Algorithmica 2001 Springer-Verlag New York Inc.

Transcription

1 Algorthmca (2001) 29: DOI: /s Algorthmca 2001 Sprnger-Verlag New York Inc. Effcent Algorthms for Integer Programs wth Two Varables per Constrant 1 R. Bar-Yehuda 2 and D. Rawtz 2 Abstract. Gven a bounded nteger program wth n varables and m constrants, each wth two varables, we present an O(mU) tme and O(m) space feasblty algorthm, where U s the maxmal varable range sze. We show that wth the same complexty we can fnd an optmal soluton for the postvely weghted mnmzaton problem for monotone systems. Usng the local-rato technque we develop an O(nmU) tme and O(m) space 2-approxmaton algorthm for the postvely weghted mnmzaton problem for the general case. We further generalze all results to nonlnear constrants (called axs-convex constrants) and to nonlnear (but monotone) weght functons. Our algorthms are not only better n complexty than other known algorthms, but also consderably smpler, and they contrbute to the understandng of these very fundamental problems. Key Words. Combnatoral optmzaton, Integer programmng, Approxmaton algorthm, Local-rato technque, 2SAT, Vertex cover. 1. Introducton. Ths paper s motvated by a paper by Hochbaum et al. [11] whch dscusses nteger programs wth two varables per constrant. The problem s defned as follows: (2VIP) mn n =1 w x s.t. a k x k + b k x k c k, k {1,...,m}, l x u, {1,...,n}, x N, {1,...,n}, where 1 k, k n, w 0, a, b, c Z m, and l, u N n. Obvously, ths problem s a generalzaton of the well-known mnmum weght vertex cover problem (VC) and the mnmum weght 2 satsfablty problem (2SAT). Both problems are known to be NP-hard [7], and the best known approxmaton rato for VC [3], [10], [14] and 2SAT [9] s asymptotcally 2. Both results are best vewed va the local-rato technque (see [2] and [4]). A 2VIP system s called monotone f each constrant s an nequalty on two varables wth coeffcents of opposte sgns. The problem of checkng whether an nteger monotone system has a feasble soluton was shown to be NP-complete by Lagaras [13]. Therefore, even for the 2VIP feasblty problem, t s natural to consder pseudopolynomal algorthms,.e., algorthms wth a runnng tme whch s polynomal n the sze of the nput and n U, where U = max {u l }. 1 A prelmnary verson of ths paper appeared n the Proceedngs of the 7th Annual European Symposum on Algorthms (1999). 2 Computer Scence Department, Technon - IIT, Hafa Israel. {reuven,rawtz}@cs.technon.ac.l. Receved June 21, 1996; revsed December 5, Communcated by N. Megddo. Onlne publcaton December 15, 2000.

