A New Approximation Method for Set Covering Problems, with Applications to Multidimensional Bin Packing

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1 A New Approxmaton Method for Set Coverng Problems, wth Applcatons to Multdmensonal Bn Pacng Nhl Bansal Alberto Caprara Maxm Svrdeno Abstract In ths paper we ntroduce a new general approxmaton method for set coverng problems, based on the combnaton of randomzed roundng of the (near-)optmal soluton of the Lnear Programmng (LP) relaxaton, leadng to a partal nteger soluton, and the applcaton of a wellbehaved approxmaton algorthm to complete ths soluton. If the value of the soluton returned by the latter can be bounded n a sutable way, as s the case for the most relevant generalzatons of bn pacng, the method leads to mproved approxmaton guarantees, along wth a proof of tghter ntegralty gaps for the LP relaxaton. For d-dmensonal vector pacng, we obtan a polynomal-tme randomzed algorthm wth asymptotc approxmaton guarantee arbtrarly close to ln d + 1. For d = 2, ths value s ,.e., we brea the natural 2 barrer for ths case. Moreover, for small values of d ths s a notable mprovement over the prevously-nown O(ln d) guarantee by Cheur and Khanna [8]. For 2-dmensonal bn pacng wth and wthout rotatons, we obtan polynomal-tme randomzed algorthms wth asymptotc approxmaton guarantee , mprovng upon prevous algorthms wth asymptotc performance guarantees arbtrarly close to 2 by Jansen and van Stee [17] for the problem wth rotatons and by Caprara [5] for the problem wthout rotatons. The prevously-unnown ey property used n our proofs follows from a retrospectve analyss of the mplcatons of the landmar bn pacng approxmaton scheme by Fernandez de la Vega and Lueer [13]. We prove that ther approxmaton scheme s subset oblvous, whch leads to numerous applcatons. 1 Introducton We analyze a smple method to fnd approxmate solutons to Set Coverng problems, showng that t leads to mproved approxmaton guarantees for the two multdmensonal generalzatons of Bn Pacng that are most relevant for practcal applcatons and whose study goes bac to the orgns of Operatons Research. The frst generalzaton s d-dm(ensonal) Vector Pacng. Here each tem and bn s a d-dmensonal vector wth non-negatve entres, and the goal s to pac the tems usng the mnmum number of bns so that for every bn the sum of the vectors paced n that bn s coordnate-wse no greater than the bn s vector. Ths problem s wdely used to model resource allocaton problems. The tems can be vewed as jobs wth requrements for d ndependent resources such as memory, CPU, hard ds,..., and the bns as machnes that have a certan amount of each resource avalable. The goal s then to place the jobs on A prelmnary verson of ths paper appeared n the Proceedngs of the 47-th Annual IEEE Symposum on Foundatons of Computer Scence (FOCS 2006). IBM T.J. Watson Research Center, P.O. Box 218, Yortown Heghts, NY 10598, e-mal: nhl@us.bm.com DEIS, Unversty of Bologna, Vale Rsorgmento 2, I Bologna, Italy, e-mal: alberto.caprara@unbo.t IBM T.J. Watson Research Center, P.O. Box 218, Yortown Heghts, NY 10598, e-mal: svr@us.bm.com 1

2 the mnmum number of machnes so that no machne s overloaded and the requrements of each job are met. The second generalzaton s 2-Dm Bn Pacng, where a gven set of rectangular tems must be paced nto the mnmum number of unt rectangular bns. The most common case s the so called orthogonal pacng, where the edges of the tems must be paced parallel to the sdes of the bn, and the tems are ether allowed to be rotated by 90 degrees or not. In the rest of the paper, by rotaton we wll always mean a 90-degree rotaton, also called orthogonal rotaton. A closely related problem s 2-Dm Strp Pacng (also nown as Cuttng-Stoc), motvated by applcatons n cloth cuttng and steel cuttng ndustry. Here we are gven a strp of nfnte wdth and fnte heght, and the goal s to pac the tems nto the strp so that the wdth occuped s mnmzed. Lterature revew For 1-Dm Bn Pacng, relevant results were obtaned as soon as the man concepts n approxmaton were defned, n the late 1970s and early 1980s. These results essentally settled the status of Bn Pacng wth the Asymptotc Polynomal-Tme Approxmaton Schemes (APTASes) due Fernandez de la Vega and Lueer [13] and Karmarar and Karp [22]. For d-dm Vector Pacng, a follore asymptotc approxmaton guarantee arbtrarly close to d follows trvally by consderng for each dmenson the tems that have the largest component n that dmenson and pacng these tems nto bns by applyng an APTAS for 1-Dm Bn Pacng. The frst non-trval result was due to Cheur and Khanna [8] who gave, for constant d, a polynomal-tme algorthm wth approxmaton guarantee O(ln d). (Though [8] state ther result as O(ln d), upon a closer loo, the asymptotc approxmaton guarantee of ther method s actually about ln d + 2 for large d.) On the other hand, Woegnger [28] ruled out an APTAS even for d = 2. For d = 2 the best nown result s an absolute approxmaton guarantee of 2 due to Kellerer and Kotov [19]. A natural queston motvated by ts ntrnsc smplcty and practcal applcatons [7] s whether there s a polynomal-tme algorthm wth asymptotc approxmaton guarantee better than 2 for d = 2 (the method of [8] has guarantee 3 for d = 2). Although 2-Dm Bn Pacng and Strp Pacng are farly complex, startng from the 1980s a slow but contnuous progress was made, whch culmnated n a seres of recent relevant results. Kenyon and Rémla [21] showed that there s an APTAS for 2-Dm Strp Pacng wthout rotatons. Ths was recently extended by Jansen and van Stee [17] to the case wth rotatons. For 2-Dm Bn Pacng, Bansal et al. [1] showed that t does not admt an APTAS unless P=NP. The best nown polynomal-tme approxmaton algorthm for 2-Dm Bn Pacng wthout rotatons s due to Caprara [5] and has asymptotc approxmaton guarantee For the case wth rotatons, an asymptotc approxmaton guarantee arbtrarly close to 2 follows from the result of [17]. An APTAS s nown for the Gullotne 2-Dm Bn Pacng [4], n whch the tems must be paced n a certan structured way. Contrbuton and outlne Our general method for a Set Coverng problem wors as follows. Frst of all, the LP relaxaton of the problem s solved. Then, a randomzed roundng procedure s appled for a few steps, after whch one s left wth a small fracton of uncovered elements (called the resdual nstance). Fnally, these elements are covered usng some approxmaton algorthm. We prove that, f the approxmaton algorthm used n the last step has asymptotc approxmaton guarantee ρ and satsfes certan propertes (beng subset oblvous, see Secton 3), then the overall method s a randomzed algorthm wth asymptotc approxmaton guarantee arbtrarly close to ln ρ+1. Roughly speang, subset oblvous means that not only the algorthm produces a soluton wth value 2

