar-5993	103	27	case	case	NOUN
gojar-5993	103	28	where	where	SCONJ
gojar-5993	103	29	𝒃	𝒃	NOUN
gojar-5993	103	30	=	=	SYM
gojar-5993	103	31	𝒆𝒊	𝒆𝒊	NOUN
gojar-5993	103	32	𝒐𝒓	𝒐𝒓	NOUN
gojar-5993	103	33	𝒃	𝒃	NOUN
gojar-5993	103	34	=	=	SYM
gojar-5993	103	35	−𝒆𝒊	−𝒆𝒊	NOUN
gojar-5993	103	36	,	,	PUNCT
gojar-5993	103	37	let	let	VERB
gojar-5993	103	38	𝑨′	𝑨′	NOUN
gojar-5993	103	39	denote	denote	VERB
gojar-5993	103	40	the	the	DET
gojar-5993	103	41	nonsingular	nonsingular	ADJ
gojar-5993	103	42	submatrix	submatrix	NOUN
gojar-5993	103	43	corresponding	correspond	VERB
gojar-5993	103	44	to	to	ADP
gojar-5993	103	45	the	the	DET
gojar-5993	103	46	active	active	ADJ
gojar-5993	103	47	constraints	constraint	NOUN
gojar-5993	103	48	at	at	ADP
gojar-5993	103	49	the	the	DET
gojar-5993	103	50	optimum	optimum	NOUN
gojar-5993	103	51	.	.	PUNCT
gojar-5993	104	1	now	now	ADV
gojar-5993	104	2	,	,	PUNCT
gojar-5993	104	3	we	we	PRON
gojar-5993	104	4	may	may	AUX
gojar-5993	104	5	assume	assume	VERB
gojar-5993	104	6	that	that	SCONJ
gojar-5993	104	7	the	the	DET
gojar-5993	104	8	rows	row	NOUN
gojar-5993	104	9	of	of	ADP
gojar-5993	104	10	𝑨′	𝑨′	NOUN
gojar-5993	104	11	are	be	AUX
gojar-5993	104	12	linearly	linearly	ADV
gojar-5993	104	13	independent	independent	ADJ
gojar-5993	104	14	.	.	PUNCT
gojar-5993	105	1	we	we	PRON
gojar-5993	105	2	then	then	ADV
gojar-5993	105	3	take	take	VERB
gojar-5993	105	4	a	a	DET
gojar-5993	105	5	maximal	maximal	ADJ
gojar-5993	105	6	set	set	NOUN
gojar-5993	105	7	of	of	ADP
gojar-5993	105	8	linear	linear	ADJ
gojar-5993	105	9	independent	independent	ADJ
gojar-5993	105	10	columns	column	NOUN
gojar-5993	105	11	(	(	PUNCT
gojar-5993	105	12	call	call	VERB
gojar-5993	105	13	this	this	DET
gojar-5993	105	14	matrix	matrix	NOUN
gojar-5993	105	15	𝑨	𝑨	NOUN
gojar-5993	105	16	”	"	PUNCT
gojar-5993	105	17	)	)	PUNCT
gojar-5993	105	18	and	and	CCONJ
gojar-5993	105	19	set	set	VERB
gojar-5993	105	20	all	all	DET
gojar-5993	105	21	other	other	ADJ
gojar-5993	105	22	components	component	NOUN
gojar-5993	105	23	to	to	ADP
gojar-5993	105	24	zero	zero	NUM
gojar-5993	105	25	.	.	PUNCT
gojar-5993	106	1	then	then	ADV
gojar-5993	106	2	𝒙	𝒙	PROPN
gojar-5993	106	3	=	=	SYM
gojar-5993	106	4	(	(	PUNCT
gojar-5993	106	5	𝑨")−𝟏𝒃′	𝑨")−𝟏𝒃′	NOUN
gojar-5993	106	6	,	,	PUNCT
gojar-5993	106	7	filled	fill	VERB
gojar-5993	106	8	up	up	ADP
gojar-5993	106	9	with	with	ADP
gojar-5993	106	10	zeros	zero	NOUN
gojar-5993	106	11	,	,	PUNCT
gojar-5993	106	12	is	be	AUX
gojar-5993	106	13	an	an	DET
gojar-5993	106	14	optimum	optimum	ADJ
gojar-5993	106	15	solution	solution	NOUN
gojar-5993	106	16	to	to	ADP
gojar-5993	106	17	our	our	PRON
gojar-5993	106	18	lp	lp	NOUN
gojar-5993	106	19	.	.	PUNCT
gojar-5993	107	1	by	by	ADP
gojar-5993	107	2	cramer	cramer	PROPN
gojar-5993	107	3	’s	’s	PART
gojar-5993	107	4	rule	rule	NOUN
gojar-5993	107	5	the	the	DET
gojar-5993	107	6	entries	entry	NOUN
gojar-5993	107	7	of	of	ADP
gojar-5993	107	8	𝒙	𝒙	PROPN
gojar-5993	107	9	are	be	AUX
gojar-5993	107	10	given	give	VERB
gojar-5993	107	11	by	by	ADP
gojar-5993	107	12	𝒙𝒊	𝒙𝒊	PROPN
gojar-5993	107	13	=	=	PUNCT
gojar-5993	107	14	𝐝𝐞𝐭	𝐝𝐞𝐭	PROPN
gojar-5993	107	15	𝑨′′′	𝑨′′′	PROPN
gojar-5993	107	16	𝐝𝐞𝐭	𝐝𝐞𝐭	NOUN
gojar-5993	107	17	𝑨′′	𝑨′′	PROPN
gojar-5993	107	18	,	,	PUNCT
gojar-5993	107	19	where	where	SCONJ
gojar-5993	107	20	𝑨′′′	𝑨′′′	PROPN
gojar-5993	107	21	arises	arise	VERB
gojar-5993	107	22	from	from	ADP
gojar-5993	107	23	𝑨′′	𝑨′′	ADV
gojar-5993	107	24	by	by	ADP
gojar-5993	107	25	replacing	replace	VERB
gojar-5993	107	26	the	the	DET
gojar-5993	107	27	𝒋	𝒋	NOUN
gojar-5993	107	28	−	−	NOUN
gojar-5993	107	29	𝒕𝒉	𝒕𝒉	DET
gojar-5993	107	30	column	column	NOUN
gojar-5993	107	31	by	by	ADP
gojar-5993	107	32	𝒃′.	𝒃′.	ADV
gojar-5993	107	33	by	by	ADP
gojar-5993	107	34	propositions	proposition	NOUN
gojar-5993	107	35	2.4	2.4	NUM
gojar-5993	107	36	and	and	CCONJ
gojar-5993	107	37	2.5	2.5	NUM
gojar-5993	107	38	respectively	respectively	ADV
gojar-5993	107	39	,	,	PUNCT
gojar-5993	107	40	we	we	PRON
gojar-5993	107	41	obtain	obtain	VERB
gojar-5993	107	42	𝒔𝒊𝒛𝒆(𝒙	𝒔𝒊𝒛𝒆(𝒙	NOUN
gojar-5993	107	43	)	)	PUNCT
gojar-5993	107	44	≤	≤	NOUN
gojar-5993	108	1	𝒏	𝒏	PROPN
gojar-5993	108	2	+	+	PROPN
gojar-5993	108	3	𝟐𝒏(𝒔𝒊𝒛𝒆(𝑨′′′	𝟐𝒏(𝒔𝒊𝒛𝒆(𝑨′′′	PUNCT
gojar-5993	108	4	)	)	PUNCT
gojar-5993	108	5	+	+	NUM
gojar-5993	108	6	𝒔𝒊𝒛𝒆(𝑨′′	𝒔𝒊𝒛𝒆(𝑨′′	NOUN
gojar-5993	108	7	)	)	PUNCT
gojar-5993	108	8	)	)	PUNCT
gojar-5993	108	9	≤	≤	NUM
gojar-5993	108	10	𝟒𝒏(𝒔𝒊𝒛𝒆(𝑨′′	𝟒𝒏(𝒔𝒊𝒛𝒆(𝑨′′	PUNCT
gojar-5993	108	11	)	)	PUNCT
gojar-5993	108	12	+	+	CCONJ
gojar-5993	108	13	𝒔𝒊𝒛𝒆(𝒃′	𝒔𝒊𝒛𝒆(𝒃′	NUM
gojar-5993	108	14	)	)	PUNCT
gojar-5993	108	15	)	)	PUNCT
gojar-5993	108	16	.	.	PUNCT
gojar-5993	109	1	if	if	SCONJ
gojar-5993	109	2	𝒃	𝒃	NOUN
gojar-5993	109	3	=	=	PUNCT
gojar-5993	109	4	±𝒆𝒊	±𝒆𝒊	VERB
gojar-5993	109	5	then	then	ADV
gojar-5993	109	6	|	|	ADV
gojar-5993	109	7	𝐝𝐞𝐭(𝑨′′′	𝐝𝐞𝐭(𝑨′′′	PUNCT
gojar-5993	109	8	)	)	PUNCT
gojar-5993	109	9	|	|	ADV
gojar-5993	109	10	is	be	AUX
gojar-5993	109	11	the	the	DET
gojar-5993	109	12	absolute	absolute	ADJ
gojar-5993	109	13	value	value	NOUN
gojar-5993	109	14	of	of	ADP
gojar-5993	109	15	a	a	DET
gojar-5993	109	16	sub	sub	NOUN
gojar-5993	109	17	determinant	determinant	ADJ
gojar-5993	109	18	of	of	ADP
gojar-5993	109	19	𝑨′′.	𝑨′′.	PROPN
gojar-5993	109	20	the	the	DET
gojar-5993	109	21	size	size	NOUN
gojar-5993	109	22	of	of	ADP
gojar-5993	109	23	x	x	PUNCT
gojar-5993	109	24	can	can	AUX
gojar-5993	109	25	then	then	ADV
gojar-5993	109	26	be	be	AUX
gojar-5993	109	27	further	far	ADV
gojar-5993	109	28	bounded	bound	VERB
gojar-5993	109	29	as	as	ADP
gojar-5993	109	30	:	:	PUNCT
gojar-5993	110	1	𝒔𝒊𝒛𝒆(𝒙	𝒔𝒊𝒛𝒆(𝒙	NOUN
gojar-5993	110	2	)	)	PUNCT
gojar-5993	110	3	≤	≤	NOUN
gojar-5993	110	4	𝟒𝒏	𝟒𝒏	ADJ
gojar-5993	110	5	⋅	⋅	PROPN
gojar-5993	110	6	𝒔𝒊𝒛𝒆(𝑨′	𝒔𝒊𝒛𝒆(𝑨′	PROPN
gojar-5993	110	7	)	)	PUNCT
gojar-5993	110	8	.	.	PUNCT
gojar-5993	111	1	this	this	PRON
gojar-5993	111	2	follows	follow	VERB
gojar-5993	111	3	because	because	SCONJ
gojar-5993	111	4	𝑨′	𝑨′	NOUN
gojar-5993	111	5	has	have	VERB
gojar-5993	111	6	fewer	few	ADJ
gojar-5993	111	7	rows	row	NOUN
gojar-5993	111	8	and	and	CCONJ
gojar-5993	111	9	columns	column	NOUN
gojar-5993	111	10	compared	compare	VERB
gojar-5993	111	11	to	to	ADP
gojar-5993	111	12	the	the	DET
gojar-5993	111	13	full	full	ADJ
gojar-5993	111	14	matrix	matrix	NOUN
gojar-5993	111	15	a	a	PRON
gojar-5993	111	16	,	,	PUNCT
gojar-5993	111	17	reducing	reduce	VERB
gojar-5993	111	18	the	the	DET
gojar-5993	111	19	maximum	maximum	ADJ
gojar-5993	111	20	size	size	NOUN
gojar-5993	111	21	contribution	contribution	NOUN
gojar-5993	111	22	.	.	PUNCT
gojar-5993	112	1	q.e.d	q.e.d	PROPN
gojar-5993	112	2	.	.	PUNCT
gojar-5993	113	1	utilizing	utilize	VERB
gojar-5993	113	2	results	result	NOUN
gojar-5993	113	3	from	from	ADP
gojar-5993	113	4	schrijver	schrijver	PROPN
gojar-5993	113	5	(	(	PUNCT
gojar-5993	113	6	1998	1998	NUM
gojar-5993	113	7	)	)	PUNCT
gojar-5993	113	8	and	and	CCONJ
gojar-5993	113	9	cook	cook	VERB
gojar-5993	113	10	et	et	PROPN
gojar-5993	113	11	al	al	PROPN
gojar-5993	113	12	.	.	PROPN
gojar-5993	113	13	,	,	PUNCT
gojar-5993	113	14	(	(	PUNCT
gojar-5993	113	15	1986	1986	NUM
gojar-5993	113	16	)	)	PUNCT
gojar-5993	113	17	,	,	PUNCT
gojar-5993	113	18	we	we	PRON
gojar-5993	113	19	derive	derive	VERB
gojar-5993	113	20	global	global	ADJ
gojar-5993	113	21	online	online	ADJ
gojar-5993	113	22	journal	journal	PROPN
gojar-5993	113	23	of	of	ADP
gojar-5993	113	24	academic	academic	ADJ
gojar-5993	113	25	research	research	NOUN
gojar-5993	113	26	(	(	PUNCT
gojar-5993	113	27	gojar	gojar	NOUN
gojar-5993	113	28	)	)	PUNCT
gojar-5993	113	29	,	,	PUNCT
gojar-5993	113	30	vol	vol	NOUN
gojar-5993	113	31	.	.	PROPN
gojar-5993	114	1	4	4	NUM
gojar-5993	114	2	,	,	PUNCT
gojar-5993	114	3	no	no	INTJ
gojar-5993	114	4	.	.	NOUN
gojar-5993	114	5	1	1	NUM
gojar-5993	114	6	february	february	NOUN
gojar-5993	114	7	2025	2025	NUM
gojar-5993	114	8	68	68	NUM
gojar-5993	114	9	bounds	bound	NOUN
gojar-5993	114	10	on	on	ADP
gojar-5993	114	11	the	the	DET
gojar-5993	114	12	size	size	NOUN
gojar-5993	114	13	of	of	ADP
gojar-5993	114	14	optimal	optimal	ADJ
gojar-5993	114	15	solutions	solution	NOUN
gojar-5993	114	16	by	by	ADP
gojar-5993	114	17	analyzing	analyze	VERB
gojar-5993	114	18	the	the	DET
gojar-5993	114	19	bit	bit	NOUN
gojar-5993	114	20	-	-	PUNCT
gojar-5993	114	21	length	length	NOUN
gojar-5993	114	22	of	of	ADP
gojar-5993	114	23	vertices	vertex	NOUN
gojar-5993	114	24	of	of	ADP
gojar-5993	114	25	and	and	CCONJ
gojar-5993	114	26	properties	property	NOUN
gojar-5993	114	27	of	of	ADP
gojar-5993	114	28	rational	rational	ADJ
gojar-5993	114	29	systems	system	NOUN
gojar-5993	114	30	.	.	PUNCT
gojar-5993	115	1	iv	iv	X
gojar-5993	115	2	.	.	PUNCT
gojar-5993	115	3	applications	application	NOUN
gojar-5993	115	4	:	:	PUNCT
gojar-5993	115	5	production	production	NOUN
gojar-5993	115	6	scheduling	scheduling	NOUN
gojar-5993	115	7	problem	problem	NOUN
gojar-5993	115	8	i	i	NOUN
gojar-5993	115	9	)	)	PUNCT
gojar-5993	115	10	integer	integer	NOUN