2 596 R. Bar-Yehuda and D. Rawtz Hochbaum and Naor [12] were the frst to consder effcent algorthms for nteger programs wth two varables per nequalty. They presented an O(mn 2 log m + nmu) tme feasblty algorthm for monotone 2VIP systems, and an optmzaton algorthm for monotone systems wth general (possbly negatve) weghts. Ths optmzaton algorthm constructs a graph representng the monotone system n queston and then uses a maxmum flow algorthm. Consequently, the tme complexty of ther algorthm s relatvely hgh,.e., when usng Goldberg and Taran s maxmum-flow algorthm [8] t s O(nmU 2 log(n 2 U/m)). Usng ths optmzaton algorthm, Hochbaum et al. [11] were able to construct an O(nmU 2 log(n 2 U/m)) tme 2-approxmaton algorthm for the 2VIP problem. Ther algorthm also uses an O(mU) tme and space 2VIP feasblty algorthm based on a transformaton to 2SAT whch they attrbute to Feder. By usng the local-rato technque, we present an O(nmU) tme and O(m) space 2- approxmaton algorthm. Ths algorthm s not only more effcent, but also more natural and smpler. In order to develop an approxmaton algorthm, t seems natural to study the feasblty problem frst. 3 Indeed, our 2-approxmaton algorthm s n fact a specfc mplementaton of a feasblty algorthm presented here. The remander of ths paper s organzed as follows: In Secton 2 we present an O(mU) tme and O(m) space feasblty algorthm for 2VIP systems. The 2-approxmaton algorthm for 2VIP systems s presented n Secton 3. In Secton 4 we show that the feasblty algorthm and the approxmaton algorthm presented n ths paper can be generalzed to some nonlnear systems wth the same tme and space complexty. We defne a generalzaton of lnear nequaltes, called axs-convex constrants, and show that the algorthms can be generalzed to work wth such constrants. We also generalze the 2-approxmaton algorthm to obectve functons of the form n =1 w (x ), where all the w s are monotone weght functons. An optmalty algorthm for monotone lnear systems appears n Secton 5. We show that ths algorthm can work wth some nonlnear constrants, and we generalze the algorthm to monotone weght functons as well. Table 1 summarzes the results for 2VIP systems. Table 1. Summary of results. Problem Prevous results (tme, space) Our results (tme, space) 2SAT feasblty O(m), O(m) [6] 2VIP feasblty O(mU), O(mU) O(mU), O(m) by usng reducton to 2SAT [11] 2SAT 2-approxmaton O(nm), O(n 2 + m) [9] O(nm), O(m) 2VIP 2-approxmaton O(nmU 2 log(n 2 U/m)), O(mU) [11] O(nmU), O(m) Monotone 2VIP O(mU), O(mU) O(mU), O(m) optmzaton by usng reducton to 2SAT 3 Ths was done by Hochbaum and Naor [12] for the monotone 2VIP problem, by Hochbaum et al. [11] for the 2VIP problem, and by Gusfeld and Ptt [9] for the 2SAT problem.

3 Effcent Algorthms for Integer Programs wth Two Varables per Constrant Feasblty Algorthm. Gven a 2VIP system, we are nterested n developng an algorthm whch fnds a feasble soluton, f such a soluton exsts. Snce the specal case where l = 0 n and u = 1 n s the known 2SAT feasblty problem, t s natural to try to extend the well known O(m) tme and space algorthm of Even et al. [6]. It s possble to transform the gven 2VIP system nto an equvalent 2SAT nstance wth nu varables and (m + n)u constrants (ths transformaton due to Feder appears n [11]). By combnng ths transformaton wth the lnear tme and space algorthm of Even et al. we get an O(mU) tme and space feasblty algorthm. In ths secton we present an O(mU) tme and O(m) space feasblty algorthm whch generalzes the algorthm by Even et al. The man dea of the algorthm from [6] s as follows: choose a varable x t, frst dscover the forced values of other varables by assgnng x t = 0, and then do the same for the assgnment x t = 1. If one of these assgnments does not lead to a contradcton, assgn ths value to x t and make the correspondng forced assgnments. The correctness of the approach of Even et al. s shown by provng that a noncontradctory assgnment preserves the feasblty property. The effcency of ther algorthm s acheved by dscoverng the forced assgnments of x t = 0 and those of x t = 1 n parallel. Our 2VIP feasblty algorthm works wth bounds, as opposed to assgnments. We choose a varable x t and an nteger α [l t, u t ] and dscover the forced bounds of other varables by consderng n turn the bound x t α and the bound x t >α. For ths purpose we use constrant propagaton. 4 Much lke Even et al. [6], we prove that a noncontradctory bound preserves the feasblty property, and use ths to show the correctness of our algorthm. Also, the effcency of our algorthm s acheved by dscoverng the forced bounds by both new bounds of x t n parallel. The purpose of ths secton s not only to show the factor (U) mprovement n space complexty, but also to lay the foundatons for the 2-approxmaton algorthm presented n the next secton. DEFINITION 1. For a gven 2VIP nstance, sat(l, u) ={x: l x u and x satsfes all 2VIP constrants}. DEFINITION 2. For the kth constrant (on the varables x k, x k ) of a gven 2VIP nstance, constrant(k) ={(α, β): x k = α, x k = β satsfy constrant k}. DEFINITION 3. Gven α, β Z we defne [α, β] ={z Z: α z β}. A lnear constrant s convex, thus the followng hold for the kth constrant on the varables x and x : OBSERVATION 1. δ [β,γ]. If (α, β), (α, γ ) constrant(k), then (α, δ) constrant(k) for all 4 Constrant propagaton was prevously used for the LP verson of the problem (e.g., see [1] and [15]), and n [12] for nteger feasblty over monotone nequaltes.