3 at most ρ opt(i) on nstance I, but also, gven a random subset S of I where each element occurs wth probablty about 1/, the value of the soluton produced by the algorthm on S s bounded by approxmately ρ opt(i)/. The ey observaton s that many nown algorthms for Bn Pacng problems are subset oblvous, or can be modfed to be such. Ths leads to the the followng results based on our general method, appled by formulatng the problem at hand as a Set Coverng problem, each set correspondng to a vald way of pacng a bn and the goal beng to cover all the tems wth the mnmum number of sets. We frst show that the classc APTAS for 1-Dm Bn Pacng due to Fernandez de la Vega and Lueer [13] s a subset-oblvous algorthm after mnor modfcatons. Based on ths, we gve a smple subset-oblvous algorthm for d-dm Vector Pacng for constant d wth asymptotc approxmaton guarantee arbtrarly close to d. Plugged nto our general method, ths leads to a polynomal-tme randomzed algorthm wth asymptotc approxmaton guarantee arbtrarly close to ln d + 1 for fxed d. For small values of d ths s a notable mprovement over the prevously nown O(ln d) guarantee [8] mentoned above. For d = 2, our result mples an asymptotc approxmaton guarantee of ln = whch breas the natural barrer of 2 for ths case. For 2-Dm Bn Pacng wth and wthout rotatons, we gve subset-oblvous algorthms wth asymptotc approxmaton guarantee Note that, n tself, ths s an mprovement for the case wth rotatons. Plugged nto our general method, these lead to a polynomal-tme randomzed algorthm wth asymptotc approxmaton guarantee arbtrarly close to ln( ) + 1 = , mprovng on the above mentoned for the case wthout rotatons [5] and (arbtrarly close to) 2 for the case wth rotatons [18]. In the descrpton above, we assumed that an optmum soluton of the Set Coverng LP relaxaton was avalable. However, snce our sets are mplctly descrbed and are typcally exponentally many, the problem of solvng ths LP relaxaton s non-trval. For the applcatons consdered n ths paper, we show that the LP relaxaton can be solved to wthn (1 + ε) accuracy for any ε > 0. For d- Dm Vector Pacng, we do ths by observng that the dual separaton problem (also nown as column generaton problem) has a Polynomal-Tme Approxmaton Scheme (PTAS), whch mples a PTAS for the LP relaxaton followng the framewor of [22, 26, 15, 16]. However, ths approach does not wor for 2-Dm Bn Pacng. In ths case the dual separaton problem s the well-nown mum 2-Dm (Geometrc) Knapsac problem, for whch the best nown algorthm, due to Jansen and Zhang [18], has a performance guarantee arbtrarly close to 2, and the exstence of a PTAS s open. However, n a companon paper [2] we llustrate a PTAS for a sutable restrcton of 2-Dm Knapsac, whch leads to an APTAS for the LP relaxaton. Ths suffces for our purposes here. Fnally, we show how to derandomze our general method. 2 Prelmnares In all the pacng problems consdered n ths paper we are gven a set I of d-dmensonal tems, the -th correspondng to a d-tuple (t 1, t2,..., td ), that must be paced nto the smallest number of unt-sze bns, correspondng to the d-tuple (1,..., 1). For the case d = 1, we let s := t 1 be the sze of tem. For the case d = 2, for I we wll wrte b for t 1 and h for t 2. The frst dmenson wll be called the wdth (or bass) and the second dmenson wll be called the heght. Moreover, we wll let a := b h denote the area of tem. For d-dm Vector Pacng, a set C of tems can be paced nto a bn f C tj 1 for each j = 1,..., d. For d-dm Bn Pacng, the tems are d-dmensonal paralleleppeds wth szes gven by the assocated tuple, the bns are d-dmensonal cubes, and a set C of tems can be paced nto a bn f the tems can be placed n the bn wthout any two overlappng wth each other. We only consder the 3

4 orthogonal pacng case, where the tems must be placed so that ther edges are parallel to the edges of the bn. We address both the classcal verson wthout rotatons, n whch, for each coordnate, all edges assocated wth that coordnate n a bn have to be parallel, and the verson wth (orthogonal) rotatons, n whch ths restrcton s not mposed. Gven an nstance I of a mnmzaton problem, we let opt(i) denote the value of the optmal soluton of the problem for I. Gven a (determnstc) algorthm for the problem, we say that t has asymptotc approxmaton guarantee ρ f there exsts a constant δ such that the value of the soluton found by the algorthm s at most ρ opt(i) + δ for each nstance I. If δ = 0, then the algorthm has (absolute) approxmaton guarantee ρ. Gven a randomzed algorthm for the problem, we say that t has asymptotc approxmaton guarantee ρ f there exsts a constant δ such that the value of the soluton found by the algorthm s at most ρ opt(i)+δ wth a probablty that tends to 1 as opt(i) tends to nfnty. An algorthm wth an asymptotc approxmaton guarantee of ρ s called an asymptotc ρ-approxmaton algorthm. An APTAS s a famly of polynomal-tme algorthms such that, for each ε > 0, there s a member of the famly wth asymptotc approxmaton guarantee 1 + ε. If δ = 0 for every ε, then ths s a PTAS. All above problems could be formulated as the followng general Set Coverng problem, n whch a set I of tems has to be covered by confguratons from the collecton C 2 I, where each confguraton C C corresponds to a set of tems that can be paced nto a bn: { } mn x C 1 ( I), x C {0, 1} (C C). (1) C C x C : C As mentoned earler, the collecton C s gven mplctly snce t s exponentally large for the applcatons we consder, and hence we need to specfy how to solve the LP relaxaton of (1). The dual of ths LP s gven by { w : } w 1 (C C), w 0 ( I). (2) C I Note that the separaton problem for the dual s the followng Knapsac-type problem: Gven weghts on tems w, fnd, f any, a feasble confguraton n whch the total weght of tems exceeds 1. By the well nown connecton between separaton and optmzaton [15, 16, 26], we have that Theorem 1 If there exsts a PTAS for the optmzaton verson of the separaton problem for (2), that s gven w R I + solve C C C w, then there exsts a PTAS for the LP relaxaton of (1). 3 The General Method Our method, hereafter called Round and Approx (R&A), constructs an approxmate soluton of the Set Coverng problem (1) by performng the followng steps, where α > 0 s a parameter whose value wll be specfed later. 1. Solve the LP relaxaton of (1) (possbly approxmately). Let x be the (near-)optmal soluton of the LP relaxaton and z := C C x C be ts value; 2. Defne the bnary vector x r startng wth x r C := 0 for C C and then repeatng the followng for αz teratons: select one confguraton C at random, lettng each C C be selected wth probablty x C /z, and let x r C := 1; 4