gojar-5993	115	11	programming	programming	NOUN
gojar-5993	115	12	:	:	PUNCT
gojar-5993	115	13	the	the	DET
gojar-5993	115	14	equivalence	equivalence	NOUN
gojar-5993	115	15	of	of	ADP
gojar-5993	115	16	boundedness	boundedness	NOUN
gojar-5993	115	17	conditions	condition	NOUN
gojar-5993	115	18	simplifies	simplifie	NOUN
gojar-5993	115	19	complexity	complexity	NOUN
gojar-5993	115	20	analyses	analysis	NOUN
gojar-5993	115	21	for	for	ADP
gojar-5993	115	22	mixed	mixed	ADJ
gojar-5993	115	23	-	-	PUNCT
gojar-5993	115	24	integer	integer	NOUN
gojar-5993	115	25	programming	programming	NOUN
gojar-5993	115	26	problems	problem	NOUN
gojar-5993	115	27	problem	problem	NOUN
gojar-5993	115	28	setup	setup	VERB
gojar-5993	115	29	a	a	DET
gojar-5993	115	30	factory	factory	NOUN
gojar-5993	115	31	produces	produce	VERB
gojar-5993	115	32	two	two	NUM
gojar-5993	115	33	products	product	NOUN
gojar-5993	115	34	,	,	PUNCT
gojar-5993	115	35	a	a	PRON
gojar-5993	115	36	and	and	CCONJ
gojar-5993	115	37	b	b	NOUN
gojar-5993	115	38	,	,	PUNCT
gojar-5993	115	39	using	use	VERB
gojar-5993	115	40	two	two	NUM
gojar-5993	115	41	resources	resource	NOUN
gojar-5993	115	42	,	,	PUNCT
gojar-5993	115	43	labor	labor	NOUN
gojar-5993	115	44	and	and	CCONJ
gojar-5993	115	45	material	material	NOUN
gojar-5993	115	46	.	.	PUNCT
gojar-5993	116	1	the	the	DET
gojar-5993	116	2	available	available	ADJ
gojar-5993	116	3	resources	resource	NOUN
gojar-5993	116	4	are	be	AUX
gojar-5993	116	5	limited	limit	VERB
gojar-5993	116	6	to	to	ADP
gojar-5993	116	7	100	100	NUM
gojar-5993	116	8	hours	hour	NOUN
gojar-5993	116	9	of	of	ADP
gojar-5993	116	10	labor	labor	NOUN
gojar-5993	116	11	and	and	CCONJ
gojar-5993	116	12	80	80	NUM
gojar-5993	116	13	units	unit	NOUN
gojar-5993	116	14	of	of	ADP
gojar-5993	116	15	material	material	NOUN
gojar-5993	116	16	.	.	PUNCT
gojar-5993	117	1	the	the	DET
gojar-5993	117	2	profit	profit	NOUN
gojar-5993	117	3	for	for	ADP
gojar-5993	117	4	producing	produce	VERB
gojar-5993	117	5	one	one	NUM
gojar-5993	117	6	unit	unit	NOUN
gojar-5993	117	7	of	of	ADP
gojar-5993	117	8	a	a	PRON
gojar-5993	117	9	is	be	AUX
gojar-5993	117	10	$	$	SYM
gojar-5993	117	11	50	50	NUM
gojar-5993	117	12	,	,	PUNCT
gojar-5993	117	13	and	and	CCONJ
gojar-5993	117	14	for	for	ADP
gojar-5993	117	15	b	b	NOUN
gojar-5993	117	16	,	,	PUNCT
gojar-5993	117	17	it	it	PRON
gojar-5993	117	18	's	be	AUX
gojar-5993	117	19	$	$	SYM
gojar-5993	117	20	40	40	NUM
gojar-5993	117	21	.	.	PUNCT
gojar-5993	118	1	the	the	DET
gojar-5993	118	2	problem	problem	NOUN
gojar-5993	118	3	is	be	AUX
gojar-5993	118	4	to	to	PART
gojar-5993	118	5	determine	determine	VERB
gojar-5993	118	6	the	the	DET
gojar-5993	118	7	production	production	NOUN
gojar-5993	118	8	quantities	quantity	NOUN
gojar-5993	118	9	𝒙𝟏	𝒙𝟏	NOUN
gojar-5993	118	10	(	(	PUNCT
gojar-5993	118	11	units	unit	NOUN
gojar-5993	118	12	of	of	ADP
gojar-5993	118	13	a	a	PRON
gojar-5993	118	14	)	)	PUNCT
gojar-5993	118	15	and	and	CCONJ
gojar-5993	118	16	𝒙𝟐	𝒙𝟐	PROPN
gojar-5993	118	17	(	(	PUNCT
gojar-5993	118	18	units	unit	NOUN
gojar-5993	118	19	of	of	ADP
gojar-5993	118	20	b	b	NOUN
gojar-5993	118	21	)	)	PUNCT
gojar-5993	118	22	to	to	PART
gojar-5993	118	23	maximize	maximize	VERB
gojar-5993	118	24	profit	profit	NOUN
gojar-5993	118	25	,	,	PUNCT
gojar-5993	118	26	subject	subject	ADJ
gojar-5993	118	27	to	to	ADP
gojar-5993	118	28	the	the	DET
gojar-5993	118	29	following	follow	VERB
gojar-5993	118	30	constraints	constraint	NOUN
gojar-5993	118	31	:	:	PUNCT
gojar-5993	118	32	labor	labor	NOUN
gojar-5993	118	33	constraint	constraint	NOUN
gojar-5993	118	34	:	:	PUNCT
gojar-5993	118	35	𝟐𝒙𝟏	𝟐𝒙𝟏	PUNCT
gojar-5993	119	1	+	+	CCONJ
gojar-5993	119	2	𝟏𝒙𝟐	𝟏𝒙𝟐	NUM
gojar-5993	119	3	≤	≤	NUM
gojar-5993	119	4	𝟏𝟎𝟎	𝟏𝟎𝟎	NUM
gojar-5993	119	5	,	,	PUNCT
gojar-5993	119	6	material	material	NOUN
gojar-5993	119	7	constraint	constraint	NOUN
gojar-5993	119	8	:	:	PUNCT
gojar-5993	119	9	𝟏𝒙𝟏	𝟏𝒙𝟏	PROPN
gojar-5993	120	1	+	+	CCONJ
gojar-5993	120	2	𝟐𝒙𝟐	𝟐𝒙𝟐	X
gojar-5993	120	3	≤	≤	NOUN
gojar-5993	120	4	𝟖𝟎.	𝟖𝟎.	CCONJ
gojar-5993	120	5	this	this	PRON
gojar-5993	120	6	is	be	AUX
gojar-5993	120	7	a	a	DET
gojar-5993	120	8	linear	linear	ADJ
gojar-5993	120	9	programming	programming	NOUN
gojar-5993	120	10	(	(	PUNCT
gojar-5993	120	11	lp	lp	NOUN
gojar-5993	120	12	)	)	PUNCT
gojar-5993	120	13	problem	problem	NOUN
gojar-5993	120	14	.	.	PUNCT
gojar-5993	121	1	however	however	ADV
gojar-5993	121	2	,	,	PUNCT
gojar-5993	121	3	if	if	SCONJ
gojar-5993	121	4	the	the	DET
gojar-5993	121	5	production	production	NOUN
gojar-5993	121	6	quantities	quantity	NOUN
gojar-5993	121	7	𝒙𝟏	𝒙𝟏	NOUN
gojar-5993	121	8	and	and	CCONJ
gojar-5993	121	9	𝒙𝟐	𝒙𝟐	NOUN
gojar-5993	121	10	must	must	AUX
gojar-5993	121	11	be	be	AUX
gojar-5993	121	12	integers	integer	NOUN
gojar-5993	121	13	(	(	PUNCT
gojar-5993	121	14	e.g.	e.g.	ADV
gojar-5993	121	15	,	,	PUNCT
gojar-5993	121	16	you	you	PRON
gojar-5993	121	17	can	can	AUX
gojar-5993	121	18	not	not	PART
gojar-5993	121	19	produce	produce	VERB
gojar-5993	121	20	fractional	fractional	ADJ
gojar-5993	121	21	units	unit	NOUN
gojar-5993	121	22	)	)	PUNCT
gojar-5993	121	23	,	,	PUNCT
gojar-5993	121	24	the	the	DET
gojar-5993	121	25	problem	problem	NOUN
gojar-5993	121	26	becomes	become	VERB
gojar-5993	121	27	a	a	DET
gojar-5993	121	28	mixed	mixed	ADJ
gojar-5993	121	29	-	-	PUNCT
gojar-5993	121	30	integer	integer	NOUN
gojar-5993	121	31	programming	programming	NOUN
gojar-5993	121	32	(	(	PUNCT
gojar-5993	121	33	mip	mip	PROPN
gojar-5993	121	34	)	)	PUNCT
gojar-5993	121	35	problem	problem	NOUN
gojar-5993	121	36	.	.	PUNCT
gojar-5993	122	1	rational	rational	ADJ
gojar-5993	122	2	polyhedron	polyhedron	NOUN
gojar-5993	122	3	and	and	CCONJ
gojar-5993	122	4	integer	integer	NOUN
gojar-5993	122	5	hull	hull	NOUN
gojar-5993	122	6			NOUN
gojar-5993	122	7	the	the	DET
gojar-5993	122	8	feasible	feasible	ADJ
gojar-5993	122	9	region	region	NOUN
gojar-5993	122	10	defined	define	VERB
gojar-5993	122	11	by	by	ADP
gojar-5993	122	12	the	the	DET
gojar-5993	122	13	constraints	constraint	NOUN
gojar-5993	122	14	is	be	AUX
gojar-5993	122	15	a	a	DET
gojar-5993	122	16	rational	rational	ADJ
gojar-5993	122	17	polyhedron	polyhedron	NOUN
gojar-5993	122	18	p	p	NOUN
gojar-5993	122	19	,	,	PUNCT
gojar-5993	122	20	containing	contain	VERB
gojar-5993	122	21	all	all	DET
gojar-5993	122	22	real	real	ADV
gojar-5993	122	23	-	-	PUNCT
gojar-5993	122	24	valued	value	VERB
gojar-5993	122	25	solutions	solution	NOUN
gojar-5993	122	26	that	that	PRON
gojar-5993	122	27	satisfy	satisfy	VERB
gojar-5993	122	28	the	the	DET
gojar-5993	122	29	constraints	constraint	NOUN
gojar-5993	122	30	.	.	PUNCT
gojar-5993	123	1			X
gojar-5993	124	1	the	the	DET
gojar-5993	124	2	integer	integer	PROPN
gojar-5993	124	3	hull	hull	NOUN
gojar-5993	124	4	𝑷𝑰	𝑷𝑰	PROPN
gojar-5993	124	5	is	be	AUX
gojar-5993	124	6	the	the	DET
gojar-5993	124	7	convex	convex	ADJ
gojar-5993	124	8	hull	hull	NOUN
gojar-5993	124	9	of	of	ADP
gojar-5993	124	10	all	all	DET
gojar-5993	124	11	integer	integer	NOUN
gojar-5993	124	12	solutions	solution	NOUN
gojar-5993	124	13	within	within	ADP
gojar-5993	124	14	p.	p.	NOUN
gojar-5993	124	15	it	it	PRON
gojar-5993	124	16	represents	represent	VERB
gojar-5993	124	17	the	the	DET
gojar-5993	124	18	feasible	feasible	ADJ
gojar-5993	124	19	region	region	NOUN
gojar-5993	124	20	for	for	ADP
gojar-5993	124	21	the	the	DET
gojar-5993	124	22	mip	mip	PROPN
gojar-5993	124	23	problem	problem	PROPN
gojar-5993	124	24	.	.	PUNCT
gojar-5993	125	1	boundedness	boundedness	NOUN
gojar-5993	125	2	analysis	analysis	NOUN
gojar-5993	125	3	1	1	NUM
gojar-5993	125	4	.	.	PUNCT
gojar-5993	125	5	boundedness	boundedness	NOUN
gojar-5993	125	6	of	of	ADP
gojar-5993	125	7	p	p	X
gojar-5993	125	8	:	:	PUNCT
gojar-5993	125	9	the	the	DET
gojar-5993	125	10	polyhedron	polyhedron	NOUN
gojar-5993	125	11	p	p	NOUN
gojar-5993	125	12	is	be	AUX
gojar-5993	125	13	bounded	bound	VERB
gojar-5993	125	14	since	since	SCONJ
gojar-5993	125	15	it	it	PRON
gojar-5993	125	16	is	be	AUX
gojar-5993	125	17	enclosed	enclose	VERB
gojar-5993	125	18	by	by	ADP
gojar-5993	125	19	the	the	DET
gojar-5993	125	20	constraints	constraint	NOUN
gojar-5993	125	21	𝟐𝒙𝟏	𝟐𝒙𝟏	PUNCT
gojar-5993	126	1	+	+	CCONJ
gojar-5993	126	2	𝟏𝒙𝟐	𝟏𝒙𝟐	NUM
gojar-5993	126	3	≤	≤	NOUN
gojar-5993	126	4	𝟏𝟎𝟎	𝟏𝟎𝟎	NUM
gojar-5993	126	5	and	and	CCONJ
gojar-5993	126	6	𝒙𝟏	𝒙𝟏	NOUN
gojar-5993	126	7	+	+	CCONJ
gojar-5993	126	8	𝟐𝒙𝟐	𝟐𝒙𝟐	X
gojar-5993	126	9	≤	≤	NUM
gojar-5993	126	10	𝟖𝟎	𝟖𝟎	NUM
gojar-5993	126	11	,	,	PUNCT
gojar-5993	126	12	which	which	PRON
gojar-5993	126	13	intersect	intersect	VERB
gojar-5993	126	14	in	in	ADP
gojar-5993	126	15	the	the	DET
gojar-5993	126	16	positive	positive	ADJ
gojar-5993	126	17	quadrant	quadrant	NOUN
gojar-5993	126	18	.	.	PUNCT
gojar-5993	127	1	2	2	X
gojar-5993	127	2	.	.	X
gojar-5993	127	3	boundedness	boundedness	NOUN
gojar-5993	127	4	of	of	ADP
gojar-5993	127	5	𝑷𝑰	𝑷𝑰	PROPN
gojar-5993	127	6	:	:	PUNCT
gojar-5993	127	7	the	the	DET