4 598 R. Bar-Yehuda and D. Rawtz Fg. 1. Routne OneOnOneImpact. OBSERVATION 2. If (α 1,α 2 ), (β 1,β 2 ), (γ 1,γ 2 ) constrant(k), then all ponts nsde the trangle nduced by (α 1,α 2 ), (β 1,β 2 ), and (γ 1,γ 2 ) satsfy constrant k. We present a routne n Fgure 1 whch wll be repeatedly used for constrant propagaton. It receves as nput two arrays l and u of sze n (passed by reference), two varables ndces,, and a constrant ndex k on these two varables. The obectve of ths routne s to fnd the mpact of constrant k and the bounds l, u on the bounds l, u. We denote by l after and u after the values of the bounds l and u after callng OneOnOne- Impact(l, u,,, k). OBSERVATION 3. OBSERVATION 4. constrant(k). sat(l after, u after ) = sat(l, u). If β [l after, u after ] there exsts α [l after, u after ] such that (α, β) The routne n Fgure 2, whch s called OneOnAllImpact, receves as nput two arrays l and u of sze n (passed by reference) and a varable ndex t, and changes l and u accordng to the mpact of l t and u t on all the ntervals. We now prove that we do not lose feasble solutons after actvatng OneOnAllImpact. Fg. 2. Routne OneOnAllImpact.

5 Effcent Algorthms for Integer Programs wth Two Varables per Constrant 599 LEMMA 1. If l after and u after are the values of l and u after callng OneOnAllImpact, then sat(l after, u after ) = sat(l, u). PROOF. All changes to l and u are made by routne OneOnOneImpact. It s easy to prove the lemma by nducton usng Observaton 3. LEMMA 2. If OneOnAllImpact(l, u, t) termnates wthout falure wth the bounds l after and u after and sat((l 1,...,l t 1,,l t+1,...,l n ), (u 1,...,u t 1,, u t+1,...,u n )), then sat(l after, u after ). PROOF. Let y sat((l 1,...,l t 1,,l t+1,...,l n ), (u 1,...,u t 1,, u t+1,..., u n )). We defne a vector y as y t, y t [l after y t t, u after t ], = l after t, y t <l after t, u after t, y t > u after t. Consder constrant k on x and x. We need to show that y, y constrant(k). Case 1: y [l after, u after ] and y [l after, u after ]. (y, y ) = (y, y ) constrant(k). Case 2: y <l after and y [l after, u after ]. y s a feasble soluton, thus (y, y ) constrant(k). When we changed the lower bound of x to l after we called OneOnOneImpact for all constrants nvolvng x ncludng constrant k. By Observaton 4 there exsts α [l after, u after ] for whch (α, y ) constrant(k). Thus, by Observaton 1 we get that, y ) constrant(k). (l after Case 3: y <l after and y <l after. y s a feasble soluton, thus (y, y ) constrant(k). When we changed the lower bound of x to l after we called OneOnOneImpact for all constrant nvolvng x ncludng constrant k. By Observaton 4 there exsts α [l after, u after ] for whch (α, l after ) constrant(k). From the same arguments we get that there exsts β [l after, u after ] for whch (l after, β) constrant(k) as well. Thus, by Observaton 2 we get that (l after,l after ) constrant(k). Other cases are smlar to Cases 2 and 3. After provng that OneOnAllImpact preserves the feasblty property t s possble to use ths routne as part of the feasblty algorthm from Fgure 3. THEOREM 1. Algorthm Feasblty returns a feasble soluton f such a soluton exsts. PROOF. Each recursve call reduces at least one of the ranges (the t th), thus the executon of the algorthm must termnate. By Lemma 1 f sat(l, u) = the algorthm wll return fal. On the other hand, for the case sat(l, u) we can prove by nducton on n =1 (u l ) that the algorthm fnds a feasble soluton. Base. n =1 (u l ) = 0 mples l = u, thus x = l s a feasble soluton.