5 3. Consder the set of tems S I that are not covered by x r, namely S f and only f = 0, and the assocated optmzaton problem for the resdual nstance: C xr C mn { C C x C : C x C 1 ( S), x C {0, 1} (C C) Apply some approxmaton algorthm to problem (3) yeldng soluton x a ; 4. Return the soluton x h := x r + x a. }. (3) Note that n Step 2 each selecton s ndependent of the others (.e., the same confguraton may be selected more than once). Of course, the qualty of the fnal soluton depends on the qualty of the approxmaton algorthm used to solve the resdual nstance. Here we focus our attenton on the case n whch ths latter qualty can be expressed n terms of a small set of weght vectors n R I, as stated n Defnton 1 below. Gven a set S I, wth a slght abuse of notaton we let S denote also the Set Coverng nstance defned by the tems n S. Moreover, we let opt(s) and appr(s) denote, respectvely, the value of the optmal soluton of (3) and the value of the heurstc soluton produced by the approxmaton algorthm that we consder. Below we defne the class of the subset-oblvous algorthms, whch are very useful for our analyss. Intutvely, snce we apply a randomzed roundng n Step 2, we do not now n advance whch wll be the subset S of remanng tems, but we stll want our approxmaton algorthm to perform well n a sutably-defned sense. The defnton below formalzes the noton of subset ndependence that we need. Defnton 1 An asymptotc ρ-approxmaton algorthm for problem (1) s called subset oblvous f, for any fxed ε > 0, there exst constants, ψ and δ (possbly dependng on ε) such that, for every nstance I of (1), there exst vectors w 1,..., w R I wth the followng propertes: () C wj ψ, for each C C and j = 1,..., ; () opt(i) I wj ; () appr(s) ρ wj + ε opt(i) + δ, for each S I. Property () says that the vectors obtaned from w 1,..., w by dvdng all the entres by constant ψ must be feasble for the dual of the LP relaxaton of (1), Property () provdes a lower bound on the value of the optmal soluton for the whole nstance I, and Property () guarantees that the value of the approxmate soluton on subset S s not sgnfcantly larger than ρ tmes the fracton of the lower bound n () assocated wth S. It s nstructve to consder an example. Suppose we have an nstance of 1-Dm Bn Pacng and we consder the Next Ft algorthm, where each tem s placed n the current bn f t fts, and placed n a new empty bn otherwse (closng the prevous bn). We wsh to show that Next Ft s an asymptotc 2-approxmaton subset-oblvous algorthm. To do ths, we let := 1 and defne the vector w 1 by w 1 := s, the sze of tem, for I. Then clearly Property () s satsfed wth ψ = 1, as no bn can contan tems wth total sze more than 1. Property () follows trvally as the number of bns used s at least equal to the total sze of the tems n the nstance. Property () holds wth ρ = 2 and δ = 1 (for any ε 0), and follows by observng that the total sze of the tems n every two consecutve bns paced by Next Ft s at least 1. 5

6 In general there are many canddates for the vectors w. In partcular, any feasble soluton w to the dual problem defned by (2) satsfes Property () wth ψ = 1, and satsfes Property () by LP dualty. Typcally, the non-trval part s to choose a small collecton of approprate vectors w and show that Property () holds wth a reasonable value of ρ. Our man result s the followng: Theorem 2 Suppose R&A uses an asymptotc µ-approxmaton algorthm to solve the LP relaxaton n Step 1, an asymptotc ρ-approxmaton subset-oblvous algorthm for problem (1) n Step 3 (wth µ < ρ), and α := ln(ρ/µ) n Step 2. Then, for any fxed γ > 0, the cost of the fnal soluton s at most (µ(ln(ρ/µ) + 1) + ε) opt(i) + δ + γz + 1 (4) wth probablty at least 1 e 2(γz ) 2 /(ψ 2 z ln ρ ). In other words, for any fxed ε > 0, R&A s a randomzed asymptotc µ(ln(ρ/µ) ε)-approxmaton algorthm for problem (1). Corollary 1 If R&A uses an APTAS to solve the LP relaxaton n Step 1, then, for any fxed ε > 0, t s a randomzed asymptotc (ln ρ ε)-approxmaton algorthm for problem (1). We need the followng concentraton nequalty n the analyss of R&A, due to McDarmd [24] (see also [25] for a nce survey on concentraton nequaltes). Lemma 1 (Independent Bounded Dfference Inequalty) Let X = (X 1,..., X n ) be a famly of ndependent random varables, wth X j A j for j = 1,..., n, and f : n A j R be a functon such that f(x) f(x ) c j whenever the vectors x and x dffer only n the j-th coordnate. Let E(f(X)) be the expected value of the random varable f(x). Then, for any t 0, P r [f(x) E(f(X)) t] e 2t2 / n c2 j. Proof of Theorem 2 The cost of the rounded soluton x r produced n Step 2 s at most αz αµ opt(i) + 1 = µ ln(ρ/µ) opt(i) + 1. We now estmate the cost of x a. Let S be the set of uncovered elements after Step 2. Note that S s a random set. Consder the random varable wj for j = 1,...,. By the structure of the algorthm and lnearty of expectaton, we now that E ( ) = I P r( S) = I ( 1 C x C/z ) αz e α I, where the last nequalty follows from C x C 1 for I and (1 1/a) αa (1 1/a) αa e α for a > 0. By the structure of the algorthm, the random varable wj s a functon of αz ndependent random varables. Changng the value of any of these random varables may lead to the selecton of a confguraton C n place of a confguraton C. Lettng S be the resultng resdual nstance n the latter case, we have wj wj { C\C wj, C \C wj } ψ, by Property (). Therefore, applyng Lemma 1, we get [ ) ] P r γz e 2(γz ) 2 /(ψ 2 αz ). (5) E ( 6

