Online Bin Packing with Cardinality Constraints Resolved

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1 Online Bin Pcing with Crdinlity Constrints Resolved János Blogh 1, József Béési, György Dós, Leh Epstein, nd Asf Levin 5 1 Deprtment of Applied Informtics, Gyul Juhász Fculty of Eduction, University of Szeged, Hungry blogh@jgyp.u-szeged.hu Deprtment of Applied Informtics, Gyul Juhász Fculty of Eduction, University of Szeged, Hungry beesi@jgyp.u-szeged.hu Deprtment of Mthemtics, University of Pnnoni, Veszprém, Hungry dosgy@lmos.vein.hu Deprtment of Mthemtics, University of Hif, Hif, Isrel le@mth.hif.c.il 5 Fculty of Industril Engineering nd Mngement, The Technion, Hif, Isrel levins@ie.technion.c.il Abstrct Crdinlity constrined bin pcing or bin pcing with crdinlity constrints is bsic bin pcing problem. In the online version with the prmeter, items hving sizes in (0, 1] ssocited with them re presented one by one to be pced into unit cpcity bins, such tht the cpcities of bins re not exceeded, nd no bin receives more thn items. We resolve the online problem in the sense tht we prove lower bound of on the overll symptotic competitive rtio. This closes the long stnding open problem of finding the vlue of the best possible overll symptotic competitive rtio, since n lgorithm of n bsolute competitive rtio for ny fixed vlue of is nown. Additionlly, we significntly improve the nown lower bounds on the symptotic competitive rtio for every specific vlue of. The novelty of our constructions is bsed on full dptivity tht cretes lrge gps between item sizes. Thus, our lower bound inputs do not follow the common prctice for online bin pcing problems of hving nown in dvnce input consisting of btches for which the lgorithm needs to be competitive on every prefix of the input. Lst, we show lower bound strictly lrger thn on the symptotic competitive rtio of the online -dimensionl vector pcing problem, nd thus provide for the first time lower bound lrger thn on the symptotic competitive rtio for the vector pcing problem in ny fixed dimension ACM Subject Clssifiction F.. Sequencing nd Scheduling, G..1 Combintoril Algorithms Keywords nd phrses Online lgorithms, bin pcing, crdinlity constrints, lower bounds Digitl Object Identifier 10.0/LIPIcs.ESA Introduction Bin pcing with crdinlity constrints (CCBP, lso clled crdinlity constrined bin pcing) is well-nown vrint of bin pcing [18, 19, 17, 9, 10, 11, 15]. In this problem, prmeter is given. Items of indices 1,,..., n, where item i hs size s i (0, 1] re to be János Blogh, József Béési, György Dós, Leh Epstein, nd Asf Levin; licensed under Cretive Commons License CC-BY 5th Annul Europen Symposium on Algorithms (ESA 017). Editors: Kir Pruhs nd Christin Sohler; Article No. 10; pp. 10:1 10:1 Leibniz Interntionl Proceedings in Informtics Schloss Dgstuhl Leibniz-Zentrum für Informti, Dgstuhl Publishing, Germny

2 10: Online Bin Pcing with Crdinlity Constrints Resolved split into subsets clled bins, such tht the totl size of items pced into ech bin is t most 1, nd no bin hs more thn items. In the stndrd bin pcing problem, only the first condition is required. CCBP is specil cse of vector pcing (VP) [1]. In VP with dimension d, set of items, where every item is non-zero d-dimensionl vector whose components re rtionl numbers in [0, 1], re to be split into subsets (clled bins in this cse s well) such tht the vector sum of every subset does not exceed 1 in ny component. Given n input for CCBP, n input for VP is creted s follows. For every item, let the first component be 1, the second component is s i, nd the remining components re equl to zero (or to 1 ). In this pper we study online lgorithms, which receive input items one by one, nd pc ech new item irrevocbly before the next item is presented, into n empty (new) bin or non-empty bin. Such lgorithms receive n input s sequence, while offline lgorithms receive n input s set. By the definition of CCBP, n item i cn be pced into non-empty bin B if the pcing is fesible both with respect to the totl