gojar-5993	127	8	integer	integer	PROPN
gojar-5993	127	9	hull	hull	NOUN
gojar-5993	127	10	𝑷𝑰	𝑷𝑰	PROPN
gojar-5993	127	11	,	,	PUNCT
gojar-5993	127	12	being	be	AUX
gojar-5993	127	13	a	a	DET
gojar-5993	127	14	subset	subset	NOUN
gojar-5993	127	15	of	of	ADP
gojar-5993	127	16	p	p	PRON
gojar-5993	127	17	,	,	PUNCT
gojar-5993	127	18	is	be	AUX
gojar-5993	127	19	also	also	ADV
gojar-5993	127	20	bounded	bound	VERB
gojar-5993	127	21	.	.	PUNCT
gojar-5993	128	1	this	this	PRON
gojar-5993	128	2	follows	follow	VERB
gojar-5993	128	3	from	from	ADP
gojar-5993	128	4	the	the	DET
gojar-5993	128	5	equivalence	equivalence	NOUN
gojar-5993	128	6	of	of	ADP
gojar-5993	128	7	boundedness	boundedness	NOUN
gojar-5993	128	8	conditions	condition	NOUN
gojar-5993	128	9	:	:	PUNCT
gojar-5993	128	10	if	if	SCONJ
gojar-5993	128	11	𝒎𝒂𝒙	𝒎𝒂𝒙	NOUN
gojar-5993	128	12	{	{	PUNCT
gojar-5993	128	13	𝒄𝑻𝒙	𝒄𝑻𝒙	NOUN
gojar-5993	128	14	:	:	PUNCT
gojar-5993	128	15	𝒙	𝒙	PROPN
gojar-5993	128	16	∈	∈	PROPN
gojar-5993	128	17	𝑷	𝑷	PROPN
gojar-5993	128	18	}	}	PUNCT
gojar-5993	128	19	is	be	AUX
gojar-5993	128	20	bounded	bound	VERB
gojar-5993	128	21	,	,	PUNCT
gojar-5993	128	22	then	then	ADV
gojar-5993	128	23	𝒎𝒂𝒙	𝒎𝒂𝒙	VERB
gojar-5993	128	24	{	{	PUNCT
gojar-5993	128	25	𝒄𝑻𝒙	𝒄𝑻𝒙	NOUN
gojar-5993	128	26	:	:	PUNCT
gojar-5993	128	27	𝒙	𝒙	PROPN
gojar-5993	128	28	∈	∈	PROPN
gojar-5993	128	29	𝑷𝑰	𝑷𝑰	PROPN
gojar-5993	128	30	,	,	PUNCT
gojar-5993	128	31	}	}	PUNCT
gojar-5993	128	32	is	be	AUX
gojar-5993	128	33	bounded	bound	VERB
gojar-5993	128	34	.	.	PUNCT
gojar-5993	129	1	simplifying	simplify	VERB
gojar-5993	129	2	the	the	DET
gojar-5993	129	3	analysis	analysis	NOUN
gojar-5993	129	4	instead	instead	ADV
gojar-5993	129	5	of	of	ADP
gojar-5993	129	6	analyzing	analyze	VERB
gojar-5993	129	7	the	the	DET
gojar-5993	129	8	mip	mip	PROPN
gojar-5993	129	9	problem	problem	NOUN
gojar-5993	129	10	directly	directly	ADV
gojar-5993	129	11	,	,	PUNCT
gojar-5993	129	12	the	the	DET
gojar-5993	129	13	equivalence	equivalence	NOUN
gojar-5993	129	14	of	of	ADP
gojar-5993	129	15	boundedness	boundedness	NOUN
gojar-5993	129	16	conditions	condition	NOUN
gojar-5993	129	17	allows	allow	VERB
gojar-5993	129	18	us	we	PRON
gojar-5993	129	19	to	to	PART
gojar-5993	129	20	focus	focus	VERB
gojar-5993	129	21	on	on	ADP
gojar-5993	129	22	the	the	DET
gojar-5993	129	23	polyhedron	polyhedron	NOUN
gojar-5993	129	24	p	p	NOUN
gojar-5993	129	25	to	to	PART
gojar-5993	129	26	verify	verify	VERB
gojar-5993	129	27	boundedness	boundedness	NOUN
gojar-5993	129	28	.	.	PUNCT
gojar-5993	130	1	once	once	ADV
gojar-5993	130	2	p	p	NOUN
gojar-5993	130	3	is	be	AUX
gojar-5993	130	4	confirmed	confirm	VERB
gojar-5993	130	5	to	to	PART
gojar-5993	130	6	be	be	AUX
gojar-5993	130	7	bounded	bound	VERB
gojar-5993	130	8	,	,	PUNCT
gojar-5993	130	9	we	we	PRON
gojar-5993	130	10	can	can	AUX
gojar-5993	130	11	conclude	conclude	VERB
gojar-5993	130	12	that	that	SCONJ
gojar-5993	130	13	𝑷𝑰	𝑷𝑰	PROPN
gojar-5993	130	14	,	,	PUNCT
gojar-5993	130	15	global	global	ADJ
gojar-5993	130	16	online	online	ADJ
gojar-5993	130	17	journal	journal	PROPN
gojar-5993	130	18	of	of	ADP
gojar-5993	130	19	academic	academic	ADJ
gojar-5993	130	20	research	research	NOUN
gojar-5993	130	21	(	(	PUNCT
gojar-5993	130	22	gojar	gojar	NOUN
gojar-5993	130	23	)	)	PUNCT
gojar-5993	130	24	,	,	PUNCT
gojar-5993	130	25	vol	vol	NOUN
gojar-5993	130	26	.	.	PROPN
gojar-5993	131	1	4	4	NUM
gojar-5993	131	2	,	,	PUNCT
gojar-5993	131	3	no	no	INTJ
gojar-5993	131	4	.	.	NOUN
gojar-5993	131	5	1	1	NUM
gojar-5993	131	6	february	february	NOUN
gojar-5993	131	7	2025	2025	NUM
gojar-5993	131	8	69	69	NUM
gojar-5993	131	9	is	be	AUX
gojar-5993	131	10	bounded	bound	VERB
gojar-5993	131	11	,	,	PUNCT
gojar-5993	131	12	avoiding	avoid	VERB
gojar-5993	131	13	the	the	DET
gojar-5993	131	14	need	need	NOUN
gojar-5993	131	15	for	for	ADP
gojar-5993	131	16	exhaustive	exhaustive	ADJ
gojar-5993	131	17	checks	check	NOUN
gojar-5993	131	18	over	over	ADP
gojar-5993	131	19	all	all	DET
gojar-5993	131	20	integer	integer	NOUN
gojar-5993	131	21	solutions	solution	NOUN
gojar-5993	131	22	.	.	PUNCT
gojar-5993	132	1	solving	solve	VERB
gojar-5993	132	2	the	the	DET
gojar-5993	132	3	problem	problem	NOUN
gojar-5993	132	4	the	the	DET
gojar-5993	132	5	integer	integer	NOUN
gojar-5993	132	6	solutions	solution	NOUN
gojar-5993	132	7	can	can	AUX
gojar-5993	132	8	then	then	ADV
gojar-5993	132	9	be	be	AUX
gojar-5993	132	10	obtained	obtain	VERB
gojar-5993	132	11	by	by	ADP
gojar-5993	132	12	applying	apply	VERB
gojar-5993	132	13	integer	integer	NOUN
gojar-5993	132	14	programming	programming	NOUN
gojar-5993	132	15	techniques	technique	NOUN
gojar-5993	132	16	,	,	PUNCT
gojar-5993	132	17	such	such	ADJ
gojar-5993	132	18	as	as	ADP
gojar-5993	132	19	branch	branch	NOUN
gojar-5993	132	20	-	-	PUNCT
gojar-5993	132	21	and	and	CCONJ
gojar-5993	132	22	-	-	PUNCT
gojar-5993	132	23	bound	bind	VERB
gojar-5993	132	24	or	or	CCONJ
gojar-5993	132	25	cutting	cut	VERB
gojar-5993	132	26	planes	plane	NOUN
gojar-5993	132	27	,	,	PUNCT
gojar-5993	132	28	which	which	PRON
gojar-5993	132	29	operate	operate	VERB
gojar-5993	132	30	within	within	ADP
gojar-5993	132	31	the	the	DET
gojar-5993	132	32	bounded	bounded	ADJ
gojar-5993	132	33	integer	integer	NOUN
gojar-5993	132	34	hull	hull	PROPN
gojar-5993	132	35	𝑷𝑰	𝑷𝑰	PROPN
gojar-5993	132	36	,	,	PUNCT
gojar-5993	132	37	.	.	PUNCT
gojar-5993	133	1	thus	thus	ADV
gojar-5993	133	2	,	,	PUNCT
gojar-5993	133	3	reducing	reduce	VERB
gojar-5993	133	4	the	the	DET
gojar-5993	133	5	boundedness	boundedness	NOUN
gojar-5993	133	6	check	check	NOUN
gojar-5993	133	7	to	to	ADP
gojar-5993	133	8	p	p	PRON
gojar-5993	133	9	,	,	PUNCT
gojar-5993	133	10	the	the	DET
gojar-5993	133	11	analysis	analysis	NOUN
gojar-5993	133	12	simplifies	simplifie	NOUN
gojar-5993	133	13	significantly	significantly	ADV
gojar-5993	133	14	,	,	PUNCT
gojar-5993	133	15	saving	save	VERB
gojar-5993	133	16	computational	computational	ADJ
gojar-5993	133	17	effort	effort	NOUN
gojar-5993	133	18	and	and	CCONJ
gojar-5993	133	19	making	make	VERB
gojar-5993	133	20	the	the	DET
gojar-5993	133	21	problem	problem	NOUN
gojar-5993	133	22	more	more	ADV
gojar-5993	133	23	tractable	tractable	ADJ
gojar-5993	133	24	.	.	PUNCT
gojar-5993	133	25	ii	ii	NOUN
gojar-5993	133	26	)	)	PUNCT
gojar-5993	133	27	computational	computational	ADJ
gojar-5993	133	28	geometry	geometry	NOUN
gojar-5993	133	29	:	:	PUNCT
gojar-5993	133	30	solution	solution	NOUN
gojar-5993	133	31	size	size	NOUN
gojar-5993	133	32	bounds	bound	VERB
gojar-5993	133	33	assist	assist	VERB
gojar-5993	133	34	in	in	ADP
gojar-5993	133	35	designing	design	VERB
gojar-5993	133	36	efficient	efficient	ADJ
gojar-5993	133	37	algorithms	algorithm	NOUN
gojar-5993	133	38	for	for	ADP
gojar-5993	133	39	convex	convex	NOUN
gojar-5993	133	40	hull	hull	NOUN
gojar-5993	133	41	and	and	CCONJ
gojar-5993	133	42	vertex	vertex	NOUN
gojar-5993	133	43	enumeration	enumeration	NOUN
gojar-5993	133	44	.	.	PUNCT
gojar-5993	134	1	application	application	NOUN
gojar-5993	134	2	context	context	NOUN
gojar-5993	134	3	consider	consider	VERB
gojar-5993	134	4	the	the	DET
gojar-5993	134	5	problem	problem	NOUN
gojar-5993	134	6	of	of	ADP
gojar-5993	134	7	computing	compute	VERB
gojar-5993	134	8	the	the	DET
gojar-5993	134	9	convex	convex	PROPN
gojar-5993	134	10	hull	hull	NOUN
gojar-5993	134	11	of	of	ADP
gojar-5993	134	12	a	a	DET
gojar-5993	134	13	set	set	NOUN
gojar-5993	134	14	of	of	ADP
gojar-5993	134	15	points	point	NOUN
gojar-5993	134	16	in	in	ADP
gojar-5993	134	17	ℝ𝒏.	ℝ𝒏.	PROPN
gojar-5993	134	18	convex	convex	NOUN
gojar-5993	134	19	hull	hull	NOUN
gojar-5993	134	20	algorithms	algorithm	NOUN
gojar-5993	134	21	,	,	PUNCT
gojar-5993	134	22	such	such	ADJ
gojar-5993	134	23	as	as	ADP
gojar-5993	134	24	quickhull	quickhull	NOUN
gojar-5993	134	25	or	or	CCONJ
gojar-5993	134	26	graham	graham	PROPN
gojar-5993	134	27	's	's	PART
gojar-5993	134	28	scan	scan	NOUN
gojar-5993	134	29	,	,	PUNCT
gojar-5993	134	30	rely	rely	VERB
gojar-5993	134	31	on	on	ADP
gojar-5993	134	32	numerical	numerical	ADJ
gojar-5993	134	33	representations	representation	NOUN
gojar-5993	134	34	of	of	ADP
gojar-5993	134	35	the	the	DET
gojar-5993	134	36	points	point	NOUN
gojar-5993	134	37	and	and	CCONJ
gojar-5993	134	38	may	may	AUX
gojar-5993	134	39	involve	involve	VERB
gojar-5993	134	40	large	large	ADJ
gojar-5993	134	41	computations	computation	NOUN
gojar-5993	134	42	when	when	SCONJ
gojar-5993	134	43	the	the	DET
gojar-5993	134	44	coordinates	coordinate	NOUN
gojar-5993	134	45	of	of	ADP
gojar-5993	134	46	the	the	DET
gojar-5993	134	47	points	point	NOUN
gojar-5993	134	48	have	have	VERB
gojar-5993	134	49	a	a	DET
gojar-5993	134	50	high	high	ADJ
gojar-5993	134	51	bit	bit	NOUN
gojar-5993	134	52	-	-	PUNCT
gojar-5993	134	53	length	length	NOUN
gojar-5993	134	54	.	.	PUNCT
gojar-5993	135	1	efficient	efficient	ADJ
gojar-5993	135	2	algorithms	algorithm	NOUN
gojar-5993	135	3	benefit	benefit	VERB
gojar-5993	135	4	from	from	ADP