7 Usng (5), the unon bound on j, and Propertes () and () of subset oblvous algorthms (Defnton 1) we obtan that, for any constant γ > 0, the cost appr(s) of the approxmate soluton x a s at most ρ w j + ε opt(i) + δ ρ e α I + ε opt(i) + δ + γz (ρ e α + ε) opt(i) + δ + γz = (µ + ε) opt(i) + δ + γz wth probablty at least 1 e 2(γz ) 2 /(ψ 2 αz ). In Secton 8 we show how to derandomze the method. In the rest of the paper, we represent the set-coverng LP relaxaton of the resdual nstance S as: { } mn x C 1 ( S), x C 0 (C C) (6) and ts dual as: C C x C : C { w : C w 1 (C C), w 0 ( I) }. (7) Note that the feasble regon of (7) s ndependent of the choce of the subset S, whch appears only n the objectve. Ths observaton wll be crucal n defnng subset-oblvous algorthms. 4 A Subset-Oblvous APTAS for 1-Dm Bn Pacng The structural property of 1-Dm Bn Pacng proved n ths secton s the ey to analyze versons of R&A for generalzatons of the problem. Recall that for an nstance I the sze of an tem I s denoted by s. Lemma 2 For any fxed ε > 0, there exsts a polynomal-tme asymptotc (1+ε)-approxmaton subsetoblvous algorthm for 1-Dm Bn Pacng. Proof We show that the APTAS of [13] wth very mnor modfcatons s a subset-oblvous algorthm. Let σ := ε/(1 + ε), M := { I : s < σ} be the set of small tems and L := { I : s σ} the set of large tems, wth l := L, assumng s 1 s 2... s l,.e., tems are ordered accordng to decreasng szes. Defne the followng reduced szes for the tems n L startng from ther orgnal real szes s 1,..., s l. If l < 2/σ 2, we let p := l and, L := {}, s := s for = 1,..., l,.e., we do not change the szes. Otherwse, usng the fundamental lnear groupng technque of [13], we defne q := lσ 2 and defne p := l/q groups L 1,..., L p of consecutve tems n L, where, for j = 1,..., p 1, L j contans tems (j 1)q + 1,..., jq, and L p contans tems (p 1)q + 1,..., l (the smallest tems n L). The reduced sze s j of each tem n group L j s gven by the sze of the smallest tem n the group, namely s j := mn Lj s. It s easy to chec that p 1 + 3/σ 2 = O(1/ε 2 ). For a gven S I, consder the followng LP, whch s the counterpart of (6) for reduced szes, where tems of the same sze are assocated wth a unque constrant. Let c 1,..., c m be the collecton of non-negatve nteger vectors c {0,..., 1/σ } p such that p c js j 1. These vectors represent the 7

8 feasble pacng confguratons of the tems n L wth reduced szes. Note that m = O(O(1/ε) O(1/ε2) ). The LP s: { m } m mn x r : c r jx r L j S (j = 1,..., p), x r 0 (r = 1,..., m) (8) and ts dual: p L j S v j : p c r jv j 1 (r = 1,..., m), v j 0 (j = 1,..., p). (9) We defne the followng approxmate soluton startng from an optmal basc soluton x of LP (8). Consder the soluton x obtaned by roundng up x. Ths corresponds to a feasble pacng of the tems n L S wth reduced szes (n case an tem s paced nto more bns, we eep t n only one of these bns). If no groupng was performed, ths s also a feasble pacng for the real szes. Otherwse, we defne the followng pacng for the real szes: n the rounded soluton, for L S q, use the space for the reduced sze of the -th largest tem n L S to pac the real sze of the ( + q)-th largest tem n L S (whch s not larger by defnton of the groupng procedure and snce each group contans at most q tems). The real szes of the q largest tems n L S are paced nto q addtonal bns, one per bn. Fnally, the small tems n M S are paced n an arbtrary order by Next Ft, startng from the bns already contanng some large tems and consderng a new bn only when the current small tem does not ft n the current bn. Let appr(s) be the value of the fnal soluton produced. We now show the subset oblvousness and approxmaton guarantee of the above algorthm. Note that the feasble regon of the dual (9) does not depend on S. Moreover ths feasble regon s defned by p varables and m lnear nequaltes plus non-negatvty condtons. Therefore, the number t of basc ), whch s constant for fxed ε. Ths mples that, for all choces of the 2 I possble subsets S, the basc optmal solutons of (9) form a constant-sze collecton v 1,..., v t. We defne the set of vectors w 1,..., w as follows, lettng := t + 1: feasble solutons satsfes t ( p+m m for f = 1,..., t, we set w f := v f j for j = 1,..., p and L j, and w f := 0 for M (n other words, w f s obtaned by expandng the vector for reduced szes v f bac to the actual szes); w t+1 := s for I. By the above defnton, w 1,..., w are solutons of (7) (notng that also s s such a soluton) and, for each S I, 1 wj s equal to the optmum of (9) for nstance L S wth reduced szes. Moreover, opt(s) s = w. Therefore, opt(s) wj for each S I. Ths mples Propertes (), wth ψ = 1, and () n Defnton 1. Fnally, we show Property () wth δ = 1 + 3/σ 2, completng the proof. If new bns are needed after pacng the small tems, we have that all the bns wth the possble excepton of the last one contan tems for a total sze of at least (1 σ). Ths mples appr(s) s 1 σ + 1 = (1 + ε) w + 1 and we are done. On the other hand, f no new bns are needed for the small tems, snce the number of fractonal components n the basc soluton x s at most p we have that m x r m x r + p. Moreover, recall that n case groupng s performed we use q addtonal bns for the q largest tems, and note that q lσ 2 ε opt(i), snce σ ε and opt(i) σl as all tems n L have sze at least σ. Now, 8

9 lettng w {w 1,..., w 1 } be the dual soluton of (7) correspondng to the optmal dual soluton n {v 1,..., v t } of (9) assocated wth S, we have: appr(s) m m x r + q x r /σ 2 + ε opt(i) = L S w /σ 2 + ε opt(i) w + ε opt(i) /σ 2. It s nterestng to note that the dependence of on ε s multply exponental. 5 Improved Approxmaton for d-dm Vector Pacng We show how to combne the results of the prevous sectons to derve a polynomal-tme randomzed algorthm for d-dm Vector Pacng wth asymptotc approxmaton guarantee arbtrarly close to ln d+ 1, whch s for d = 2. Recall that each tem I corresponds to a 2-dmensonal vector (b, h ). Lemma 3 For any fxed ε > 0, there exsts a polynomal-tme asymptotc (d+ε)-approxmaton subsetoblvous algorthm for d-dm Vector Pacng, for constant d. Proof To avod confuson, n ths proof we wll denote by opt BP (I) the value of the optmal 1-Dm Bn Pacng soluton for a generc nstance I and appr BP (I) the value of the soluton obtaned by the subset-oblvous APTAS of Lemma 2 on nstance I. We gve the proof n the case d = 2. The general case s proved analogously. Consder the followng smple approxmaton algorthm analogous to the one n [13]. We partton the set I of tems nto sets B := { I : b h } and H := I \ B, and for a gven S I we pac the tems n B S (resp., H S) near-optmally nto bns by applyng the subsetoblvous APTAS of Lemma 2 to the Bn Pacng nstance wth szes {b : B} (resp., wth szes {h : H}). Note that each feasble pacng nto one bn of one-dmensonal tems wth szes n {b : B} (resp., {h : H}) corresponds to a feasble pacng nto one two-dmensonal bn of the correspondng set of two-dmensonal tems from B (resp., H). Fnally, we return the pacng of the tems n S defned by the bns n the two solutons obtaned. We now show that ths algorthm s subset-oblvous. By Lemma 2, we have that for any ζ > 0, there exst constants B, H, ξ and vectors u 1,..., u B R B and v 1,..., v H R H wth the followng propertes: opt BP (B) B u j, appr BP (B S) (1 + ζ) B B opt BP (H) H v j, appr BP (H S) (1 + ζ) H H B S H S u j v j + ζ opt(i) + ξ, (10) + ζ opt(i) + ξ. (11) Moreover, u 1,..., u B R B and v 1,..., v H R H are solutons of (2) for Bn Pacng. The requred vectors w 1,..., w are the followng, lettng := B + H : for f = 1,..., B, we set w f := u f for B, and wf := 0 for H; for f = 1,..., H, we set w B+f := 0 for B, and w B+f := v f for H. 9