size of items lredy pced into tht bin nd with respect to the number of pced items (i.e., the bin contins items of totl size t most 1 s i nd it contins t most 1 items). An optiml offline lgorithm, which uses minimum number of bins for pcing the items, is denoted by OP T. For n input L nd lgorithm A, we let A(L) denote the number of bins tht A uses to pc L. We lso use OP T (L) to denote the number of bins tht OP T uses for given input L. The bsolute competitive rtio of n lgorithm A is the supremum rtio over ll inputs L between the number of its bins A(L) nd the number of the bins of OP T, OP T (L). The symptotic pproximtion rtio is the limit of bsolute pproximtion rtios R K when K tends to infinity nd R K tes into ccount only inputs for which OP T uses t lest K bins, tht is the symptotic competitive rtio of A is lim sup K OP T (L) K A(L) OP T (L). The term competitive rtio is used for online lgorithms insted of pproximtion rtio nd it is equivlent. In this pper we mostly del with the symptotic competitive rtio, nd lso refer to it by the term competitive rtio. When we discuss the bsolute competitive rtio, we use this lst term explicitly. In this pper, we resolve the long stnding open problem of online CCBP, in the sense tht we find the best overll symptotic competitive rtio nd the best overll bsolute competitive rtio. An lgorithm with n symptotic competitive rtios of hs been designed by Bbel et l. [], nd similr lgorithm ws shown to hve n bsolute competitive rtio of [6] (erlier, it ws nown tht the competitive rtio of suitble vrint of First Fit is below.7 for ny [18]). However, prior to this wor, ll lower bounds were strictly smller thn the best lower bounds for stndrd bin pcing [, 5]. With the exception of the cse = for which simple lgorithms hve competitive rtios of 1.5 [18, 10], nd more sophisticted lgorithm hs competitive rtio of t most 1.71 [], ll lower bounds on the competitive rtio were implied by prtil inputs of ones used to prove lower bounds for stndrd bin pcing [,, 5] (such lower bounds cn be used for 1 δ when ll items hve sizes no smller thn δ, for fixed vlue δ > 0), nd modifictions of such inputs [, 1, 6]. Tht is, ll lower bounds hd the form where number of lists my be presented, ech list hs lrge number of items of certin size (the sequence of sizes of the different lists is incresing, nd the numbers of items in the lists re not necessrily equl). The unnown fctor is the number of presented lists, tht is, the input cn stop fter ny of the lists. See Tble 1 for vlues of previously nown lower bounds (nd note tht for =,, 5, 6 lgorithms with competitive rtios strictly below re nown [10]).

3 J. Blogh, J. Béési, Gy. Dós, L. Epstein, nd A. Levin 10: Tble 1 Bounds for 10. The middle column contins the previously best nown symptotic lower bounds on the symptotic competitive rtio for CCBP with prmeter. The right column contins our improved lower bounds. Vlue of previous lower bound new lower bound [1] = 1.5 [] = 1.5 [1] 1.60 = 1.5 [6] = 1.5 [] [6] [6] [6] [6] [5] [5] In this wor, we te different pproch for proving lower bounds, where mny of the item sizes re bsed on the complete nd precise ction of the lgorithm up to the time it is presented. While some ingredients of our pproch were used for the very limited specil cse of = in the pst [7,, 1], it ws uncler how nd if it could be used for >. In nutshell, in these lower bound sequences for =, sub-inputs were constructed such tht items pced in certin wys (for exmple, s the second item of bin) hd much lrger sizes thn items of the sme sub-input pced in other wys. Here, we generlize the pproch for lrger vlues of by defining creful constructions where sufficiently lrge multiplictive gps re creted. This requires much more delicte procedures where item sizes re defined. Additionlly, we improve the lower bounds for ll vlues of, nd in prticulr, prove lower bounds bove the best nown lower bound on the competitive rtio for stndrd online bin pcing, [5] for. Alredy for = our lower bound is bove 1.55, nd lredy for =, our lower bound