6 600 R. Bar-Yehuda and D. Rawtz Fg. 3. Feasblty algorthm. Step. By Lemma 1 at least one of the calls to OneOnAllImpact termnates wthout falure. If call left was chosen, then by Lemma 2 we know that sat(l left, u left ). Therefore, by the nducton hypothess we can fnd a feasble soluton for l left, u left. Obvously, a feasble soluton x sat(l left, u left ) satsfes x sat(l, u). The same apples for call rght. Ths concludes the proof. THEOREM 2. O(m). Algorthm Feasblty can be mplemented n tme O(mU) and space PROOF. To acheve a tme complexty of O(mU), we run both calls to OneOnAllImpact n parallel (ths approach was used for 2SAT by Even et al. [6]), and prefer the faster opton of the two, f such a choce exsts. After every change n the range of a varable x, we need to check the m constrants nvolvng ths varable, n order to dscover the mpact of the change. To perform ths task effcently we can store the nput n an ncdence lst, where every varable has ts constrants lst. As l and u can change up to (u l ) tmes, we conclude that the total tme complexty of the changes s O( n =1 m (u l )) = O(mU) (the tme wasted on uncompleted trals s bounded by the tme complexty of the chosen trals). The algorthm uses O(m) space for the nput and a constant number of arrays of sze n, thus uses lnear space. 3. From Feasblty to Approxmaton. Before presentng our approxmaton algorthm, we frst dscuss the specal case where U = 2, whch s the mnmum 2SAT problem, and ts specal case, the vertex cover problem. The man dea of Bar-Yehuda and Even s 2-approxmaton algorthm for the vertex cover problem [2] [4] s as follows: choose an edge e = (u,v) and subtract ε =

7 Effcent Algorthms for Integer Programs wth Two Varables per Constrant 601 mn{w(u), w(v)} from both w(u) and w(v). Every vertex cover C V must cover e, therefore the subtracton of ε from w(u), w(v), whch may cost up to 2 ε, reduces the optmum by at least ε. After repeatng ths process untl such subtractons are no longer possble, the resultng cover s C ={v: w(v) = 0}. When consderng a 2CNF formula, 5 wth respect to the mnmum 2SAT problem, monotone clauses (e.g., x x ) can be treated as n the vertex cover case. However, what about clauses wth negatve lterals (of the form x x or x x )? Also, even f we knew how to make weght subtractons, how would we choose a truth assgnment? Thus, the above algorthm should be modfed n order to be employed for the mnmum 2SAT problem. The dea s that for a gven 2CNF formula, f x 1 x 2 and x 1 x 3 we can subtract ε = mn{w 2,w 3 } from w 2 and w 3. Ths leaves us wth the task of choosng a feasble truth assgnment: when t s possble, assgn a zero cost partal assgnment, whle relyng upon the consstency property (Lemma 2). Ths approach was used by Gusfeld and Ptt [9] for approxmatng the mnmum 2SAT problem. The 2CNF formula can be presented as a dgraph where each vertex represents a boolean varable or ts negaton, and an edge represents a OneOnOneImpact propagaton (logcal ). A propagaton of an assgnment can be vewed as a traversal (e.g., BFS, DFS) of the dgraph. In order to be able to propagate forced assgnments, Gusfeld and Ptt s algorthm starts wth a preprocess phase of constructng a transtve closure. Ths phase uses (n 2 ) extra memory, whch s expensve. It s much more crtcal when tryng to approxmate 2VIP by usng the transformaton to 2SAT from [11], and then Gusfeld and Ptt s [9] algorthm. In ths case the preprocess uses (n 2 U 2 ) extra memory. 6 It seems only natural to try to extend the prevous deas when approxmatng the 2VIP problem. In the prevous secton we have seen that t s possble to generalze the 2CNF assgnment propagaton to a 2VIP bound propagaton, but how should we mplement weght subtractons? We can vew a boolean assgnment as a bound change, thus the cost of the assgnment x = 1 s actually the cost of rasng the lower bound of x from 0 to 1. For a 2VIP nstance, every unt ncrease of the varable x would cost w,orn other words, an ncrease of the lower bound l to l costs w (l l ). Bearng that n mnd, we defne an array called ˆl, whch holds the values (of the varables), for whch we have already pad. Ths means that nstead of reducng a weght due to a rse of l we ncrease ˆl. Note that, unlke l, ˆl can hold nonntegral values. Ths allows us to avod usng drect weght reductons and the consequent reducton to 2SAT. We present an O(nmU) tme and O(m) space 2-approxmaton algorthm. Ths approxmaton algorthm s a specfc mplementaton of our feasblty algorthm (namely, the algorthm chooses varables and bounds n a specfc order to get a 2-approxmaton). Not only does ths algorthm seem natural, but also ts complexty, O(nm) tme and O(m) space, n the case of 2SAT s lower than that of Gusfeld and Ptt s 2SAT algorthm (O(nm) and O(n 2 + m) correspondngly). In order to use the local-rato technque [2] we extend the 2VIP defnton. 5 Conunctve Normal Form, see, e.g., [5]. 6 Ths s n addton to the (mu 2 ) extra memory needed for the transformaton. As far as we know, every algorthm whch reles upon drect 2SAT transformaton suffers from ths drawback.