10 Property () s trvally satsfed by the vectors w 1,..., w wth ψ = 1. Moreover, Propertes () and () are now smple to prove, lettng ζ := ε/2: opt(i) {opt BP (B), opt BP (H)} { B u j, H v j } = and, for each S I, by (10) and (11) we obtan that B H, I appr(s) = appr BP (B S) + appr BP (H S) B (1 + ζ) u j + (1 + ζ) H 2(1 + ζ) B S w j + 2ζ opt(i) + 2ξ. H S v j + 2ζ opt(i) + 2ξ Snce the separaton problem for the dual of the confguraton LP of the d-dm Vector Pacng s a mum d-dm (non-geometrc) Knapsac Problem, whch admts a PTAS for constant d [14], Theorem 1 mples the exstence of a PTAS for the confguraton LP for d-dm Vector Pacng. Thus, combnng Lemma 3 and Theorem 2, we obtan: Theorem 3 For any fxed ε > 0, usng a PTAS for the LP relaxaton n Step 1 and the algorthm of Lemma 3 n Step 3, method R&A s a randomzed polynomal-tme asymptotc (ln(d + ε) ε)- approxmaton algorthm for d-dm Vector Pacng, for constant d. In Table 1 we report the asymptotc approxmaton guarantees of R&A, of the smple approxmaton algorthm used n Step 3 of R&A, due to [13], and of the method by [8] for varous values of d. d [13] [8] R&A Table 1: Asymptotc approxmaton guarantees for d-dm Vector Pacng (ε omtted for brevty). 6 Improved Approxmaton for 2-Dm Bn Pacng wthout Rotatons We now show the mplcatons of our approach for 2-Dm Bn Pacng, recallng that the tems n an nstance I correspond to rectangles wth szes {(b, h ) : I}. Ths case s much more nvolved than the one of the prevous secton. The subset-oblvous algorthm that we present s essentally coped from [3], and analogous to the approxmaton algorthm of [5], both havng asymptotc approxmaton guarantee arbtrarly close to The man dfference between our algorthm and the one of [5] s that the latter s a very fast combnatoral algorthm combned wth an APTAS for 1-Dm Bn Pacng, whereas ours s based on the soluton of a certan LP, whch s useful to derve the vectors that we need to show subset oblvousness. In the followng sectons, we wll extensvely use the Next Ft Decreasng Heght (NFDH) procedure ntroduced by [9]. NFDH consders the tems n decreasng order of heght and greedly pacs them n 10

11 ths order nto shelves. A set of tems s sad to be paced nto a shelf f ther bottom edges all le on the same horzontal lne. More specfcally, startng from the bottom of the frst bn, the tems are paced left justfed nto a shelf untl the next tem does not ft. The shelf s then closed (.e., no further tem wll be paced nto the shelf), and the next tem s used to defne a new shelf whose bottom lne touches the top edge of the tallest (frst) tem n the shelf below. If the shelf does not ft n the bn,.e., f the next tem does not ft on top of the tallest tem n the shelf below, the bn s closed and a new bn s started. The procedure contnues untl all tems are paced. The ey propertes of NFDH that we need are: Lemma 4 ([9]) Gven a set of tems of heght at most h, f NFDH s used to pac these tems nto a rectangle of wdth B and heght H, and not all the tems ft, lettng B be the mnmum total wdth of the tems n a shelf n the rectangle, then the total area not occuped n the rectangle s at most hb + (B B)H. Lemma 5 ([9]) Gven a set of tems S, the total number of bns used by NFDH to pac the tems s at most 4 b h + 2. Another standard tool n the desgn of bn pacng algorthms s the so-called harmonc transformaton, frst ntroduced by Lee and Lee [23]. Let t be a postve nteger and x be a postve real n (0, 1]. The harmonc transformaton f t wth parameter t s defned as follows: f t (x) := 1/q f x (1/(q + 1), 1/q] for an nteger q t 1; f t (x) := tx/(t 1) f x (0, 1/t]. The crucal property of ths transformaton s that, for any sequence x 1,..., x n wth x (0, 1] for = 1,..., n and n =1 x 1, we have n =1 f t(x ) Π + 1/(t 1). Here, Π = s the harmonc constant defned n [23]. In order to get rd of the nconvenent 1/(t 1) term, we wll use a slght varant of f t, called g t, defned by g t (x) := f t (x) f x (1/t, 1]; g t (x) := x f x (0, 1/t]. (12) Lemma 6 For any postve nteger t and for any sequence x 1,..., x n wth x (0, 1] for = 1,..., n and n =1 x 1, t holds that n =1 g t(x ) Π. Proof Snce g t (x) f u (x) for every x (0, 1] and for every postve nteger u t, we have n =1 g t(x ) lm n u =1 f u(x ) Π. Fnally, we need a result that was the ey to prove the approxmaton guarantee n [5], for whch we restate an explct (easy) proof for the sae of completeness: Lemma 7 ([12, 27]) Let S be a set of tems wth szes {(b, h ) : S} that fts nto a bn and let w be an arbtrary dual feasble soluton of (2) for the 1-Dm Bn Pacng nstance wth szes {h : S}. Then the set of tems wth szes {(b, w ) : S} also fts nto a bn. Proof Consder a feasble pacng of the tems n S wth heghts h nto a bn. Consder the tems n ncreasng order of dstance of ther bottom edge from the bottom edge of the bn (breang tes arbtrarly). For each tem n ths order, wthout changng the horzontal coordnates, frst move down as far as possble wthout overlappng other tems and then change the heght from h to w, greedly movng up some tems f overlaps wth them after ths ncrease. If the correspondng pacng s not feasble, there must be some sequence 1,..., m of tems that are on top of each other such that m l=1 w l > 1. Gven that we dd not change the wdths and the 11