is bove the competitive rtio of mny lgorithms for stndrd online bin pcing (see for exmple [1, ]). Our result for CCBP provides, in prticulr, lower bound of for the symptotic competitive rtio of VP in two dimensions. The previously nown lower bounds for VP re s follows. The best results for constnt dimensions re firly low, nd tending to s the dimension d grows to infinity [1, 8, 7], while lower bound of Ω(d 1 ε ) ws given by Azr et l. [] for the cse where both d nd the optiml cost re functions of common prmeter n tht grow to infinity when n grows to infinity, nd thus this result does not give ny lower bound on the competitive rtio for constnt vlues of d (see lso [1, ] for results on vectors with smll components). In prticulr, the best lower bound for d = prior to this wor ws [1, 8, 7]. An upper bound of d on the competitive rtio is nown [1]. We conclude this wor by estblishing lower bound strictly lrger thn on the competitive rtio of -dimensionl VP, nd thus we show here for the first time tht the -dimensionl VP is provbly hrder for online lgorithms thn its specil cse of CCBP. Note tht the offline CCBP problem is NP-hrd in the strong sense, nd pproximtion schemes re nown for it [9, 11, 15]. We note tht for online CCBP, it is sometimes the E S A 0 1 7

4 10: Online Bin Pcing with Crdinlity Constrints Resolved cse tht the competitive rtio for some specific lgorithms for CCBP is lrger by 1 with comprison to tht of the corresponding lgorithms for stndrd bin pcing [18, 16, 0, 10]. Interestingly, this is not the cse with respect to the results shown in this pper. 1.1 Pper outline We discuss generl properties in Section, nd we define procedures for constructing subinputs in Section. Our min result, n overll lower bound of on the competitive rtio of ny online lgorithm for CCBP is proved in Section, nd improved lower bounds for fixed vlues of re given in Section 5. Our result for VP is estblished in Section 6. Omitted proofs pper in the full version of this wor. Preliminries The nlysis of the lower bounds on the symptotic competitive rtio of online lgorithms will be bsed on the following lemm tht bsiclly llows us to disregrd constnt number of bins in the costs of the optiml solution nd the solution returned by the lgorithm. Lemm 1. Consider n lgorithm ALG, such tht the symptotic competitive rtio of the lgorithm ALG is t most R, where R 1 is fixed vlue, nd let f(n) denote positive function such tht f(n) = o(n) nd for ny input, ALG(I) R OP T (I) + f(op T (I)). Let C 0, C b 0 be constnts. Assume tht for given integer N 0, for ny integer n N 0 there is n input I n for which OP T (I n ) = Ω(n), then we hve R lim sup n Proof. We hve ALG(I n ) + C n ALG(I n ) + C OP T (I n ) C b. R OP T (In ) C b n + C + R C b n + f(op T (In )) n for ny n N 0. Since ALG(I n ) + C OP T (I n ) C b nd OP T (I n ) C b = Ω(n) while C + R C b + f(op T (I n )) = o(n), letting n grow to infinity implies tht R lim sup n ALG(I n ) + C OP T (I n ) C b. In wht follows, we will use Lemm 1 s follows. We construct inputs whose size depends on prmeter N, so tht the costs of optiml solutions increse with the input size. We will compre the cost of fixed online lgorithm ALG plus suitble non-negtive constnt to the optiml cost minus suitble non-negtive constnt by considering their rtio. Constructions of sub-inputs In this section we introduce the core of our lower bound constructions. In such constructions, we dptively present inputs tht re bsed on the behvior of the lgorithm. More specificlly, we define severl procedures tht construct sub-inputs ccording to certin conditions. Similrly to [, 1, 7] (nd other wor on online problems), new input item is presented t ech time, where its size is bsed on the ction of the lgorithm on previous items. For exmple, if the previous item ws pced into n empty bin, the size of the next item is