gojar-5993	135	5	guarantees	guarantee	NOUN
gojar-5993	135	6	about	about	ADP
gojar-5993	135	7	the	the	DET
gojar-5993	135	8	size	size	NOUN
gojar-5993	135	9	of	of	ADP
gojar-5993	135	10	intermediate	intermediate	ADJ
gojar-5993	135	11	and	and	CCONJ
gojar-5993	135	12	final	final	ADJ
gojar-5993	135	13	solutions	solution	NOUN
gojar-5993	135	14	,	,	PUNCT
gojar-5993	135	15	which	which	PRON
gojar-5993	135	16	directly	directly	ADV
gojar-5993	135	17	impacts	impact	VERB
gojar-5993	135	18	computation	computation	NOUN
gojar-5993	135	19	time	time	NOUN
gojar-5993	135	20	and	and	CCONJ
gojar-5993	135	21	memory	memory	NOUN
gojar-5993	135	22	usage	usage	NOUN
gojar-5993	135	23	.	.	PUNCT
gojar-5993	136	1	problem	problem	NOUN
gojar-5993	136	2	setup	setup	NOUN
gojar-5993	136	3	let	let	VERB
gojar-5993	136	4	𝑷	𝑷	PROPN
gojar-5993	136	5	=	=	PRON
gojar-5993	136	6	{	{	PUNCT
gojar-5993	136	7	𝒙	𝒙	PROPN
gojar-5993	136	8	∈	∈	PROPN
gojar-5993	136	9	ℝ𝒏	ℝ𝒏	NOUN
gojar-5993	136	10	:	:	PUNCT
gojar-5993	136	11	𝑨𝒙	𝑨𝒙	PROPN
gojar-5993	136	12	≤	≤	NOUN
gojar-5993	136	13	𝒃	𝒃	AUX
gojar-5993	136	14	}	}	PUNCT
gojar-5993	136	15	be	be	AUX
gojar-5993	136	16	a	a	DET
gojar-5993	136	17	rational	rational	ADJ
gojar-5993	136	18	polyhedron	polyhedron	NOUN
gojar-5993	136	19	defined	define	VERB
gojar-5993	136	20	by	by	ADP
gojar-5993	136	21	m	m	PROPN
gojar-5993	136	22	linear	linear	PROPN
gojar-5993	136	23	inequalities	inequality	NOUN
gojar-5993	136	24	,	,	PUNCT
gojar-5993	136	25	where	where	SCONJ
gojar-5993	136	26	𝑨	𝑨	PROPN
gojar-5993	136	27	∈	∈	PROPN
gojar-5993	136	28	ℚ𝒎×𝒏.	ℚ𝒎×𝒏.	NOUN
gojar-5993	136	29	the	the	DET
gojar-5993	136	30	goal	goal	NOUN
gojar-5993	136	31	is	be	AUX
gojar-5993	136	32	to	to	PART
gojar-5993	136	33	compute	compute	VERB
gojar-5993	136	34	the	the	DET
gojar-5993	136	35	convex	convex	PROPN
gojar-5993	136	36	hull	hull	NOUN
gojar-5993	136	37	of	of	ADP
gojar-5993	136	38	the	the	DET
gojar-5993	136	39	integer	integer	NOUN
gojar-5993	136	40	points	point	NOUN
gojar-5993	136	41	in	in	ADP
gojar-5993	136	42	p	p	NOUN
gojar-5993	136	43	,	,	PUNCT
gojar-5993	136	44	denoted	denote	VERB
gojar-5993	136	45	𝒄𝒐𝒏𝒗(𝑷𝑰	𝒄𝒐𝒏𝒗(𝑷𝑰	NOUN
gojar-5993	136	46	)	)	PUNCT
gojar-5993	136	47	.	.	PUNCT
gojar-5993	137	1	solution	solution	NOUN
gojar-5993	137	2	size	size	NOUN
gojar-5993	137	3	bounds	bound	VERB
gojar-5993	137	4	from	from	ADP
gojar-5993	137	5	theoretical	theoretical	ADJ
gojar-5993	137	6	results	result	NOUN
gojar-5993	137	7	,	,	PUNCT
gojar-5993	137	8	if	if	SCONJ
gojar-5993	137	9	an	an	DET
gojar-5993	137	10	optimal	optimal	ADJ
gojar-5993	137	11	solution	solution	NOUN
gojar-5993	137	12	x	x	X
gojar-5993	137	13	to	to	ADP
gojar-5993	137	14	a	a	DET
gojar-5993	137	15	linear	linear	ADJ
gojar-5993	137	16	program	program	NOUN
gojar-5993	137	17	over	over	ADP
gojar-5993	137	18	p	p	NOUN
gojar-5993	137	19	exists	exist	VERB
gojar-5993	137	20	,	,	PUNCT
gojar-5993	137	21	its	its	PRON
gojar-5993	137	22	size	size	NOUN
gojar-5993	137	23	is	be	AUX
gojar-5993	137	24	bounded	bound	VERB
gojar-5993	137	25	as	as	ADP
gojar-5993	137	26	:	:	PUNCT
gojar-5993	137	27	𝒔𝒊𝒛𝒆(𝒙	𝒔𝒊𝒛𝒆(𝒙	NOUN
gojar-5993	137	28	)	)	PUNCT
gojar-5993	137	29	≤	≤	NUM
gojar-5993	137	30	𝟒𝒏(𝒔𝒊𝒛𝒆(𝑨	𝟒𝒏(𝒔𝒊𝒛𝒆(𝑨	NOUN
gojar-5993	137	31	)	)	PUNCT
gojar-5993	137	32	+	+	CCONJ
gojar-5993	138	1	𝒔𝒊𝒛𝒆(𝒃	𝒔𝒊𝒛𝒆(𝒃	NOUN
gojar-5993	138	2	)	)	PUNCT
gojar-5993	138	3	)	)	PUNCT
gojar-5993	138	4	.	.	PUNCT
gojar-5993	139	1	this	this	PRON
gojar-5993	139	2	means	mean	VERB
gojar-5993	139	3	each	each	DET
gojar-5993	139	4	vertex	vertex	NOUN
gojar-5993	139	5	of	of	ADP
gojar-5993	139	6	the	the	DET
gojar-5993	139	7	convex	convex	PROPN
gojar-5993	139	8	hull	hull	NOUN
gojar-5993	139	9	𝒄𝒐𝒏𝒗(𝑷𝑰)has	𝒄𝒐𝒏𝒗(𝑷𝑰)has	PROPN
gojar-5993	139	10	coordinates	coordinate	NOUN
gojar-5993	139	11	with	with	ADP
gojar-5993	139	12	a	a	DET
gojar-5993	139	13	bit	bit	NOUN
gojar-5993	139	14	-	-	PUNCT
gojar-5993	139	15	length	length	NOUN
gojar-5993	139	16	constrained	constrain	VERB
gojar-5993	139	17	by	by	ADP
gojar-5993	139	18	this	this	DET
gojar-5993	139	19	bound	bind	VERB
gojar-5993	139	20	.	.	PUNCT
gojar-5993	140	1	application	application	NOUN
gojar-5993	140	2	to	to	PART
gojar-5993	140	3	convex	convex	VERB
gojar-5993	140	4	hull	hull	NOUN
gojar-5993	140	5	algorithms	algorithm	NOUN
gojar-5993	140	6	a.	a.	PROPN
gojar-5993	140	7	numerical	numerical	PROPN
gojar-5993	140	8	stability	stability	PROPN
gojar-5993	140	9	:	:	PUNCT
gojar-5993	140	10	o	o	NOUN
gojar-5993	140	11	algorithms	algorithm	NOUN
gojar-5993	140	12	like	like	ADP
gojar-5993	140	13	quickhull	quickhull	NOUN
gojar-5993	140	14	require	require	NOUN
gojar-5993	140	15	operations	operation	NOUN
gojar-5993	140	16	on	on	ADP
gojar-5993	140	17	vertex	vertex	NOUN
gojar-5993	140	18	coordinates	coordinate	NOUN
gojar-5993	140	19	,	,	PUNCT
gojar-5993	140	20	such	such	ADJ
gojar-5993	140	21	as	as	ADP
gojar-5993	140	22	comparing	compare	VERB
gojar-5993	140	23	slopes	slope	NOUN
gojar-5993	140	24	or	or	CCONJ
gojar-5993	140	25	calculating	calculate	VERB
gojar-5993	140	26	determinants	determinant	NOUN
gojar-5993	140	27	.	.	PUNCT
gojar-5993	141	1	knowing	know	VERB
gojar-5993	141	2	the	the	DET
gojar-5993	141	3	bounds	bound	NOUN
gojar-5993	141	4	on	on	ADP
gojar-5993	141	5	the	the	DET
gojar-5993	141	6	size	size	NOUN
gojar-5993	141	7	of	of	ADP
gojar-5993	141	8	x	x	PUNCT
gojar-5993	141	9	ensures	ensure	VERB
gojar-5993	141	10	that	that	SCONJ
gojar-5993	141	11	these	these	DET
gojar-5993	141	12	operations	operation	NOUN
gojar-5993	141	13	remain	remain	VERB
gojar-5993	141	14	numerically	numerically	ADV
gojar-5993	141	15	stable	stable	ADJ
gojar-5993	141	16	and	and	CCONJ
gojar-5993	141	17	feasible	feasible	ADJ
gojar-5993	141	18	on	on	ADP
gojar-5993	141	19	finite	finite	ADJ
gojar-5993	141	20	-	-	ADJ
gojar-5993	141	21	precision	precision	NOUN
gojar-5993	141	22	systems	system	NOUN
gojar-5993	141	23	.	.	PUNCT
gojar-5993	142	1	global	global	ADJ
gojar-5993	142	2	online	online	PROPN
gojar-5993	142	3	journal	journal	PROPN
gojar-5993	142	4	of	of	ADP
gojar-5993	142	5	academic	academic	ADJ
gojar-5993	142	6	research	research	NOUN
gojar-5993	142	7	(	(	PUNCT
gojar-5993	142	8	gojar	gojar	NOUN
gojar-5993	142	9	)	)	PUNCT
gojar-5993	142	10	,	,	PUNCT
gojar-5993	142	11	vol	vol	NOUN
gojar-5993	142	12	.	.	PROPN
gojar-5993	142	13	4	4	NUM
gojar-5993	142	14	,	,	PUNCT
gojar-5993	142	15	no	no	INTJ
gojar-5993	142	16	.	.	NOUN
gojar-5993	142	17	1	1	NUM
gojar-5993	142	18	february	february	NOUN
gojar-5993	142	19	2025	2025	NUM
gojar-5993	142	20	70	70	NUM
gojar-5993	142	21	b.	b.	PROPN
gojar-5993	142	22	efficient	efficient	ADJ
gojar-5993	142	23	data	datum	NOUN
gojar-5993	142	24	structures	structure	NOUN
gojar-5993	142	25	:	:	PUNCT
gojar-5993	142	26	o	o	NOUN
gojar-5993	142	27	solution	solution	NOUN
gojar-5993	142	28	size	size	NOUN
gojar-5993	142	29	bounds	bound	NOUN
gojar-5993	142	30	guide	guide	VERB
gojar-5993	142	31	the	the	DET
gojar-5993	142	32	choice	choice	NOUN
gojar-5993	142	33	of	of	ADP
gojar-5993	142	34	data	datum	NOUN
gojar-5993	142	35	structures	structure	NOUN
gojar-5993	142	36	.	.	PUNCT
gojar-5993	143	1	for	for	ADP
gojar-5993	143	2	example	example	NOUN
gojar-5993	143	3	,	,	PUNCT
gojar-5993	143	4	if	if	SCONJ
gojar-5993	143	5	the	the	DET
gojar-5993	143	6	bound	bind	VERB
gojar-5993	143	7	indicates	indicate	VERB
gojar-5993	143	8	small	small	ADJ
gojar-5993	143	9	bit	bit	NOUN
gojar-5993	143	10	-	-	PUNCT
gojar-5993	143	11	lengths	length	NOUN
gojar-5993	143	12	,	,	PUNCT
gojar-5993	143	13	lightweight	lightweight	ADJ
gojar-5993	143	14	data	data	NOUN
gojar-5993	143	15	structures	structure	NOUN
gojar-5993	143	16	(	(	PUNCT
gojar-5993	143	17	e.g.	e.g.	ADV
gojar-5993	143	18	,	,	PUNCT
gojar-5993	143	19	arrays	array	VERB
gojar-5993	143	20	with	with	ADP
gojar-5993	143	21	fixed	fix	VERB
gojar-5993	143	22	-	-	PUNCT
gojar-5993	143	23	width	width	NOUN
gojar-5993	143	24	integers	integer	NOUN
gojar-5993	143	25	)	)	PUNCT
gojar-5993	143	26	can	can	AUX
gojar-5993	143	27	be	be	AUX
gojar-5993	143	28	used	use	VERB
gojar-5993	143	29	,	,	PUNCT
gojar-5993	143	30	reducing	reduce	VERB
gojar-5993	143	31	memory	memory	NOUN
gojar-5993	143	32	overhead	overhead	NOUN
gojar-5993	143	33	.	.	PUNCT
gojar-5993	144	1	c.	c.	PROPN
gojar-5993	144	2	algorithm	algorithm	PROPN
gojar-5993	144	3	design	design	NOUN
gojar-5993	144	4	:	:	PUNCT
gojar-5993	144	5	o	o	NOUN
gojar-5993	145	1	when	when	SCONJ
gojar-5993	145	2	enumerating	enumerate	VERB
gojar-5993	145	3	vertices	vertex	NOUN
gojar-5993	145	4	of	of	ADP
gojar-5993	145	5	𝒄𝒐𝒏𝒗(𝑷𝑰	𝒄𝒐𝒏𝒗(𝑷𝑰	NOUN
gojar-5993	145	6	)	)	PUNCT
gojar-5993	145	7	,	,	PUNCT
gojar-5993	145	8	solution	solution	NOUN
gojar-5993	145	9	size	size	NOUN
gojar-5993	145	10	bounds	bound	NOUN
gojar-5993	145	11	restrict	restrict	VERB
gojar-5993	145	12	the	the	DET
gojar-5993	145	13	search	search	NOUN
gojar-5993	145	14	space	space	NOUN
gojar-5993	145	15	,	,	PUNCT
gojar-5993	145	16	enabling	enable	VERB
gojar-5993	145	17	pruning	pruning	NOUN
gojar-5993	145	18	strategies	strategy	NOUN