12 horzontal coordnates, these tems are on top of each other also n the ntal pacng,.e., m l=1 h l 1, or, equvalently, the set { 1,..., m } s a feasble confguraton n C for the prmal LP assocated wth (2) for the 1-Dm Bn Pacng Problem. Gven that w s a feasble soluton of (2), we must have m l=1 w l 1, yeldng a contradcton. The man dea of [3] s the followng: Consder the relaxaton n whch we wsh to pac the tems nto a strp of heght 1 and nfnte wdth, wth the goal of mnmzng the total wdth. Clearly, f the tems can be paced nto opt(i) bns, then they can also be paced nto a strp of wdth opt(i). In [3] t s shown that f the wdths of tems are harmonc,.e., f all wdths ε are of the form 1/q for some nteger q, then there s an APTAS for strp pacng wth the addtonal property that, f we cut the strp vertcally at some x = where s an nteger, then each tem that s cut has wdth at most ε. Ths mples that ths 2-Dm Strp Pacng soluton can be converted nto a 2-Dm Bn Pacng soluton whle ncreasng the cost by a factor of at most 1 + ε. In other words, f the wdths are harmonc, then there s an APTAS for 2-Dm Bn Pacng. Thus an asymptotc Π approxmaton for general nstances follows by frst applyng the harmonc transformaton to the wdths, ncreasng the optmal value by a factor factor of at most Π (see below for detals), and then applyng the above algorthm. The APTAS for harmonc wdths wth the specal property mentoned above s based on a slght modfcaton of the APTAS for strp pacng due to [21]. In what follows, we gve a self-contaned proof of ths result, presented n a way so that t s easy to show subset-oblvousness. Lemma 8 For any fxed ε > 0, there exsts a polynomal-tme asymptotc (Π + ε)-approxmaton subset-oblvous algorthm for 2-Dm Bn Pacng wthout rotatons. Proof The proof s very smlar n some parts to the subset oblvous APTAS for 1-Dm Bn Pacng llustrated n Lemma 2. However, at the cost of some repettons, we gve full detals also n ths case to avod possble confuson. Agan, we ntroduce an nternal parameter σ > 0, and show n the end that by defnng approprately σ as a functon of ε we acheve the requred accuracy. Gven the orgnal nstance I, frst of all we apply the harmonc transformaton (12) to the tem wdths, by defnng t := 1/σ and replacng each orgnal wdth b wth the ncreased wdth b := g t (b ). For the rest of the algorthm, we wll only consder the ncreased wdths of the tems, called smply wdths n the followng, and forget about ther orgnal wdths. We let M := { I : h < σ} be the set of short tems and L := { I : h σ} be the set of tall tems. Furthermore, we let l := L, b(l) := L b denote the total ncreased wdth of the tall tems, and assume that these tems are ordered accordng to decreasng heghts. If b(l) < 2/σ 2, we let p := l and, L := {}, h := h for = 1,..., l. Otherwse, we defne groups L 1,..., L p of consecutve tems n L, so that, for j = 1,..., p 1, the tems n each group L j have total wdth n (σ 2 b(l) 1, σ 2 b(l)], by nsertng tems n the group untl the total wdth exceeds σ 2 b(l) 1, and the tems n group L p have total wdth at most σ 2 b(l). Note that we have p 2/σ 2. We let the reduced heght h j of each tem n group L j be the heght of the shortest tem n L j. Let c 1,..., c m be the collecton of the vectors c {0,..., 1/σ } p such that p c jh j 1. Note that such a c corresponds to a set of tems that may be placed one on top of the other n a bn (wth respect to the reduced heghts), called a slce. Followng the ey observaton n [21], f we allow tems to be slced vertcally, for a gven S I the resultng smplfed 2-Dm Bn Pacng problem s to assgn wdths to all possble slces so that, for each group L j, the total wdth of the slces contanng heght h j s at least equal to the total wdth of the tems n L j S. If we let x r denote the wdth of slce c r, we have the followng LP, whch s completely analogous to LP (8): m m mn x r : c r jx r b (j = 1,..., p), x r 0 (r = 1,..., m). (13) L j S 12

13 We defne the followng approxmate soluton startng from an optmal basc soluton x of LP (13), havng at most p postve components. In fact, we start from a basc soluton of the LP n whch the n the coverng constrants s replaced by =, whch s easly seen to be equvalent to (13). Startng from such a soluton, n whch there are no useless spaces allocated for the tems n L, smplfes both the descrpton and the analyss of the algorthm. For each postve component x r assocated wth vector c r, we ntroduce x r bns n the soluton. In each of these bns, we defne c r j shelves of heght h j, and use these shelves to pac the tems as follows. The coverng constrants n the LP ensure that the tems n L S wth ther reduced heghts can be paced nto these shelves f they can be slced vertcally (pacng slces nto dstnct shelves/bns). More precsely, such a pacng can be obtaned by consderng, for j = 1,..., p, the shelves of heght h j and pacng the tems n L j S nto these shelves n decreasng order of (orgnal) heght (.e., n ncreasng order of ndex) when an tem does not ft, the slce that fts s paced nto the current shelf and the other slce s paced nto the next shelf. We call ths the slced pacng for the reduced heghts. We then partton the tems n L S nto new groups L 1,..., L p (agan consderng the tems n decreasng order of orgnal heght), wth p 1/σ 2, so that the total wdth of the tems n each group s exactly σ 2 b(l), wth the possble excepton of the tems n the last group, by possbly slcng vertcally tems between groups (note that slcng was not allowed n the defnton of the old groups L 1,..., L p ). We put asde the tems n L 1 and, startng from the slced pacng for the reduced heghts above, we defne the slced pacng for the orgnal heghts by pacng the tems n each group L j wth ther orgnal heghts nto the space used by the tems n L j 1 n the prevous pacng wth reduced heghts, for j 2. Note that the defnton of reduced heghts guarantees that ths pacng s feasble. Moreover, ths pacng of tems n L j s done n decreasng order of wdth, rather than n decreasng order of heght. After havng defned the slced pacng for the orgnal heghts, n order to have a pacng n whch tems are not slced, the tems n L 1 together wth all tems that were slced by formng the groups L 1,..., L p or n defnng the pacng are paced nto separate bns by NFDH. Fnally, we pac the short tems n M S usng the rectangles of wdth 1 and heght 1 p cr j h j that are left free for each bn assocated wth a postve component x r n the LP, as well as addtonal bns f needed. The pacng of these tems s done by consderng separately each wdth class (.e., tems wth b = 1, tems wth b = 1/2,..., and tems wth b 1/t), and, for each class, pacng the assocated tems by NFDH. We now show the subset oblvousness and approxmaton guarantee of the above algorthm. By reasonng as n the proof of Lemma 2, we have a constant-sze collecton v 1,..., v s of solutons of the dual of (13), and we defne the set of vectors w 1,..., w as follows, lettng := s + 1: for f = 1,..., s, we set w f := (b v f j )/Π for j = 1,..., p and L j, and w f := 0 for M; w s+1 := (b h )/Π for I. Intutvely, each quantty w f s an analog of the area of tem I, computed wth respect to the ncreased wdths (and the modfed heghts v f j for f s) and scaled by the factor Π. We frst show that w 1,..., w are solutons of (7) for 2-Dm Bn Pacng. By Lemma 6 we have that the vector (b 1 /Π, b 2 /Π,...) = (g t (b 1 )/Π, g t (b 2 )/Π,...) s a feasble soluton of (2) for the 1-Dm Bn Pacng nstance wth one tem of sze b for I. Therefore, by Lemma 7, for each set C C of tems that ft nto a bn wth ther orgnal szes, the same tems wth the modfed wdths b /Π also ft nto a bn. Ths mples C (b h )/Π 1. Moreover, f the tems ft wth ther orgnal heghts, they also ft wth ther reduced heghts. Then, by Lemma 7, gven that v f s a feasble 13