5 J. Blogh, J. Béési, Gy. Dós, L. Epstein, nd A. Levin 10:5 different from the size tht would be used if the previous item is dded to non-empty bin. In order to ensure tht the properties re stisfied, we will define invrints, nd we will prove the specific properties tht we need in the sequel vi induction. The constructions use s prmeter since they re defined to be used for CCBP. However, they cn be used for ny pcing problem of items into bins nd the property tht is the crdinlity constrint is not used in the constructions of sub-inputs (it is used lter in the nlysis of inputs constructed using these sub-inputs). Thus, if the constructions re used for other problems lie we do for VP, the prmeter should be specified. In the first procedure, the most importnt property is tht there will be gp between two types of items constructed by pplying the procedure, in the sense tht the procedure cretes items tht will be clled smll nd items tht will be clled lrge, ny lrge item is lrger thn ny smll item, nd there is requirement on the size rtio tht will be stisfied ( multiplictive gp between the size of the smllest lrge item nd the lrgest smll item). Such constructions differ from previous wor [, 1, 7] where only n dditive gp ws creted. The gp ws lwys positive, but it could be rbitrrily smll. In prticulr, one limittion ws tht it ws unnown how such n pproch could be used for CCBP with prmeter >. We will lso use this method to construct sub-inputs with lrge items, such tht there is multiplictive gp in the differences between 1 nd the items sizes. This new method will llow us to provide tight overll result for CCBP, new nd significntly improved lower bounds on the symptotic competitive rtio for CCBP with fixed vlues of, nd our improved lower bound for VP..1 Procedure SMALL In this first procedure clled SMALL, rtionl vlue 0 < ε 1, nd n integer upper bound N on the number of items to be presented re given. The gol is to present (t most) N items of sizes in (0, ε], such tht every item will be seen s either smll item or lrge item, nd such tht ny lrge item is more thn times lrger thn ny smll item. In fct, stronger requirement on the item sizes will hold. Moreover, ll item sizes will be rtionl. Given two logicl conditions, C 1 nd C specified for ech construction (such tht for every pced item, exctly one of them holds), new item will be defined s smll if C 1 holds nd it will be defined s lrge if C holds. There is third condition C tht is bsed on the pcing of the prefix of items introduced so fr, nd the sub-input is stopped if C holds. Let N be n upper bound on the number of items tht will be creted by the procedure. Let N N be the ctul number of items (where N is nown in dvnce nd used for the sequence construction, while N is not necessrily nown in dvnce nd it becomes nown when C holds for the first time). The item sizes 1,,..., N will be defined bsed on nother sequence x 1, x,..., x N, such tht i = ε xi for 1 i N. The vlues x i will be integrl in order to ensure tht the vlues i will be rtionl. There will lso be two sequences of vlues τ 1,..., τ N nd ρ 1,..., ρ N, representing thresholds on item sizes of further items. Let τ 0 = N+, ρ 0 = N+, nd i = 1. The process is defined s follows for ny given vlue of i (such tht 1 i N ). Let x i = τi 1+ρi 1 (we will show tht these vlues re integers). After the lgorithm pcs item i, if C 1 holds, let τ i = τ i 1 nd ρ i = x i nd if C holds, let τ i = x i nd ρ i = ρ i 1. If C holds or i = N, stop nd otherwise increse i by 1. Intuitively, the process is s follows. The intervl (τ i, ρ i ) contins the x j vlues of ll further items (with j > i), nd for j i, ll items stisfying C 1 hve x j vlues in [ρ i, ρ 0 ) nd ll items stisfying C hve x j vlues in (τ 0, τ i ]. In ech itertion i, the new vlues τ i, ρ i re E S A 0 1 7