gojar-5993	145	19	in	in	ADP
gojar-5993	145	20	branch	branch	NOUN
gojar-5993	145	21	-	-	PUNCT
gojar-5993	145	22	and	and	CCONJ
gojar-5993	145	23	-	-	PUNCT
gojar-5993	145	24	bound	bind	VERB
gojar-5993	145	25	algorithms	algorithm	NOUN
gojar-5993	145	26	.	.	PUNCT
gojar-5993	146	1	for	for	ADP
gojar-5993	146	2	example	example	NOUN
gojar-5993	146	3	,	,	PUNCT
gojar-5993	146	4	if	if	SCONJ
gojar-5993	146	5	a	a	DET
gojar-5993	146	6	candidate	candidate	NOUN
gojar-5993	146	7	vertex	vertex	NOUN
gojar-5993	146	8	exceeds	exceed	VERB
gojar-5993	146	9	the	the	DET
gojar-5993	146	10	size	size	NOUN
gojar-5993	146	11	bounds	bound	NOUN
gojar-5993	146	12	,	,	PUNCT
gojar-5993	146	13	it	it	PRON
gojar-5993	146	14	can	can	AUX
gojar-5993	146	15	be	be	AUX
gojar-5993	146	16	discarded	discard	VERB
gojar-5993	146	17	without	without	ADP
gojar-5993	146	18	further	further	ADJ
gojar-5993	146	19	computation	computation	NOUN
gojar-5993	146	20	.	.	PUNCT
gojar-5993	147	1	application	application	NOUN
gojar-5993	147	2	in	in	ADP
gojar-5993	147	3	ℝ𝟐	ℝ𝟐	PROPN
gojar-5993	147	4	suppose	suppose	VERB
gojar-5993	147	5	p	p	X
gojar-5993	147	6	is	be	AUX
gojar-5993	147	7	a	a	DET
gojar-5993	147	8	polygon	polygon	NOUN
gojar-5993	147	9	defined	define	VERB
gojar-5993	147	10	by	by	ADP
gojar-5993	147	11	:	:	PUNCT
gojar-5993	147	12	𝑷	𝑷	PROPN
gojar-5993	147	13	=	=	PUNCT
gojar-5993	147	14	{	{	PUNCT
gojar-5993	147	15	𝒙	𝒙	PROPN
gojar-5993	147	16	∈	∈	PROPN
gojar-5993	147	17	ℝ𝟐	ℝ𝟐	NOUN
gojar-5993	147	18	:	:	PUNCT
gojar-5993	147	19	𝟐𝒙𝟏	𝟐𝒙𝟏	X
gojar-5993	148	1	+	+	CCONJ
gojar-5993	148	2	𝒙𝟐	𝒙𝟐	NOUN
gojar-5993	148	3	≤	≤	NUM
gojar-5993	148	4	𝟏𝟎	𝟏𝟎	NUM
gojar-5993	148	5	,	,	PUNCT
gojar-5993	148	6	 	 	SPACE
gojar-5993	148	7	𝒙𝟏	𝒙𝟏	NOUN
gojar-5993	148	8	+	+	CCONJ
gojar-5993	148	9	𝟑𝒙𝟐	𝟑𝒙𝟐	X
gojar-5993	148	10	≤	≤	NUM
gojar-5993	148	11	𝟏𝟓	𝟏𝟓	NUM
gojar-5993	148	12	,	,	PUNCT
gojar-5993	148	13	 	 	SPACE
gojar-5993	148	14	𝒙𝟏	𝒙𝟏	NOUN
gojar-5993	148	15	,	,	PUNCT
gojar-5993	148	16	𝒙𝟐	𝒙𝟐	NOUN
gojar-5993	148	17	≥	≥	NOUN
gojar-5993	148	18	𝟎	𝟎	NUM
gojar-5993	148	19	}	}	PUNCT
gojar-5993	148	20	.	.	PUNCT
gojar-5993	149	1	the	the	DET
gojar-5993	149	2	integer	integer	NOUN
gojar-5993	149	3	points	point	NOUN
gojar-5993	149	4	in	in	ADP
gojar-5993	149	5	p	p	NOUN
gojar-5993	149	6	are	be	AUX
gojar-5993	149	7	(	(	PUNCT
gojar-5993	149	8	𝟎	𝟎	PROPN
gojar-5993	149	9	,	,	PUNCT
gojar-5993	149	10	𝟎	𝟎	NUM
gojar-5993	149	11	)	)	PUNCT
gojar-5993	149	12	,	,	PUNCT
gojar-5993	149	13	(	(	PUNCT
gojar-5993	149	14	𝟏	𝟏	NUM
gojar-5993	149	15	,	,	PUNCT
gojar-5993	149	16	𝟎	𝟎	NUM
gojar-5993	149	17	)	)	PUNCT
gojar-5993	149	18	,	,	PUNCT
gojar-5993	149	19	(	(	PUNCT
gojar-5993	149	20	𝟐	𝟐	NUM
gojar-5993	149	21	,	,	PUNCT
gojar-5993	149	22	𝟎	𝟎	NUM
gojar-5993	149	23	)	)	PUNCT
gojar-5993	149	24	,	,	PUNCT
gojar-5993	149	25	…	…	PUNCT
gojar-5993	149	26	,	,	PUNCT
gojar-5993	149	27	(	(	PUNCT
gojar-5993	149	28	𝟒	𝟒	NUM
gojar-5993	149	29	,	,	PUNCT
gojar-5993	149	30	𝟑	𝟑	NUM
gojar-5993	149	31	)	)	PUNCT
gojar-5993	149	32	.	.	PUNCT
gojar-5993	150	1			PUNCT
gojar-5993	151	1	the	the	DET
gojar-5993	151	2	convex	convex	PROPN
gojar-5993	151	3	hull	hull	NOUN
gojar-5993	151	4	of	of	ADP
gojar-5993	151	5	these	these	DET
gojar-5993	151	6	points	point	NOUN
gojar-5993	151	7	forms	form	NOUN
gojar-5993	151	8	a	a	DET
gojar-5993	151	9	polygon	polygon	NOUN
gojar-5993	151	10	whose	whose	DET
gojar-5993	151	11	vertices	vertex	NOUN
gojar-5993	151	12	are	be	AUX
gojar-5993	151	13	subsets	subset	NOUN
gojar-5993	151	14	of	of	ADP
gojar-5993	151	15	the	the	DET
gojar-5993	151	16	integer	integer	NOUN
gojar-5993	151	17	points	point	NOUN
gojar-5993	151	18	.	.	PUNCT
gojar-5993	152	1			NOUN
gojar-5993	152	2	using	use	VERB
gojar-5993	152	3	the	the	DET
gojar-5993	152	4	size	size	NOUN
gojar-5993	152	5	bounds	bound	NOUN
gojar-5993	152	6	,	,	PUNCT
gojar-5993	152	7	we	we	PRON
gojar-5993	152	8	confirm	confirm	VERB
gojar-5993	152	9	that	that	SCONJ
gojar-5993	152	10	all	all	DET
gojar-5993	152	11	integer	integer	NOUN
gojar-5993	152	12	solutions	solution	NOUN
gojar-5993	152	13	𝒙	𝒙	NOUN
gojar-5993	152	14	=	=	SYM
gojar-5993	152	15	(	(	PUNCT
gojar-5993	152	16	𝒙𝟏	𝒙𝟏	NOUN
gojar-5993	152	17	,	,	PUNCT
gojar-5993	152	18	𝒙𝟐	𝒙𝟐	NOUN
gojar-5993	152	19	)	)	PUNCT
gojar-5993	152	20	satisfy	satisfy	NOUN
gojar-5993	152	21	𝒔𝒊𝒛𝒆(𝒙	𝒔𝒊𝒛𝒆(𝒙	NOUN
gojar-5993	152	22	)	)	PUNCT
gojar-5993	152	23	≤	≤	NOUN
gojar-5993	152	24	𝟒(𝟐	𝟒(𝟐	PUNCT
gojar-5993	153	1	+	+	CCONJ
gojar-5993	153	2	𝟐	𝟐	X
gojar-5993	153	3	)	)	PUNCT
gojar-5993	153	4	=	=	SYM
gojar-5993	153	5	𝟏𝟔	𝟏𝟔	PROPN
gojar-5993	153	6	,	,	PUNCT
gojar-5993	153	7	ensuring	ensure	VERB
gojar-5993	153	8	efficient	efficient	ADJ
gojar-5993	153	9	computations	computation	NOUN
gojar-5993	153	10	.	.	PUNCT
gojar-5993	154	1	iii	iii	X
gojar-5993	154	2	)	)	PUNCT
gojar-5993	154	3	impact	impact	NOUN
gojar-5993	154	4	on	on	ADP
gojar-5993	154	5	algorithms	algorithm	NOUN
gojar-5993	154	6	:	:	PUNCT
gojar-5993	154	7	with	with	ADP
gojar-5993	154	8	these	these	DET
gojar-5993	154	9	bounds	bound	NOUN
gojar-5993	154	10	:	:	PUNCT
gojar-5993	154	11			NOUN
gojar-5993	154	12	vertex	vertex	NOUN
gojar-5993	154	13	enumeration	enumeration	NOUN
gojar-5993	154	14	:	:	PUNCT
gojar-5993	154	15	we	we	PRON
gojar-5993	154	16	avoid	avoid	VERB
gojar-5993	154	17	considering	consider	VERB
gojar-5993	154	18	infeasible	infeasible	ADJ
gojar-5993	154	19	points	point	NOUN
gojar-5993	154	20	with	with	ADP
gojar-5993	154	21	excessively	excessively	ADV
gojar-5993	154	22	large	large	ADJ
gojar-5993	154	23	coordinates	coordinate	NOUN
gojar-5993	154	24	.	.	PUNCT
gojar-5993	155	1			PROPN
gojar-5993	155	2	convex	convex	PROPN
gojar-5993	155	3	hull	hull	NOUN
gojar-5993	155	4	computation	computation	NOUN
gojar-5993	155	5	:	:	PUNCT
gojar-5993	155	6	ensures	ensure	VERB
gojar-5993	155	7	that	that	SCONJ
gojar-5993	155	8	the	the	DET
gojar-5993	155	9	algorithm	algorithm	NOUN
gojar-5993	155	10	’s	’s	PART
gojar-5993	155	11	runtime	runtime	NOUN
gojar-5993	155	12	is	be	AUX
gojar-5993	155	13	proportional	proportional	ADJ
gojar-5993	155	14	to	to	ADP
gojar-5993	155	15	the	the	DET
gojar-5993	155	16	actual	actual	ADJ
gojar-5993	155	17	feasible	feasible	ADJ
gojar-5993	155	18	vertices	vertex	NOUN
gojar-5993	155	19	,	,	PUNCT
gojar-5993	155	20	reducing	reduce	VERB
gojar-5993	155	21	unnecessary	unnecessary	ADJ
gojar-5993	155	22	overhead	overhead	NOUN
gojar-5993	155	23	.	.	PUNCT
gojar-5993	156	1	this	this	DET
gojar-5993	156	2	example	example	NOUN
gojar-5993	156	3	demonstrates	demonstrate	VERB
gojar-5993	156	4	how	how	SCONJ
gojar-5993	156	5	solution	solution	NOUN
gojar-5993	156	6	size	size	NOUN
gojar-5993	156	7	bounds	bound	NOUN
gojar-5993	156	8	provide	provide	VERB
gojar-5993	156	9	theoretical	theoretical	ADJ
gojar-5993	156	10	guarantees	guarantee	NOUN
gojar-5993	156	11	that	that	PRON
gojar-5993	156	12	directly	directly	ADV
gojar-5993	156	13	improve	improve	VERB
gojar-5993	156	14	the	the	DET
gojar-5993	156	15	efficiency	efficiency	NOUN
gojar-5993	156	16	and	and	CCONJ
gojar-5993	156	17	practicality	practicality	NOUN
gojar-5993	156	18	of	of	ADP
gojar-5993	156	19	convex	convex	PROPN
gojar-5993	156	20	hull	hull	NOUN
gojar-5993	156	21	and	and	CCONJ
gojar-5993	156	22	vertex	vertex	NOUN
gojar-5993	156	23	enumeration	enumeration	NOUN
gojar-5993	156	24	algorithms	algorithm	NOUN
gojar-5993	156	25	.	.	PUNCT
gojar-5993	157	1	thus	thus	ADV
gojar-5993	157	2	,	,	PUNCT
gojar-5993	157	3	it	it	PRON
gojar-5993	157	4	improves	improve	VERB
gojar-5993	157	5	the	the	DET
gojar-5993	157	6	bounds	bound	NOUN
gojar-5993	157	7	that	that	PRON
gojar-5993	157	8	contribute	contribute	VERB
gojar-5993	157	9	to	to	ADP
gojar-5993	157	10	better	well	ADJ
gojar-5993	157	11	prerecession	prerecession	NOUN
gojar-5993	157	12	and	and	CCONJ
gojar-5993	157	13	numerical	numerical	ADJ
gojar-5993	157	14	stability	stability	NOUN
gojar-5993	157	15	in	in	ADP
gojar-5993	157	16	lp	lp	ADJ
gojar-5993	157	17	solvers	solver	NOUN
gojar-5993	157	18	.	.	PUNCT
gojar-5993	158	1	v.	v.	ADP
gojar-5993	158	2	conclusion	conclusion	NOUN
gojar-5993	158	3	this	this	DET
gojar-5993	158	4	paper	paper	NOUN
gojar-5993	158	5	establishes	establish	VERB
gojar-5993	158	6	critical	critical	ADJ
gojar-5993	158	7	theoretical	theoretical	ADJ
gojar-5993	158	8	results	result	NOUN
gojar-5993	158	9	in	in	ADP
gojar-5993	158	10	rational	rational	ADJ
gojar-5993	158	11	linear	linear	ADJ
gojar-5993	158	12	programming	programming	NOUN
gojar-5993	158	13	and	and	CCONJ
gojar-5993	158	14	polyhedral	polyhedral	ADJ
gojar-5993	158	15	optimization	optimization	NOUN
gojar-5993	158	16	,	,	PUNCT
gojar-5993	158	17	emphasizing	emphasize	VERB
gojar-5993	158	18	boundedness	boundedness	NOUN
gojar-5993	158	19	equivalence	equivalence	NOUN
gojar-5993	158	20	and	and	CCONJ
gojar-5993	158	21	solution	solution	NOUN
gojar-5993	158	22	size	size	NOUN
gojar-5993	158	23	constraints	constraint	NOUN
gojar-5993	158	24	.	.	PUNCT
gojar-5993	159	1	by	by	ADP