14 soluton of the dual of (13),.e., a feasble soluton of (2) for the 1-Dm Bn Pacng nstance wth L j tems of sze h j for j = 1,..., p, these tems also ft f each (reduced) heght h j s replaced by v f j. Ths mples C (b v f j )/Π 1 for f = 1,..., s, and therefore Propertes (), wth ψ = 1, and () n Defnton 1. Fnally, we show Property (). Let z be the optmal value of LP (13) and w {w 1,..., w 1 } be the vector correspondng to the optmal dual soluton v of (13), for whch z = p L j S b v j = Π We have that the number of bns ntally ntroduced s L S w = Π w. m x h z + p Π w + 2/σ 2. h=1 We now bound the number of addtonal bns needed for the tall tems. The structure of the wdths guarantees that, n the slced pacng for the orgnal heghts, there are no tems splt n the shelves that contan only tems from the same group L j havng the same wdth value 1/q wth q {1,..., t 1}. (Ths s the other ey property of the harmonc transformaton.) Accordngly, the total area of the tall tems paced nto addtonal bns s the sum of the followng contrbutons: the total area of the tems n L 1, whch s at most σ2 b(l) 2σ 2 L b 2σ opt(i), where the frst nequalty follows from b 2b for I (by defnton of harmonc transformaton) and the second from opt(i) L b h σ L b (by defnton of L); the total area of the tems slced by formng the groups, whch s at most p 1/σ 2 ; the total area of the tems splt n the shelves that contan tems from dfferent groups L j and L j+1, whch s at most p 1/σ 2 as these shelves are not more than p due to the structure of the slced pacng for the reduced heghts, whch pacs the tems n decreasng order of heght; the total area of the tems splt n the shelves that contan tems from the same group L j but wth dstnct wdth classes, whch s at most t p 1/σ 1/σ 2, as these shelves are not more than t p due to the structure of the slced pacng for the orgnal heghts, whch pacs the tems n decreasng order of wdth; the total area of the tems splt n the shelves that contan tems from the same group L j wth wdths n (0, 1/t], whch s at most (1/t) m h=1 x h σπ w + 2/σ snce the area of the possble tem splt n each shelf s at most 1/t tmes the heght of the shelf. By Lemma 5, ths mples that the number m t of addtonal bns needed for the tall tems s [ ] m t 4 2σ opt(i) + 2/σ 2 + 1/σ 1/σ 2 + σπ w + 2/σ + 2. (14) Therefore, f no addtonal bns are needed for the short tems, we have appr(s) m x r + m t Π w + 2/σ 2 + m t. (15) 14

15 If addtonal bns are needed for the short tems, the structure of LP (13), recallng that we are solvng the verson wth equalty n the coverng constrants, guarantees that the area of the tems n L S s p p b h b h j = h j ( p m ) b = h j c r jx r L S L j S m p c r jh j x r L j S m p c r jh j x r 2/σ 2. Moreover, wth the excepton of the last addtonal bn and the, at most t, bns contanng short tems assocated wth dstnct wdth classes, all spaces occuped by the short tems are nearly completely flled. More specfcally, n all bns except at most t + 1, by Lemma 4 the area devoted to short tems and not occuped s at most σ + 1/t 2σ. Specfcally: n the bns contanng only short tems havng wdth 1/q, q {1,..., t 1}, the total area not occuped by these tems s at most σ (gven that the total wdth of the tems n each shelf s exactly 1), n the bns contanng only short tems havng wdth 1/t, the total area not occuped by these tems s at most σ + 1/t 2σ (gven that the total wdth of the tems n each shelf s at least 1 1/t). Therefore, lettng m s denote the number of addtonal bns needed for the short tems, the area of the tems n M S s m p b h m s (1 2σ) + x r 1 c r jh j 2σ (t + 1) and therefore M S m (1 2σ) 1 p c r jh j x r + m s (1/σ + 2), [ m ] b h (1 2σ) x r + m s (1/σ + 2/σ 2 + 2). Recallng that b h = Π w appr(s) = for I, we have m x r + m s + m t Π b h + (1/σ + 2/σ 2 + 2) + m t (1 2σ) w + (1/σ + 2/σ2 + 2) + m t. (16) (1 2σ) Combnng (14), (15) and (16) (and recallng Lemma 5), by defnng σ approprately we have n both cases appr(s) (Π + ε) + ε opt(i) + O(1). wj Recall that for 2-Dm Bn Pacng we cannot use Theorem 1 as the separaton problem for the dual of the LP relaxaton s a 2-Dm (Geometrc) Knapsac, for whch the exstence of a PTAS s open for d = 2. However, we can show that: 15