6 10:6 Online Bin Pcing with Crdinlity Constrints Resolved defined such tht these requirements re stisfied. In prticulr, the x i vlues of ny item stisfying C 1 re lrger thn those of items stisfying C. Next, we estblish the invrints of this procedure. Lemm. Let N be the number of items. For ny i such tht 1 i N, ρ i ρ i 1 nd τ i τ i 1. Additionlly, we hve ρ i τ i = N+ i, ll x i vlues re integrl, if item i stisfies C 1, x i ρ N nd otherwise x i τ N. Proof. We strt with showing tht the x i vlues s well s ρ i nd τ i re integrl nd ρ i τ i = N+ i. We prove this by induction. Indeed ρ 0 = N+ tht is integrl, τ 0 = N+ tht is n integer s well. Furthermore, ρ 0 τ 0 = N+, nd x 1 = N+1 tht is n integer, nd no mtter if the first item stisfies C 1 or C, we hve tht both ρ 1 nd τ 1 re integers, nd ρ 1 τ 1 = N+1. Thus, the cses i = 0 nd i = 1 for the induction clim hold. Assume tht ρ i 1 τ i 1 = N+ i holds for some i where 1 i N 1. Then, x i = τ i 1 + ρ i 1 = τ i 1 + ρ i 1 τ i 1 = τ i 1 + N+ i, which is n integer for 1 i N, since τ i 1 is n integer. Moreover, if τ i = τ i 1 nd ρ i = x i, then ρ i τ i = x i τ i 1 = ρi 1 τi 1, nd otherwise τ i = x i nd ρ i = ρ i 1, then ρ i τ i = ρ i 1 x i = ρi 1 τi 1. In both cses, ρ i τ i = N+ i nd both τ i nd ρ i re integers. Since, in prticulr, for ny i, ρ i > τ i holds nd x i+1 is their verge, we find τ i < x i+1 < ρ i. Thus, ρ i ρ i 1 nd τ i τ i 1 holds for ny i. Finlly, since in the cse tht item i stisfies C 1, we let ρ i = x i, nd in the cse tht item i stisfies C, we let τ i = x i, we get x i = ρ i ρ i+1... ρ N in the first cse, nd x i = τ i τ i+1... τ N in the second cse. ( Corollry. For ny item i, i ε N+, ε N+), nd in prticulr i 1. For ny item i 1 stisfying C 1 nd ny item i stisfying C, it holds tht i i1 >. Note tht it is possible tht the constructed input is such tht there re only items stisfying C 1 or only items stisfying C. Proof. The first clim holds by definition. Since we hve x i1 ρ N nd x i τ N, we get i i1 > ρ N τ N, Using ρ N τ N = N+ N s N N, we find i i1 >.. Procedure LARGE The second type of input is such tht ll items hve sizes in (1 ε, 1) for given vlue ε > 0. The construction is the sme s before, but the size of the ith item is b i = 1 i. The terms smll nd lrge refer to the difference between the size of the item nd 1. Corollry. All b i for 1 i N re in (1 ε N+, 1 ε N+ ). The sizes of ny smll item i s nd ny lrge item i l stisfy 1 b il > (1 b is ).. Procedure SMALLndLARGE We will lso use procedure where the conditions C 1 nd C re not fixed, nd they re bsed on dditionl properties of the pcing nd the input tht hs been presented so fr. Moreover, in this cse the size of ech item is bsed on i, but it is fixed for ech item seprtely (it will be either i or 1 i ). In this construction the sub-input will be decomposed into prts where for n item of n odd indexed prt the size of the item will be 1 i, wheres for n item of n even indexed prt the size of the item will be i. The definitions of C 1 nd C will lso depend on the prity of the index of the prt contining the item. This procedure is clled SMALLndLARGE.

7 J. Blogh, J. Béési, Gy. Dós, L. Epstein, nd A. Levin 10:7 A lower bound of for CCBP The generl structure of inputs constructed in this section is s follows. There re lrge number of very smll items, such tht the first item pced into bin by the lgorithm is significntly lrger thn smll items pced s second item or lter. Afterwrds, there re two cses. In the first cse there re very lrge items (of sizes lmost 1) tht cn be combined with 1 items tht rrived erlier, but only with those tht re smller. Thus, n optiml solution cn pc ll items densely except for those items tht re first in their bins (for the lgorithm). The lgorithm cnnot use its previously pced bins gin to pc new items, nd therefore the best pproch is to pc lrge number of items into ech bin (otherwise the percentge of lrger smll items is lrger, which mes the optiml pcing more sprse, but the lgorithm hs n even lrger number of bins, nd the effect of the lst property is more significnt). Another option is tht insted of the very lrge items, items slightly lrger thn 1 will rrive, in which cse it turns out tht the lgorithm should hve pced 1 items into ech bin (so tht new item could be still pced there). For very lrge vlues of, the two vlues 1 nd re not very different, nd since the lgorithm does not now which items will rrive, pcing 1 items into ech bin (if is very lrge) is good strtegy. The result of pcing 1 items into