gojar-5993	159	2	proving	prove	VERB
gojar-5993	159	3	the	the	DET
gojar-5993	159	4	equivalence	equivalence	NOUN
gojar-5993	159	5	of	of	ADP
gojar-5993	159	6	boundedness	boundedness	NOUN
gojar-5993	159	7	between	between	ADP
gojar-5993	159	8	rational	rational	ADJ
gojar-5993	159	9	polyhedra	polyhedra	NOUN
gojar-5993	159	10	and	and	CCONJ
gojar-5993	159	11	their	their	PRON
gojar-5993	159	12	integer	integer	NOUN
gojar-5993	159	13	hulls	hull	NOUN
gojar-5993	159	14	,	,	PUNCT
gojar-5993	159	15	as	as	ADV
gojar-5993	159	16	well	well	ADV
gojar-5993	159	17	as	as	ADP
gojar-5993	159	18	global	global	ADJ
gojar-5993	159	19	online	online	ADJ
gojar-5993	159	20	journal	journal	PROPN
gojar-5993	159	21	of	of	ADP
gojar-5993	159	22	academic	academic	ADJ
gojar-5993	159	23	research	research	NOUN
gojar-5993	159	24	(	(	PUNCT
gojar-5993	159	25	gojar	gojar	NOUN
gojar-5993	159	26	)	)	PUNCT
gojar-5993	159	27	,	,	PUNCT
gojar-5993	159	28	vol	vol	NOUN
gojar-5993	159	29	.	.	PROPN
gojar-5993	159	30	4	4	NUM
gojar-5993	159	31	,	,	PUNCT
gojar-5993	159	32	no	no	INTJ
gojar-5993	159	33	.	.	NOUN
gojar-5993	159	34	1	1	NUM
gojar-5993	159	35	february	february	NOUN
gojar-5993	159	36	2025	2025	NUM
gojar-5993	159	37	71	71	NUM
gojar-5993	159	38	deriving	derive	VERB
gojar-5993	159	39	explicit	explicit	ADJ
gojar-5993	159	40	bounds	bound	NOUN
gojar-5993	159	41	on	on	ADP
gojar-5993	159	42	the	the	DET
gojar-5993	159	43	size	size	NOUN
gojar-5993	159	44	of	of	ADP
gojar-5993	159	45	optimal	optimal	ADJ
gojar-5993	159	46	solutions	solution	NOUN
gojar-5993	159	47	,	,	PUNCT
gojar-5993	159	48	this	this	DET
gojar-5993	159	49	work	work	NOUN
gojar-5993	159	50	contributes	contribute	VERB
gojar-5993	159	51	to	to	ADP
gojar-5993	159	52	a	a	DET
gojar-5993	159	53	deeper	deep	ADJ
gojar-5993	159	54	understanding	understanding	NOUN
gojar-5993	159	55	of	of	ADP
gojar-5993	159	56	the	the	DET
gojar-5993	159	57	structural	structural	ADJ
gojar-5993	159	58	and	and	CCONJ
gojar-5993	159	59	numerical	numerical	ADJ
gojar-5993	159	60	properties	property	NOUN
gojar-5993	159	61	of	of	ADP
gojar-5993	159	62	optimization	optimization	NOUN
gojar-5993	159	63	problems	problem	NOUN
gojar-5993	159	64	.	.	PUNCT
gojar-5993	160	1	these	these	DET
gojar-5993	160	2	findings	finding	NOUN
gojar-5993	160	3	are	be	AUX
gojar-5993	160	4	not	not	PART
gojar-5993	160	5	only	only	ADV
gojar-5993	160	6	of	of	ADP
gojar-5993	160	7	theoretical	theoretical	ADJ
gojar-5993	160	8	interest	interest	NOUN
gojar-5993	160	9	but	but	CCONJ
gojar-5993	160	10	also	also	ADV
gojar-5993	160	11	pave	pave	VERB
gojar-5993	160	12	the	the	DET
gojar-5993	160	13	way	way	NOUN
gojar-5993	160	14	for	for	ADP
gojar-5993	160	15	advancements	advancement	NOUN
gojar-5993	160	16	in	in	ADP
gojar-5993	160	17	computational	computational	ADJ
gojar-5993	160	18	optimization	optimization	NOUN
gojar-5993	160	19	,	,	PUNCT
gojar-5993	160	20	particularly	particularly	ADV
gojar-5993	160	21	in	in	ADP
gojar-5993	160	22	improving	improve	VERB
gojar-5993	160	23	algorithmic	algorithmic	ADJ
gojar-5993	160	24	efficiency	efficiency	NOUN
gojar-5993	160	25	and	and	CCONJ
gojar-5993	160	26	ensuring	ensure	VERB
gojar-5993	160	27	numerical	numerical	ADJ
gojar-5993	160	28	stability	stability	NOUN
gojar-5993	160	29	.	.	PUNCT
gojar-5993	161	1	vi	vi	X
gojar-5993	161	2	.	.	PUNCT
gojar-5993	162	1	recommendations	recommendation	NOUN
gojar-5993	162	2	future	future	ADJ
gojar-5993	162	3	work	work	NOUN
gojar-5993	162	4	may	may	AUX
gojar-5993	162	5	explore	explore	VERB
gojar-5993	162	6	extensions	extension	NOUN
gojar-5993	162	7	to	to	ADP
gojar-5993	162	8	non	non	ADJ
gojar-5993	162	9	-	-	ADJ
gojar-5993	162	10	convex	convex	ADJ
gojar-5993	162	11	settings	setting	NOUN
gojar-5993	162	12	,	,	PUNCT
gojar-5993	162	13	where	where	SCONJ
gojar-5993	162	14	the	the	DET
gojar-5993	162	15	feasible	feasible	ADJ
gojar-5993	162	16	regions	region	NOUN
gojar-5993	162	17	are	be	AUX
gojar-5993	162	18	no	no	ADV
gojar-5993	162	19	longer	long	ADV
gojar-5993	162	20	polyhedral	polyhedral	ADJ
gojar-5993	162	21	,	,	PUNCT
gojar-5993	162	22	presenting	present	VERB
gojar-5993	162	23	new	new	ADJ
gojar-5993	162	24	challenges	challenge	NOUN
gojar-5993	162	25	in	in	ADP
gojar-5993	162	26	understanding	understand	VERB
gojar-5993	162	27	boundedness	boundedness	NOUN
gojar-5993	162	28	and	and	CCONJ
gojar-5993	162	29	solution	solution	NOUN
gojar-5993	162	30	representation	representation	NOUN
gojar-5993	162	31	.	.	PUNCT
gojar-5993	163	1	another	another	DET
gojar-5993	163	2	promising	promising	ADJ
gojar-5993	163	3	direction	direction	NOUN
gojar-5993	163	4	involves	involve	VERB
gojar-5993	163	5	generalizations	generalization	NOUN
gojar-5993	163	6	to	to	ADP
gojar-5993	163	7	cases	case	NOUN
gojar-5993	163	8	with	with	ADP
gojar-5993	163	9	irrational	irrational	ADJ
gojar-5993	163	10	coefficients	coefficient	NOUN
gojar-5993	163	11	,	,	PUNCT
gojar-5993	163	12	which	which	PRON
gojar-5993	163	13	require	require	VERB
gojar-5993	163	14	advanced	advanced	ADJ
gojar-5993	163	15	techniques	technique	NOUN
gojar-5993	163	16	to	to	PART
gojar-5993	163	17	address	address	VERB
gojar-5993	163	18	the	the	DET
gojar-5993	163	19	complexities	complexity	NOUN
gojar-5993	163	20	introduced	introduce	VERB
gojar-5993	163	21	by	by	ADP
gojar-5993	163	22	nonrational	nonrational	ADJ
gojar-5993	163	23	systems	system	NOUN
gojar-5993	163	24	.	.	PUNCT
gojar-5993	164	1	furthermore	furthermore	ADV
gojar-5993	164	2	,	,	PUNCT
gojar-5993	164	3	integrating	integrate	VERB
gojar-5993	164	4	these	these	DET
gojar-5993	164	5	theoretical	theoretical	ADJ
gojar-5993	164	6	insights	insight	NOUN
gojar-5993	164	7	into	into	ADP
gojar-5993	164	8	practical	practical	ADJ
gojar-5993	164	9	optimization	optimization	NOUN
gojar-5993	164	10	software	software	NOUN
gojar-5993	164	11	and	and	CCONJ
gojar-5993	164	12	exploring	explore	VERB
gojar-5993	164	13	their	their	PRON
gojar-5993	164	14	impact	impact	NOUN
gojar-5993	164	15	on	on	ADP
gojar-5993	164	16	real	real	ADJ
gojar-5993	164	17	-	-	PUNCT
gojar-5993	164	18	world	world	NOUN
gojar-5993	164	19	applications	application	NOUN
gojar-5993	164	20	,	,	PUNCT
gojar-5993	164	21	such	such	ADJ
gojar-5993	164	22	as	as	ADP
gojar-5993	164	23	logistics	logistic	NOUN
gojar-5993	164	24	,	,	PUNCT
gojar-5993	164	25	network	network	NOUN
gojar-5993	164	26	design	design	NOUN
gojar-5993	164	27	,	,	PUNCT
gojar-5993	164	28	and	and	CCONJ
gojar-5993	164	29	machine	machine	NOUN
gojar-5993	164	30	learning	learning	NOUN
gojar-5993	164	31	,	,	PUNCT
gojar-5993	164	32	could	could	AUX
gojar-5993	164	33	significantly	significantly	ADV
gojar-5993	164	34	enhance	enhance	VERB
gojar-5993	164	35	the	the	DET
gojar-5993	164	36	utility	utility	NOUN
gojar-5993	164	37	and	and	CCONJ
gojar-5993	164	38	scope	scope	NOUN
gojar-5993	164	39	of	of	ADP
gojar-5993	164	40	rational	rational	ADJ
gojar-5993	164	41	lp	lp	ADJ
gojar-5993	164	42	and	and	CCONJ
gojar-5993	164	43	polyhedral	polyhedral	ADJ
gojar-5993	164	44	optimization	optimization	NOUN
gojar-5993	164	45	.	.	PUNCT
gojar-5993	165	1	such	such	ADJ
gojar-5993	165	2	efforts	effort	NOUN
gojar-5993	165	3	would	would	AUX
gojar-5993	165	4	bridge	bridge	VERB
gojar-5993	165	5	the	the	DET
gojar-5993	165	6	gap	gap	NOUN
gojar-5993	165	7	between	between	ADP
gojar-5993	165	8	theoretical	theoretical	ADJ
gojar-5993	165	9	advancements	advancement	NOUN
gojar-5993	165	10	and	and	CCONJ
gojar-5993	165	11	their	their	PRON
gojar-5993	165	12	practical	practical	ADJ
gojar-5993	165	13	implementations	implementation	NOUN
gojar-5993	165	14	,	,	PUNCT
gojar-5993	165	15	fostering	foster	VERB
gojar-5993	165	16	innovation	innovation	NOUN
gojar-5993	165	17	in	in	ADP
gojar-5993	165	18	both	both	CCONJ
gojar-5993	165	19	academic	academic	ADJ
gojar-5993	165	20	and	and	CCONJ
gojar-5993	165	21	industrial	industrial	ADJ
gojar-5993	165	22	domains	domain	NOUN
gojar-5993	165	23	.	.	PUNCT
gojar-5993	166	1	references	reference	NOUN
gojar-5993	166	2	akif	akif	PROPN
gojar-5993	166	3	,	,	PUNCT
gojar-5993	166	4	m	m	PROPN
gojar-5993	166	5	b.	b.	PROPN
gojar-5993	166	6	and	and	CCONJ
gojar-5993	166	7	cihan	cihan	PROPN
gojar-5993	166	8	,	,	PUNCT
gojar-5993	166	9	a.	a.	NOUN
gojar-5993	166	10	(	(	PUNCT
gojar-5993	166	11	2008	2008	NUM
gojar-5993	166	12	)	)	PUNCT
gojar-5993	166	13	.	.	PUNCT
gojar-5993	167	1	a	a	DET
gojar-5993	167	2	0	0	NUM
gojar-5993	167	3	-	-	SYM
gojar-5993	167	4	1	1	NUM
gojar-5993	167	5	integer	integer	NOUN
gojar-5993	167	6	programming	programming	NOUN
gojar-5993	167	7	approach	approach	NOUN
gojar-5993	167	8	to	to	ADP
gojar-5993	167	9	a	a	DET
gojar-5993	167	10	university	university	NOUN
gojar-5993	167	11	timetabling	timetabling	NOUN
gojar-5993	167	12	problem	problem	NOUN
gojar-5993	167	13	.	.	PUNCT
gojar-5993	168	1	hacettepe	hacettepe	ADJ
gojar-5993	168	2	journal	journal	PROPN
gojar-5993	168	3	of	of	ADP
gojar-5993	168	4	mathematics	mathematic	NOUN
gojar-5993	168	5	and	and	CCONJ
gojar-5993	168	6	statistics	statistic	NOUN