8 602 R. Bar-Yehuda and D. Rawtz DEFINITION 4. Gven a, b R n, we wrte a b f a b. Also, we defne max{a, b} =(max{a 1, b 1 },...,max{a n, b n }), mn{a, b} =(mn{a 1, b 1 },...,mn{a n, b n }). Gven l, u N n and ˆl, û R n for whch l ˆl û u, we defne the followng Extended 2VIP problem: (E2VIP) mn n =1 (x, ˆl, û )w s.t. a k x k + b k x k c k, k {1,...,m}, x [l, u ], {1,...,n}, where 0, x < ˆl, (x, ˆl, û ) = (x ˆl ), x [ ˆl, û ], (û ˆl ), x > û, and 1 k, k n, w 0, a, b, c Z m, and l, u N n. We defne W (x, ˆl, û) = n =1 (x, ˆl, û )w. A feasble soluton x s called an optmal soluton f, for every feasble soluton x, W (x, ˆl, û) W (x, ˆl, û). Wedenote W ( ˆl, û) = W (x, ˆl, û). A feasble soluton x s called an r-approxmaton f W (x, ˆl, û) r W ( ˆl, û). OBSERVATION 5. Gven ˆl, û, ˆm R n for whch ˆl ˆm û, we get W (x, ˆl, û) = W (x, ˆl, ˆm) + W (x, ˆm, û). Smlarly to the Decomposton Observaton from [2] we have: OBSERVATION 6 (Decomposton Observaton). ˆm û, W ( ˆl, ˆm) + W ( ˆm, û) W ( ˆl, û). Gven ˆl, û, ˆm R n such that ˆl PROOF. Let x, y, and z be optmal solutons for the system wth respect to ˆl, ˆm; ˆm, û; and ˆl, û correspondngly. W ( ˆl, ˆm) + W ( ˆm, û) = W (x, ˆl, ˆm) + W (y, ˆm, û) (by defnton) W (z, ˆl, ˆm) + W (z, ˆm, û) (optmalty of x, y ) W (z, ˆl, û) (observaton 5) = W ( ˆl, û) (by defnton). The followng s ths paper s Local-Rato Theorem (see [2] and [4]): THEOREM 3. If x s an r-approxmaton wth respect to ˆl, ˆm and wth respect to ˆm, û, then x s an r-approxmaton wth respect to ˆl, û.

9 Effcent Algorthms for Integer Programs wth Two Varables per Constrant 603 Fg Approxmaton algorthm. PROOF. W (x, ˆl, û) = W (x, ˆl, ˆm) + W (x, ˆm, û) (Observaton 5) r W ( ˆl, ˆm) + r W ( ˆl, û) (gven) r W ( ˆl, û) (Decomposton Observaton). We are ready to present the 2-approxmaton algorthm see Fgure 4. OBSERVATION 7. Feasblty. Algorthm Approxmate s a specfc mplementaton of Algorthm THEOREM 4. systems. Algorthm Approxmate s a 2-approxmaton algorthm for E2VIP PROOF. By Observaton 7 Algorthm Approxmate returns a feasble soluton. We prove by nducton on the depth of the recurson that the algorthm fnds a 2-approxmaton.