16 Theorem 4 There exsts an APTAS for the LP relaxaton of (1) for 2-Dm Bn Pacng wth and wthout rotatons. The proof of ths result s rather long and techncal, and the deas requred are orthogonal to those consdered n ths paper. Thus we descrbe ths proof n a companon paper [2]. The man pont s that, n order to obtan an APTAS for the LP relaxaton, t s suffcent to have a PTAS for the specal case of 2-Dm Knapsac n whch, roughly speang, the profts of the tems are close to ther areas. Combnng Lemma 8, Theorem 4 and Theorem 2 we obtan: Theorem 5 For any fxed ε > 0, usng the APTAS of Theorem 4 n Step 1 and the algorthm of Lemma 8 n Step 3, method R&A s a randomzed polynomal-tme asymptotc (ln(π +ε)+1+ε)-approxmaton algorthm for 2-Dm Bn Pacng wthout rotatons. 7 Improved Approxmaton for 2-Dm Bn Pacng wth Rotatons In ths secton we desgn a polynomal-tme (determnstc) subset-oblvous approxmaton algorthm for 2-Dm Bn Pacng wth (orthogonal) rotatons wth asymptotc approxmaton guarantee arbtrarly close to Π, mprovng on the prevously-nown 2. An nterestng aspect of ths algorthm s the fact that, for the frst tme n ths paper, a set of vectors satsfyng the requrements n Defnton 1 are used for algorthmc purposes and not only for the sae of the analyss. Moreover, ths algorthm can be plugged nto the R&A method, leadng to an asymptotc approxmaton guarantee arbtrarly close to ln Π + 1. The results presented hold also for the case n whch the bn sze s not the same for both dmensons, and we address the case of unt square bns only for smplcty of presentaton. Lemma 9 For any fxed ε > 0, there exsts a polynomal-tme asymptotc (Π + ε)-approxmaton subset-oblvous algorthm for 2-Dm Bn Pacng wth rotatons. Proof In the proof, gven a 2-Dm Bn Pacng nstance I, we let opt(i) denote the optmal value for the problem we consder, n whch rotatons are allowed, and opt 2BP (I) the optmal value for the case n whch rotatons are not allowed. Gven an tem subset S I, a rotaton of S s represented by a partton S N S R of S, where S N s the subset of tems that are not rotated and S R the subset of tems that are rotated. A trval exponentaltme algorthm wth asymptotc approxmaton guarantee arbtrarly close to Π s the followng: Gven S I, try all the 2 S rotatons of S and, for each of them, apply the algorthm of Lemma 8 to the tems rotated accordngly (the proof below shows that ths trval algorthm s subset oblvous). The ey pont of the polynomal-tme verson s to avod tryng all the rotatons. Let σ be an nternal parameter dependng on the requred accuracy ε. Gven the orgnal nstance I, we defne the tem set I := {(b, h ) : I} {(h, b ) : I}, correspondng to the unon of the non-rotated and rotated tems n I. We consder the 2-Dm Bn Pacng (wthout rotatons) nstance I, and apply Lemma 8 wth nput accuracy σ n a constructve way, explctly computng the vectors w 1,..., w R 2 I as n the proof of that lemma, called w 1,..., w n ths proof. For each I and j = 1,...,, we let u j be the component of wj assocated wth tem non-rotated and v j be the component of w j assocated wth tem rotated, respectvely. After havng computed w 1,..., w, gven S I, we fnd the rotaton S N S R of S that approxmately mnmzes u j + v j, (17) N R as llustrated below, and then apply the algorthm of Lemma 8 to ths rotaton. 16

17 We complete the descrpton of the algorthm by showng how we fnd a near-optmal rotaton wth respect to (17). If a rotaton s represented by bnary varables y, S, where y = 1 f S N and y = 0 f S R, (17) s equvalent to the followng Integer LP: { } mn z : z u j y + v j (1 y ) (j = 1,..., ), y {0, 1} ( S). (18) It s easy to show that ths problem s wealy NP-hard and solvable n pseudo-polynomal tme by dynamc programmng (gven that s fxed). In our case we solve the assocated LP relaxaton, fndng an optmal basc soluton, and then return the nteger soluton obtaned by roundng the fractonal y varables arbtrarly. We now show the subset oblvousness and approxmaton guarantee of the algorthm. Note that, gven that Property () n Lemma 8 holds wth ψ = 1 for vectors w 1,..., w R 2 I, we have that for each S I opt 2BP (S). (19) Let IN I R be the rotaton of I assocated wth the optmal soluton of 2-Dm Bn Pacng wth rotatons for I, where opt(i) = opt 2BP (IN I R ) s the correspondng number of bns. We defne the vectors w 1,..., w R I as follows: for j = 1,...,, we set := uj for I N, and wj := vj for I R. In other words, we eep only the components of w 1,..., w assocated wth the tems rotated as n IN I R. (Note that ths defnton of w1,..., w s non-constructve.) In order to show Property (), consder an arbtrary feasble confguraton C C for the case wth rotatons. Gven that the total area of the tems n C s at most one, by Lemma 5 the tems n the rotaton CN C R of C correspondng to I N I R can be paced nto at most 6 bns (very rough estmate), whch mples = u j + v j = opt 2BP (C N CR) 6, C C N C R C N C R where the nequalty s mpled by (19), yeldng Property () wth ψ = 6. As to Property (), we have opt(i) = opt 2BP (IN IR) = I N u j + I R v j, where the nequalty s agan mpled by (19). Fnally, we show Property (). Gven S I and lettng S N S R be the rotaton found by the I 17

18 algorthm and SN S R be the rotaton of S correspondng to I N I R, we have: appr(s) (Π + σ) u j + v j + σ opt 2BP (I) + O(1) N R (Π + σ) u j σ opt 2BP (I) + O(1) = (Π + σ) (Π + σ) [ [ N R v j ] + + σ opt 2BP (I) + O(1) ] + + 8σ opt(i) + σ + O(1), recallng that the nput accuracy of the algorthm n Lemma 8 s σ. Here, the frst nequalty follows from Lemma 8. The second nequalty follows from the fact that rotaton S N S R s an approxmate soluton of the optmzaton problem (17): lettng F S be the set of ndces of the varables y that are fractonal n ths optmal basc soluton of the LP relaxaton of (18), we have F, so by roundng the soluton value ncreases by at most snce each objectve functon coeffcent s at most 1. Fnally, the last nequalty follows from opt 2BP (I) 8opt(I) + 2, n turn mpled by the trval bound I b h opt(i), by I b h = 2 I b h, and by opt 2BP (I) 4 I b h +2 by Lemma 5. Combnng Lemma 9, Theorem 4 and Theorem 2 we obtan: Theorem 6 For any fxed ε > 0, usng the APTAS of Theorem 4 n Step 1 and the algorthm of Lemma 9 n Step 3, method R&A s a randomzed polynomal-tme asymptotc (ln(π +ε)+1+ε)-approxmaton algorthm for 2-Dm Bn Pacng wth rotatons. 8 Derandomzaton In ths secton we present a determnstc varant of method R&A n whch Step 2 s replaced by a greedy procedure that defnes x r guded by a sutable potental functon. Let ψ be the constant and w 1,..., w be the vectors n Defnton 1 for the algorthm n Step 3. Moreover, let x be the (near-)optmal soluton of the LP relaxaton of (1) and C 1,..., C m C be the confguratons assocated wth the nonzero components of x, wth z = m l=1 x C l. Roughly speang, the proof of Theorem 2 says that f we select αz of these confguratons randomly accordng to probabltes x C l /z, lettng S be the set of uncovered tems after the selecton, then wj s concentrated around e α I wj for j = 1,...,. The determnstc varant s a greedy procedure that, at each teraton, selects a confguraton from C 1,..., C m. The ey part s the score accordng to whch ths confguraton s selected, whch s defned as follows. Let σ be a (small) parameter such that σψ < 1 to be specfed later. For an arbtrary set of tems S, consder the potental functon ( Φ(S) := ln exp σ ) w j. (20) (To mprove readablty, n ths secton we wll often use the notaton exp(x) n place of e x.) Our determnstc varant of Step 2 s: 18

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