gojar-5993	168	7	,	,	PUNCT
gojar-5993	168	8	37	37	NUM
gojar-5993	168	9	:	:	SYM
gojar-5993	168	10	41	41	NUM
gojar-5993	168	11	-	-	SYM
gojar-5993	168	12	55	55	NUM
gojar-5993	168	13	.	.	PUNCT
gojar-5993	169	1	cook	cook	PROPN
gojar-5993	169	2	,	,	PUNCT
gojar-5993	169	3	w.	w.	PROPN
gojar-5993	169	4	,	,	PUNCT
gojar-5993	169	5	cunningham	cunningham	PROPN
gojar-5993	169	6	,	,	PUNCT
gojar-5993	169	7	w.	w.	PROPN
gojar-5993	169	8	h.	h.	PROPN
gojar-5993	169	9	,	,	PUNCT
gojar-5993	169	10	pulleyblank	pulleyblank	PROPN
gojar-5993	169	11	,	,	PUNCT
gojar-5993	169	12	w.	w.	PROPN
gojar-5993	169	13	r.	r.	PROPN
gojar-5993	169	14	,	,	PUNCT
gojar-5993	169	15	&	&	CCONJ
gojar-5993	169	16	schrijver	schrijver	NOUN
gojar-5993	169	17	,	,	PUNCT
gojar-5993	169	18	a.	a.	NOUN
gojar-5993	169	19	(	(	PUNCT
gojar-5993	169	20	1986	1986	NUM
gojar-5993	169	21	)	)	PUNCT
gojar-5993	169	22	.	.	PUNCT
gojar-5993	170	1	combinatorial	combinatorial	ADJ
gojar-5993	170	2	optimization	optimization	NOUN
gojar-5993	170	3	.	.	PUNCT
gojar-5993	171	1	wiley	wiley	PROPN
gojar-5993	171	2	.	.	PUNCT
gojar-5993	172	1	dantzig	dantzig	PROPN
gojar-5993	172	2	,	,	PUNCT
gojar-5993	172	3	g.	g.	PROPN
gojar-5993	172	4	b.	b.	PROPN
gojar-5993	172	5	(	(	PUNCT
gojar-5993	172	6	1947	1947	NUM
gojar-5993	172	7	)	)	PUNCT
gojar-5993	172	8	.	.	PUNCT
gojar-5993	173	1	linear	linear	PROPN
gojar-5993	173	2	programming	programming	NOUN
gojar-5993	173	3	and	and	CCONJ
gojar-5993	173	4	extensions	extension	NOUN
gojar-5993	173	5	.	.	PUNCT
gojar-5993	174	1	princeton	princeton	PROPN
gojar-5993	174	2	university	university	PROPN
gojar-5993	174	3	press	press	NOUN
gojar-5993	174	4	.	.	PUNCT
gojar-5993	175	1	dantzig	dantzig	PROPN
gojar-5993	175	2	,	,	PUNCT
gojar-5993	175	3	g.	g.	PROPN
gojar-5993	175	4	b.	b.	PROPN
gojar-5993	175	5	(	(	PUNCT
gojar-5993	175	6	1947	1947	NUM
gojar-5993	175	7	)	)	PUNCT
gojar-5993	175	8	.	.	PUNCT
gojar-5993	176	1	maximization	maximization	NOUN
gojar-5993	176	2	of	of	ADP
gojar-5993	176	3	a	a	DET
gojar-5993	176	4	linear	linear	ADJ
gojar-5993	176	5	function	function	NOUN
gojar-5993	176	6	of	of	ADP
gojar-5993	176	7	variables	variable	NOUN
gojar-5993	176	8	subject	subject	ADJ
gojar-5993	176	9	to	to	ADP
gojar-5993	176	10	linear	linear	PROPN
gojar-5993	176	11	inequalities	inequality	NOUN
gojar-5993	176	12	.	.	PUNCT
gojar-5993	177	1	in	in	ADP
gojar-5993	177	2	t.	t.	PROPN
gojar-5993	177	3	c.	c.	PROPN
gojar-5993	177	4	koopmans	koopmans	PROPN
gojar-5993	177	5	(	(	PUNCT
gojar-5993	177	6	ed	ed	NOUN
gojar-5993	177	7	.	.	PUNCT
gojar-5993	177	8	)	)	PUNCT
gojar-5993	177	9	,	,	PUNCT
gojar-5993	177	10	activity	activity	NOUN
gojar-5993	177	11	analysis	analysis	NOUN
gojar-5993	177	12	of	of	ADP
gojar-5993	177	13	production	production	NOUN
gojar-5993	177	14	and	and	CCONJ
gojar-5993	177	15	allocation	allocation	NOUN
gojar-5993	177	16	(	(	PUNCT
gojar-5993	177	17	pp	pp	ADJ
gojar-5993	177	18	.	.	PUNCT
gojar-5993	177	19	339	339	NUM
gojar-5993	177	20	-	-	SYM
gojar-5993	177	21	347	347	NUM
gojar-5993	177	22	)	)	PUNCT
gojar-5993	177	23	.	.	PUNCT
gojar-5993	178	1	wiley	wiley	PROPN
gojar-5993	178	2	.	.	PUNCT
gojar-5993	179	1	elmuti	elmuti	PROPN
gojar-5993	179	2	,	,	PUNCT
gojar-5993	179	3	d.	d.	PROPN
gojar-5993	179	4	(	(	PUNCT
gojar-5993	179	5	2003	2003	NUM
gojar-5993	179	6	)	)	PUNCT
gojar-5993	179	7	.	.	PUNCT
gojar-5993	180	1	the	the	DET
gojar-5993	180	2	perceived	perceive	VERB
gojar-5993	180	3	impact	impact	NOUN
gojar-5993	180	4	of	of	ADP
gojar-5993	180	5	outsourcing	outsource	VERB
gojar-5993	180	6	on	on	ADP
gojar-5993	180	7	organizational	organizational	ADJ
gojar-5993	180	8	performance	performance	NOUN
gojar-5993	180	9	.	.	PUNCT
gojar-5993	181	1	american	american	ADJ
gojar-5993	181	2	journal	journal	PROPN
gojar-5993	181	3	of	of	ADP
gojar-5993	181	4	business	business	NOUN
gojar-5993	181	5	,	,	PUNCT
gojar-5993	181	6	18	18	NUM
gojar-5993	181	7	:	:	SYM
gojar-5993	181	8	33	33	NUM
gojar-5993	181	9	-	-	SYM
gojar-5993	181	10	42	42	NUM
gojar-5993	181	11	.	.	PUNCT
gojar-5993	181	12	genova	genova	PROPN
gojar-5993	181	13	,	,	PUNCT
gojar-5993	181	14	k.	k.	PROPN
gojar-5993	181	15	and	and	CCONJ
gojar-5993	181	16	guliashki	guliashki	PROPN
gojar-5993	181	17	,	,	PUNCT
gojar-5993	182	1	v.	v.	PROPN
gojar-5993	182	2	(	(	PUNCT
gojar-5993	182	3	2011	2011	NUM
gojar-5993	182	4	)	)	PUNCT
gojar-5993	182	5	.	.	PUNCT
gojar-5993	183	1	linear	linear	ADJ
gojar-5993	183	2	integer	integer	NOUN
gojar-5993	183	3	programming	programming	NOUN
gojar-5993	183	4	methods	method	NOUN
gojar-5993	183	5	and	and	CCONJ
gojar-5993	183	6	approaches	approach	VERB
gojar-5993	183	7	a	a	DET
gojar-5993	183	8	survey	survey	NOUN
gojar-5993	183	9	.	.	PUNCT
gojar-5993	184	1	journal	journal	NOUN
gojar-5993	184	2	of	of	ADP
gojar-5993	184	3	cybernetics	cybernetic	NOUN
gojar-5993	184	4	and	and	CCONJ
gojar-5993	184	5	information	information	NOUN
gojar-5993	184	6	technologies	technology	NOUN
gojar-5993	184	7	.	.	PUNCT
gojar-5993	185	1	vol	vol	NOUN
gojar-5993	185	2	11	11	NUM
gojar-5993	185	3	.	.	PUNCT
gojar-5993	186	1	gupta	gupta	PROPN
gojar-5993	186	2	,	,	PUNCT
gojar-5993	186	3	prem	prem	PROPN
gojar-5993	186	4	kumar	kumar	PROPN
gojar-5993	186	5	(	(	PUNCT
gojar-5993	186	6	er	er	INTJ
gojar-5993	186	7	.	.	PUNCT
gojar-5993	186	8	)	)	PUNCT
gojar-5993	186	9	and	and	CCONJ
gojar-5993	186	10	d.	d.	PROPN
gojar-5993	186	11	s.	s.	PROPN
gojar-5993	186	12	hira	hira	PROPN
gojar-5993	186	13	(	(	PUNCT
gojar-5993	186	14	2014	2014	NUM
gojar-5993	186	15	)	)	PUNCT
gojar-5993	186	16	;	;	PUNCT
gojar-5993	186	17	operations	operation	NOUN
gojar-5993	186	18	research	research	VERB
gojar-5993	186	19	seventh	seventh	ADJ
gojar-5993	186	20	global	global	ADJ
gojar-5993	186	21	online	online	PROPN
gojar-5993	186	22	journal	journal	PROPN
gojar-5993	186	23	of	of	ADP
gojar-5993	186	24	academic	academic	ADJ
gojar-5993	186	25	research	research	NOUN
gojar-5993	186	26	(	(	PUNCT
gojar-5993	186	27	gojar	gojar	NOUN
gojar-5993	186	28	)	)	PUNCT
gojar-5993	186	29	,	,	PUNCT
gojar-5993	186	30	vol	vol	NOUN
gojar-5993	186	31	.	.	PROPN
gojar-5993	186	32	4	4	NUM
gojar-5993	186	33	,	,	PUNCT
gojar-5993	186	34	no	no	INTJ
gojar-5993	186	35	.	.	NOUN
gojar-5993	186	36	1	1	NUM
gojar-5993	186	37	february	february	NOUN
gojar-5993	186	38	2025	2025	NUM
gojar-5993	186	39	72	72	NUM
gojar-5993	186	40	revised	revise	VERB
gojar-5993	186	41	edition	edition	NOUN
gojar-5993	186	42	.	.	PUNCT
gojar-5993	187	1	by	by	ADP
gojar-5993	187	2	rajendra	rajendra	PROPN
gojar-5993	187	3	ravindra	ravindra	PROPN
gojar-5993	187	4	pvt	pvt	PROPN
gojar-5993	187	5	ltd	ltd	PROPN
gojar-5993	187	6	,	,	PUNCT
gojar-5993	187	7	ram	ram	VERB
gojar-5993	187	8	nagar	nagar	NOUN
gojar-5993	187	9	new	new	ADJ
gojar-5993	187	10	delhi110055	delhi110055	PROPN
gojar-5993	187	11	and	and	CCONJ
gojar-5993	187	12	published	publish	VERB
gojar-5993	187	13	by	by	ADP
gojar-5993	187	14	s.	s.	PROPN
gojar-5993	187	15	chand	chand	PROPN
gojar-5993	187	16	&	&	CCONJ
gojar-5993	187	17	company	company	PROPN
gojar-5993	187	18	pvt	pvt	PROPN
gojar-5993	187	19	ltd	ltd	PROPN
gojar-5993	187	20	,	,	PUNCT
gojar-5993	187	21	india	india	PROPN
gojar-5993	187	22	kantorovich	kantorovich	PROPN
gojar-5993	187	23	,	,	PUNCT
gojar-5993	187	24	l.	l.	PROPN
gojar-5993	187	25	v.	v.	PROPN
gojar-5993	187	26	(	(	PUNCT
gojar-5993	187	27	1939	1939	NUM
gojar-5993	187	28	)	)	PUNCT
gojar-5993	187	29	.	.	PUNCT
gojar-5993	188	1	mathematical	mathematical	ADJ
gojar-5993	188	2	methods	method	NOUN
gojar-5993	188	3	in	in	ADP
gojar-5993	188	4	the	the	DET
gojar-5993	188	5	organization	organization	NOUN
gojar-5993	188	6	and	and	CCONJ
gojar-5993	188	7	planning	planning	NOUN
gojar-5993	188	8	of	of	ADP
gojar-5993	188	9	production	production	NOUN
gojar-5993	188	10	.	.	PUNCT
gojar-5993	189	1	leningrad	leningrad	PROPN
gojar-5993	189	2	state	state	PROPN
gojar-5993	189	3	university	university	PROPN
gojar-5993	189	4	.	.	PUNCT
gojar-5993	190	1	karmarkar	karmarkar	NOUN
gojar-5993	190	2	,	,	PUNCT
gojar-5993	190	3	n.	n.	PROPN
gojar-5993	190	4	(	(	PUNCT
gojar-5993	190	5	1984	1984	NUM
gojar-5993	190	6	)	)	PUNCT
gojar-5993	190	7	.	.	PUNCT
gojar-5993	191	1	a	a	DET
gojar-5993	191	2	new	new	ADJ
gojar-5993	191	3	polynomial	polynomial	ADJ
gojar-5993	191	4	-	-	PUNCT
gojar-5993	191	5	time	time	NOUN
gojar-5993	191	6	algorithm	algorithm	NOUN
gojar-5993	191	7	for	for	ADP
gojar-5993	191	8	linear	linear	PROPN
gojar-5993	191	9	programming	programming	NOUN
gojar-5993	191	10	.	.	PUNCT
gojar-5993	192	1	combinatorica	combinatorica	PROPN
gojar-5993	192	2	,	,	PUNCT
gojar-5993	192	3	4(4	4(4	NUM
gojar-5993	192	4	)	)	PUNCT
gojar-5993	192	5	,	,	PUNCT
gojar-5993	192	6	373	373	NUM
gojar-5993	192	7	-	-	SYM
gojar-5993	192	8	395	395	NUM
gojar-5993	192	9	.	.	PUNCT
gojar-5993	193	1	laisin	laisin	PROPN
gojar-5993	193	2	,	,	PUNCT
gojar-5993	193	3	m.	m.	NOUN
gojar-5993	193	4	,	,	PUNCT
gojar-5993	193	5	edike	edike	ADJ
gojar-5993	193	6	,	,	PUNCT
gojar-5993	193	7	c.	c.	PROPN