10 604 R. Bar-Yehuda and D. Rawtz Base. u = l mples W (l, ˆl, û) = 0. Step. There are several cases: Case 1: ˆl l. A 2-approxmaton wth respect to max{ ˆl, l} s obvously a 2- approxmaton soluton wth respect to ˆl. Case 2: û u. Trval. Case 3: Call rght faled. By Lemma 1 there s no feasble soluton whch satsfes x t α + 1, therefore we do not change the problem by addng the constrant x y α. By Lemma 1 callng OneOnAllImpact(l left, u left, t) does not change the problem ether. Case 4: Call left faled. Smlar to Case 3. Case 5: Both calls succeeded and W (l left, ˆl, û) W (l rght, ˆl, û). We frst show that every feasble soluton s a 2-approxmaton wth respect to ˆl and ˆm. We examne an optmal soluton x wth respect to ˆl and ˆm. By the constructon of ˆm we know that ˆm l left. Thus, f l left x u left, then the monotoncty of W (, ˆl, ˆm) mples If l rght x u rght, then W (x, ˆl, ˆm) W (l left, ˆl, ˆm). W (x, ˆl, ˆm) W (l rght, ˆl, ˆm) (x l rght ) W (l left, ˆl, û) (by the constructon of ˆm) = W (l left, ˆl, ˆm) (l left ˆm). On the other hand, by the constructon of ˆm: W ( ˆm, ˆl, ˆm) W (l left, ˆl, ˆm) + W (l rght, ˆl, ˆm) 2 W (l left, ˆl, ˆm). Thus, for every feasble soluton x: W (x, ˆl, ˆm) 2 W (l left, ˆl, ˆm). Therefore, by Theorem 3 a 2-approxmaton wth respect to ˆm and û s a 2-approxmaton wth respect to ˆl and û. We need to show that there exsts an optmal soluton x wth respect to ˆm and û for whch x α. For every feasble soluton y such that y α + 1 we defne y as y, y = l left, y <l left u left, y > u left y [l left, u left By Lemma 2 y s a feasble soluton. l left ˆm mples W (y, ˆm, û) W (y, ˆm, û), thus there s an optmal soluton wth respect to ˆm and û wthn the bounds l left, u left. Therefore, a 2-approxmaton wthn the bounds l left, u left s a 2-approxmaton wth respect to ˆm and û. Case 6: Both calls succeeded and W (l left, ˆl, û) >W (l rght, ˆl, û). Smlar to Case 5. Ths concludes the proof.,. ],

11 Effcent Algorthms for Integer Programs wth Two Varables per Constrant 605 COROLLARY 5. systems. THEOREM 6. O(m). Algorthm Approxmate s a 2-approxmaton algorthm for 2VIP Algorthm Approxmate can be mplemented n tme O(nmU) and space PROOF. In order to get the requred tme complexty, we must choose the x t s carefully. One possblty s to choose the varables n an ncreasng order,.e., x 1, x 2,...,x n, and to restart from the begnnng after reachng x n. We call such n teratons on all n varables a pass. As stated before, changng the range of x mght cause changes n the ranges of other varables. The exstence of a constrant on x and another varable x makes x a canddate for a range update. Ths means that we have to check the m constrants nvolvng x n order to dscover the consequences of changng ts range each tme ths range changes. l and u can change up to (u l ) tmes, therefore we get that the tme complexty of a sngle teraton s O(mU + n) = O(mU). One pass may nvolve all n varables, so the tme complexty of one pass s O(nmU). By choosng α = 1 2 (l t + u t ), we reduce the possble range for x t at least by half. Therefore, n a sngle pass we reduce the possble ranges for all varables at least by half. Thus, we get that the total tme complexty s log U k=1 O(mn(U/2k )) = O(mnU). As before, the algorthm uses an ncdence lst data structure and a constant number of arrays of sze n, thus uses lnear space. 4. Generalzatons. What f the constrants are not lnear? Also, what f the weght functons of the varables are not lnear? Can we avod the expensve transformaton to the 2SAT problem? In ths secton we try to extend our approach to a wder famly of problems. In order to do so, we had to dentfy the propertes of lnear constrants and lnear weght functons whch are suffcent for the correctness of our clams Generalzed Constrants. In ths subsecton we are nterested n problems of the form: (G2VIP) mn n =1 w x s.t. (x k, x k ) constrant(k), k {1,...,m}, x [l, u ], {1,...,n}, where 1 k, k n, w 0, and l, u N n. Also each constrant(k) satsfes the followng: f (α 1,α 2 ), (β 1,β 2 ) constrant(k), then there exsts a shortest path n N 2 (lattce) connectng (α 1,α 2 ) and (β 1,β 2 ) such that all ts ponts satsfy constrant k. A constrant whch satsfes ths property s called an axs-convex constrant. We assume that for an axs-convex constrant we have an O(1) tme oracle OneOnOneImpact whch returns a tght range on x when gven a range for x. The followng s mpled by the defnton of an axs-convex constrant: OBSERVATION 8. Gven an axs-convex constrant C, f (α 1,α 2 ), (β 1,β 2 ) C, then for all β [β 1,β 2 ] there exsts α [α 1,α 2 ] such that (α, β) C.