gojar-5993	193	8	and	and	CCONJ
gojar-5993	193	9	bright	bright	ADJ
gojar-5993	193	10	o.	o.	NOUN
gojar-5993	193	11	osu	osu	NOUN
gojar-5993	193	12	(	(	PUNCT
gojar-5993	193	13	2024	2024	NUM
gojar-5993	193	14	)	)	PUNCT
gojar-5993	193	15	;	;	PUNCT
gojar-5993	193	16	the	the	DET
gojar-5993	193	17	construction	construction	NOUN
gojar-5993	193	18	of	of	ADP
gojar-5993	193	19	rational	rational	ADJ
gojar-5993	193	20	polyhedron	polyhedron	NOUN
gojar-5993	193	21	on	on	ADP
gojar-5993	193	22	an	an	DET
gojar-5993	193	23	𝒏	𝒏	PROPN
gojar-5993	193	24	×	×	PROPN
gojar-5993	193	25	𝒏	𝒏	PROPN
gojar-5993	193	26	board	board	NOUN
gojar-5993	193	27	with	with	ADP
gojar-5993	193	28	some	some	DET
gojar-5993	193	29	application	application	NOUN
gojar-5993	193	30	on	on	ADP
gojar-5993	193	31	integral	integral	ADJ
gojar-5993	193	32	polyhedral	polyhedral	ADJ
gojar-5993	193	33	.	.	PUNCT
gojar-5993	194	1	tijer	tijer	NOUN
gojar-5993	194	2	,	,	PUNCT
gojar-5993	194	3	vol	vol	NOUN
gojar-5993	194	4	11	11	NUM
gojar-5993	194	5	,	,	PUNCT
gojar-5993	194	6	issue	issue	NOUN
gojar-5993	194	7	11	11	NUM
gojar-5993	194	8	,	,	PUNCT
gojar-5993	194	9	www.tijer.org	www.tijer.org	PROPN
gojar-5993	194	10	nemhauser	nemhauser	NOUN
gojar-5993	194	11	,	,	PUNCT
gojar-5993	194	12	g.	g.	PROPN
gojar-5993	194	13	l.	l.	PROPN
gojar-5993	194	14	,	,	PUNCT
gojar-5993	194	15	&	&	CCONJ
gojar-5993	194	16	wolsey	wolsey	PROPN
gojar-5993	194	17	,	,	PUNCT
gojar-5993	194	18	l.	l.	PROPN
gojar-5993	194	19	a.	a.	PROPN
gojar-5993	194	20	(	(	PUNCT
gojar-5993	194	21	1999	1999	NUM
gojar-5993	194	22	)	)	PUNCT
gojar-5993	194	23	.	.	PUNCT
gojar-5993	195	1	integer	integer	NOUN
gojar-5993	195	2	and	and	CCONJ
gojar-5993	195	3	combinatorial	combinatorial	ADJ
gojar-5993	195	4	optimization	optimization	NOUN
gojar-5993	195	5	.	.	PUNCT
gojar-5993	196	1	wiley	wiley	PROPN
gojar-5993	196	2	.	.	PUNCT
gojar-5993	197	1	schrijver	schrijver	PROPN
gojar-5993	197	2	,	,	PUNCT
gojar-5993	197	3	a.	a.	NOUN
gojar-5993	197	4	(	(	PUNCT
gojar-5993	197	5	1998	1998	NUM
gojar-5993	197	6	)	)	PUNCT
gojar-5993	197	7	.	.	PUNCT
gojar-5993	198	1	theory	theory	NOUN
gojar-5993	198	2	of	of	ADP
gojar-5993	198	3	linear	linear	PROPN
gojar-5993	198	4	and	and	CCONJ
gojar-5993	198	5	integer	integer	NOUN
gojar-5993	198	6	programming	programming	NOUN
gojar-5993	198	7	.	.	PUNCT
gojar-5993	199	1	wiley	wiley	PROPN
gojar-5993	199	2	.	.	PUNCT
gojar-5993	200	1	author	author	NOUN
gojar-5993	200	2	information	information	NOUN
gojar-5993	200	3	:	:	PUNCT
gojar-5993	201	1	prof	prof	PROPN
gojar-5993	201	2	.	.	PROPN
gojar-5993	202	1	mark	mark	PROPN
gojar-5993	202	2	laisin	laisin	PROPN
gojar-5993	202	3	is	be	AUX
gojar-5993	202	4	of	of	ADP
gojar-5993	202	5	the	the	DET
gojar-5993	202	6	department	department	NOUN
gojar-5993	202	7	of	of	ADP
gojar-5993	202	8	mathematics	mathematics	PROPN
gojar-5993	202	9	,	,	PUNCT
gojar-5993	202	10	chukwuemeka	chukwuemeka	PROPN
gojar-5993	202	11	odumegwu	odumegwu	PROPN
gojar-5993	202	12	ojukwu	ojukwu	PROPN
gojar-5993	202	13	university	university	PROPN
gojar-5993	202	14	,	,	PUNCT
gojar-5993	202	15	uli	uli	PROPN
gojar-5993	202	16	,	,	PUNCT
gojar-5993	202	17	anambra	anambra	PROPN
gojar-5993	202	18	state	state	PROPN
gojar-5993	202	19	,	,	PUNCT
gojar-5993	202	20	nigeria	nigeria	PROPN
gojar-5993	202	21	.	.	PUNCT
gojar-5993	203	1	email	email	NOUN
gojar-5993	203	2	:	:	PUNCT
gojar-5993	204	1	laisinmark@gmail.com	laisinmark@gmail.com	PROPN
gojar-5993	204	2	collins	collins	PROPN
gojar-5993	204	3	edike	edike	PROPN
gojar-5993	204	4	is	be	AUX
gojar-5993	204	5	of	of	ADP
gojar-5993	204	6	the	the	DET
gojar-5993	204	7	department	department	NOUN
gojar-5993	204	8	of	of	ADP
gojar-5993	204	9	mathematics	mathematics	PROPN
gojar-5993	204	10	,	,	PUNCT
gojar-5993	204	11	chukwuemeka	chukwuemeka	PROPN
gojar-5993	204	12	odumegwu	odumegwu	PROPN
gojar-5993	204	13	ojukwu	ojukwu	PROPN
gojar-5993	204	14	university	university	PROPN
gojar-5993	204	15	,	,	PUNCT
gojar-5993	204	16	uli	uli	PROPN
gojar-5993	204	17	,	,	PUNCT
gojar-5993	204	18	anambra	anambra	PROPN
gojar-5993	204	19	state	state	PROPN
gojar-5993	204	20	,	,	PUNCT
gojar-5993	204	21	nigeria	nigeria	PROPN
gojar-5993	204	22	.	.	PUNCT
gojar-5993	205	1	email	email	NOUN
gojar-5993	205	2	:	:	PUNCT
gojar-5993	205	3	edikecollins505@gmail	edikecollins505@gmail	NOUN
gojar-5993	205	4	.	.	PUNCT
gojar-5993	205	5	com	com	PROPN
gojar-5993	205	6	dr	dr	PROPN
gojar-5993	205	7	r.	r.	PROPN
gojar-5993	205	8	n.	n.	PROPN
gojar-5993	205	9	ujumadu	ujumadu	PROPN
gojar-5993	205	10	is	be	AUX
gojar-5993	205	11	of	of	ADP
gojar-5993	205	12	the	the	DET
gojar-5993	205	13	department	department	NOUN
gojar-5993	205	14	of	of	ADP
gojar-5993	205	15	mathematics	mathematics	PROPN
gojar-5993	205	16	,	,	PUNCT
gojar-5993	205	17	chukwuemeka	chukwuemeka	PROPN
gojar-5993	205	18	odumegwu	odumegwu	PROPN
gojar-5993	205	19	ojukwu	ojukwu	PROPN
gojar-5993	205	20	university	university	PROPN
gojar-5993	205	21	,	,	PUNCT
gojar-5993	205	22	uli	uli	PROPN
gojar-5993	205	23	,	,	PUNCT
gojar-5993	205	24	anambra	anambra	PROPN
gojar-5993	205	25	state	state	PROPN
gojar-5993	205	26	,	,	PUNCT
gojar-5993	205	27	nigeria	nigeria	PROPN
gojar-5993	205	28	.	.	PUNCT
gojar-5993	206	1	email	email	NOUN
gojar-5993	206	2	:	:	PUNCT
gojar-5993	206	3	rozyngujmadu@yahoo	rozyngujmadu@yahoo	PROPN
gojar-5993	206	4	.	.	PROPN
gojar-5993	206	5	com	com	PROPN
gojar-5993	206	6	apa	apa	PROPN
gojar-5993	206	7	laisin	laisin	PROPN
gojar-5993	206	8	,	,	PUNCT
gojar-5993	206	9	m.	m.	NOUN
gojar-5993	206	10	,	,	PUNCT
gojar-5993	206	11	edike	edike	ADJ
gojar-5993	206	12	,	,	PUNCT
gojar-5993	206	13	c.	c.	PROPN
gojar-5993	206	14	,	,	PUNCT
gojar-5993	206	15	&	&	CCONJ
gojar-5993	206	16	ujumadu	ujumadu	PROPN
gojar-5993	206	17	,	,	PUNCT
gojar-5993	206	18	r.	r.	PROPN
gojar-5993	206	19	n.	n.	PROPN
gojar-5993	206	20	(	(	PUNCT
gojar-5993	206	21	2025	2025	NUM
gojar-5993	206	22	)	)	PUNCT
gojar-5993	206	23	.	.	PUNCT
gojar-5993	207	1	on	on	ADP
gojar-5993	207	2	boundedness	boundedness	NOUN
gojar-5993	207	3	and	and	CCONJ
gojar-5993	207	4	solution	solution	NOUN
gojar-5993	207	5	size	size	NOUN
gojar-5993	207	6	in	in	ADP
gojar-5993	207	7	rational	rational	ADJ
gojar-5993	207	8	linear	linear	ADJ
gojar-5993	207	9	programming	programming	NOUN
gojar-5993	207	10	and	and	CCONJ
gojar-5993	207	11	polyhedral	polyhedral	ADJ
gojar-5993	207	12	optimization	optimization	NOUN
gojar-5993	207	13	.	.	PUNCT
gojar-5993	208	1	global	global	ADJ
gojar-5993	208	2	online	online	PROPN
gojar-5993	208	3	journal	journal	PROPN
gojar-5993	208	4	of	of	ADP
gojar-5993	208	5	academic	academic	ADJ
gojar-5993	208	6	research	research	NOUN
gojar-5993	208	7	(	(	PUNCT
gojar-5993	208	8	gojar	gojar	NOUN
gojar-5993	208	9	)	)	PUNCT
gojar-5993	208	10	,	,	PUNCT
gojar-5993	208	11	4(1	4(1	NOUN
gojar-5993	208	12	)	)	PUNCT
gojar-5993	208	13	,	,	PUNCT
gojar-5993	208	14	60	60	NUM
gojar-5993	208	15	-	-	SYM
gojar-5993	208	16	72	72	NUM
gojar-5993	208	17	.	.	PUNCT
gojar-5993	209	1	https://klamidas.com/gojar-v4n1-2025-04/.	https://klamidas.com/gojar-v4n1-2025-04/.	X
gojar-5993	209	2	mla	mla	PROPN
gojar-5993	209	3	laisin	laisin	VERB
gojar-5993	209	4	,	,	PUNCT
gojar-5993	209	5	mark	mark	NOUN
gojar-5993	209	6	,	,	PUNCT
gojar-5993	209	7	edike	edike	ADJ
gojar-5993	209	8	,	,	PUNCT
gojar-5993	209	9	collins	collin	NOUN
gojar-5993	209	10	,	,	PUNCT
gojar-5993	209	11	&	&	CCONJ
gojar-5993	209	12	ujumadu	ujumadu	PROPN
gojar-5993	209	13	,	,	PUNCT
gojar-5993	209	14	r.	r.	PROPN
gojar-5993	209	15	n.	n.	PROPN
gojar-5993	209	16	“	"	PUNCT
gojar-5993	209	17	on	on	ADP
gojar-5993	209	18	boundedness	boundedness	NOUN
gojar-5993	209	19	and	and	CCONJ
gojar-5993	209	20	solution	solution	NOUN
gojar-5993	209	21	size	size	NOUN
gojar-5993	209	22	in	in	ADP
gojar-5993	209	23	rational	rational	ADJ
gojar-5993	209	24	linear	linear	ADJ
gojar-5993	209	25	programming	programming	NOUN
gojar-5993	209	26	and	and	CCONJ
gojar-5993	209	27	polyhedral	polyhedral	ADJ
gojar-5993	209	28	optimization	optimization	NOUN
gojar-5993	209	29	”	"	PUNCT
gojar-5993	209	30	.	.	PUNCT
gojar-5993	210	1	global	global	ADJ
gojar-5993	210	2	online	online	PROPN
gojar-5993	210	3	journal	journal	PROPN
gojar-5993	210	4	of	of	ADP
gojar-5993	210	5	academic	academic	ADJ
gojar-5993	210	6	research	research	NOUN
gojar-5993	210	7	(	(	PUNCT
gojar-5993	210	8	gojar	gojar	NOUN
gojar-5993	210	9	)	)	PUNCT
gojar-5993	210	10	,	,	PUNCT
gojar-5993	210	11	vol	vol	NOUN
gojar-5993	210	12	.	.	PROPN
gojar-5993	210	13	4	4	NUM
gojar-5993	210	14	,	,	PUNCT
gojar-5993	210	15	no	no	INTJ
gojar-5993	210	16	.	.	NOUN
gojar-5993	210	17	1	1	NUM
gojar-5993	210	18	,	,	PUNCT
gojar-5993	210	19	2025	2025	NUM
gojar-5993	210	20	,	,	PUNCT
gojar-5993	210	21	pp	pp	ADJ
gojar-5993	210	22	.	.	PUNCT
gojar-5993	211	1	60	60	NUM
gojar-5993	211	2	-	-	SYM
gojar-5993	211	3	72	72	NUM
gojar-5993	211	4	.	.	PUNCT
gojar-5993	212	1	https://klamidas.com/gojarv4n1-2025-04/.	https://klamidas.com/gojarv4n1-2025-04/.	NOUN