12 606 R. Bar-Yehuda and D. Rawtz Observatons 1 and 4 hold for axs-convex constrants. A weaker verson of Observaton 2 also holds for axs-convex constrants. The proof uses Observaton 8. OBSERVATION 9. Gven an axs-convex constrant C and (α 1,α 2 ), (β 1,β 2 ), (γ 1,γ 2 ) C, f (α 1,β 2 ) s nsde the trangle nduced by (α 1,α 2 ), (β 1,β 2 ), and (γ 1,γ 2 ), then (α 1,β 2 ) C. It s easy to see that Lemmas 1 and 2 reman vald wth axs-convex constrants. Thus, Algorthm Feasblty and Algorthm Approxmate can be appled to G2VIP systems. COROLLARY 7. A feasble soluton, f such a soluton exsts, can be found for G2VIP systems n tme O(mU) and space O(m). COROLLARY 8. A 2-approxmaton, f a feasble soluton exsts, can be found for G2VIP systems n tme O(nmU) and space O(m) Generalzed Weght Functons. In the orgnal defnton of a 2VIP system we used lnear weght functons (w x for every varable x ). The followng defnton generalzes the weght functon of a varable x : DEFINITION 5. A nonnegatve weght functon ω s called a monotone weght functon wth respect to an nterval f ω(α) ω(α + 1) for all α Z n the nterval. We assume, wthout loss of generalty, that a monotone weght functon s monotone over the real numbers and s nvertble. If t s not we can always defne ω (z) = ω( z )+ (ω( z ) ω( z )) (z z ). For a lnear weght functon ω we get that α 0 ω (α+1) ω (α) = w. For a general monotone weght functon ths s not necessarly the case. Therefore, n order to make Algorthm Approxmate applcable to such weght functons we replace (x, ˆl, û )w n the defnton of E2VIP systems by ω (x ) ω ( ˆl ), x [ ˆl, û ]. (x, ˆl, û ) = ω (û ) ω ( ˆl ), x > û, 0, x < ˆl. Now Algorthm Approxmate s applcable as s. COROLLARY 9. A 2-approxmaton, f a feasble soluton exsts, can be found for systems wth monotone weght functons n tme O(nmU) and space O(m). REMARK 1. If x { s,1,...,s,n }, we can defne a new varable x {0,...,n 1} and a new monotone weght functon w (α) = w (s,α+1 ). Thus, we get that the same results hold for l = 0 and u = n 1.

13 Effcent Algorthms for Integer Programs wth Two Varables per Constrant Optmzaton Algorthm for Monotone Systems. In ths secton we show how to mprove the complexty of Hochbaum and Naor s [12] optmzaton algorthm for a specfc case of 2VIP systems called monotone systems: DEFINITION 6. A monotone nequalty s an nequalty on two varables wth coeffcents of opposte sgns. A system wth two varables per nequalty s called monotone f all the constrants n the system are monotone. We formulate the followng monotone system: (M2VIP) mn n =1 w x s.t. a k x k b k x k c k, k {1,...,m}, x [l, u ], {1,...,n}, where 1 k, k n, w 0, a, b N n, c Z n, and l, u N. Lagaras [13] proved that decdng whether a monotone nteger program has a feasble soluton s NP-complete. Hochbaum and Naor [12] showed that the set of all feasble solutons of a monotone system form a dstrbuton lattce (ths has been observed before by Venott [16]). Usng ths property they presented an O(mn 2 log m + nmu) tme feasblty algorthm for monotone systems, whch fnds the top (or bottom) of ths lattce. They also presented an O(nmU 2 log(n 2 U/m)) tme optmzaton algorthm for such systems wth general (possbly negatve) weghts. By lmtng the dscusson to nonnegatve weghts, one can construct an O(mU) tme optmzaton algorthm: use the transformaton to a 2SAT nstance [11], then assgn 0 to all boolean varables whch dd not get an assgnment by the transformaton. As before, ths transformaton uses O(mU) space. In Fgure 5 we present an O(mU) tme and O(m) space optmalty algorthm for monotone systems wth nonnegatve weghts. Ths algorthm s actually a feasblty algorthm whch fnds the bottom of the feasble solutons lattce. 7 Due to the nonnegatve weghts, the bottom of the lattce s also an optmal soluton. THEOREM 10. An optmal soluton can be found for nonnegatve monotone systems n tme O(mU) and space O(m). Fg. 5. Optmzaton algorthm for monotone systems. 7 Note that when gven l as the ntal value, the feasblty algorthm from [12] can be mplemented n tme O(mU) and space O(m).

14 608 R. Bar-Yehuda and D. Rawtz PROOF. We prove that Algorthm Monotone fnds an optmal soluton for M2VIP systems wth nonnegatve weghts n tme O(mU) and space O(m). It s easy to prove by nducton usng Lemma 1 that sat(l after, u after ) = sat(l, u), where l after and u after are the values of l and u after the loop. Thus, f one of the calls to OneOnAllImpact fals, then no feasble soluton exsts. The lemma also mples that f the nstance of M2VIP s satsfable, then the above algorthm termnates wthout falure. We now prove that x = l s a feasble soluton by showng that every constrant k: a k x k b k x k c k s satsfed by x = l. We examne the value of the lower bound of x k after the last call of OneOnOneImpact(l, u, k, k, k) durng the algorthm (the call s made at least once). After ths call we get l k (c k + b k l k )/a k. l k wll not change from ths pont untl the algorthm termnates (as a change of l k mples another call), and other lower bounds only ncrease durng the executon algorthm, thus l k,l k must satsfy constrant k. Due to Lemma 1, we get that x = l s an optmal soluton (and t s the only optmal soluton f w > 0). Based on the same arguments as n Theorem 6 we get that the tme complexty of fndng an optmal soluton for a monotone system s O( n =1 m (u l )) = O(mU). The algorthm uses an ncdence lst data structure and therefore uses lnear space. As n the prevous secton we try to extend our approach to some nonlnear systems. We use an mportant property of monotone constrants: DEFINITION 7. Let constrant k be an axs-convex constrant on x and x. Constrant k s called a monotone axs-convex constrant f for every (α 1,α 2 ), (β 1,β 2 ) constrant(k) also (mn{α 1,β 1 }, mn{α 2,β 2 }) constrant(k). A system of monotone axs-convex constrants s called a generalzed monotone system. By usng smlar arguments to those used n the prevous secton we get: COROLLARY 11. An optmal soluton can be found for generalzed monotone systems wth monotone weght functons n tme O(mU) and space O(m). Acknowledgments. We thank Ar Freund, Nv Glboa, Avgal Orn, and especally Hadas Heer for ther careful readng and suggestons. References [1] B. Aspvall and Y. Shloach. A polynomal tme algorthm for solvng systems of lnear nequaltes wth two varables per nequalty. SIAM Journal on Computng, 9: , [2] R. Bar-Yehuda. One for the prce of two: a unfed approach for approxmatng coverng problems. In Proceedngs of the 1st Internatonal Workshop on Approxmaton Algorthms for Combnatoral Optmzaton Problems, pages 49 62, July Also n Algorthmca, 27: , [3] R. Bar-Yehuda and S. Even. A lnear tme approxmaton algorthm for the weghted vertex cover problem. Journal of Algorthms, 2: , 1981.

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