ech bin is tht in the first cse the very lrge items increse the number of bins roughly by fctor of, while n optiml solution hs reltively few bins with smll items. Note tht the order of options in the construction below is reversed for the se of convenience. Let N be lrge integer. Apply procedure SMALL with ε = 1 for the construction of N items (i.e., condition C never hppens). The condition C is tht the item is pced s the first item of some bin (into n empty bin), nd the condition C 1 is tht the item is pced into non-empty bin. The item sizes re no lrger thn 1. The multiplictive gp between the smllest lrge item nd the lrgest smll item is lrger thn. The N items presented so fr will be clled the first phse items. Let δ > 0 denote the lrgest size of ny first phse item pced not s first item of bin (the lrgest smll item). Let α = δ. Any first phse item tht is pced s the first item of bin ( lrge item) hs size strictly bove α. Let < 1 be the lrgest size of ny first phse item. Obviously, 1 > 1 1 > 1. For the first phse items, let X denote the number of bins pced by the lgorithm tht contin items, nd let Y denote the number of other bins (such tht there re X + Y bins in totl fter N items hve been presented). The first phse items re followed by nother set of items clled the second phse items. This set of items is selected out of two possible options. The first option is tht N 1 items of size 1 rrive, nd the second option is tht N X Y 1 items of size 1 α = 1 δ rrive. In both cses it is possible to crete n offline solution such tht ech bin (except for possibly two bins) hs items. In the first cse, n offline solution hs N 1 bins, ech with one item of size 1 nd n rbitrry subset of 1 first phse items (the lst bin my hve smller number of such items). Such solution is optiml. In the second cse, n offline solution hs N X Y 1 bins, ech with one item of size 1 δ nd 1 smll first phse items, nd X +Y bins with lrge first phse items (for ech one of these two bin types, the lst bin my hve smller number of such items). Indeed the lst solution is n optiml solution though we will only use tht it is fesible solution. In the first cse, the lgorithm cnnot use the bins tht lredy hve items for pcing second phse items, nd its cost is t lest X + N 1 X + N 1. In the second cse, the lgorithm cnnot use ny of its bins to pc ny second phse item, s ech bin hs lrge E S A 0 1 7

8 10:8 Online Bin Pcing with Crdinlity Constrints Resolved first phse item of size bove α, so its cost is N X Y X + Y + X + Y + N X Y 1 1. We cll the two inputs (of the two cses) I 1 nd I. Obviously, since ech input consists of more thn N items, OP T (I 1 ) = Ω( N ) nd OP T (I ) = Ω( N ) hold. Letting N = n provides n input I n s required. By Lemm 1, we will nlyze modified competitive rtios of the form ALG(I)+C OP T (I) C b for fixed constnts C nd C b. For the input I 1, OP T (I 1 ) 1 N 1 nd ALG(I 1) X + N 1. For the input I, OP T (I ) N X Y 1 + X +Y nd ALG(I ) X + Y + N X Y 1. First, we nlyze the competitive rtio r for input I nd show tht it tends to s grows to infinity. Let Z = X + Y. We hve OP T (I ) N Z 1 + Z nd ALG(I ) Z + N Z 1. Thus, r Z( 1) + (N Z) (N Z) + ( 1)Z = Z( ) + N N Z Since Z N nd the lst lower bound on r is rtio between n incresing function of Z nd decresing function of Z, we conclude tht by substituting N insted of Z in the lst bound, we chieve vlid lower bound on r. Thus, we hve r N( )+N = / N N 1 1/( ) = +1 nd the lst bound tends to when grows to infinity. By Lemm 1, the overll (symptotic) competitive rtio is t lest. Since there is -competitive lgorithm for ny vlue of [] (even for the bsolute competitive rtio [6]), we estblish the following.. Theorem 5. The overll best possible symptotic nd bsolute competitive rtios for bin pcing with crdinlity constrints re equl to. To obtin better lower bound on the symptotic competitive rtio r for fixed vlue of, we use I 1 s well. By r ALG(I1) OP T (I X +N/( 1) 1) 1 N/( 1) we hve ( 1)X (r 1) N. By counting rguments, N X + ( 1)Y holds, nd we get X N ( 1)Z, nd (r 1)N ( 1)X ( 1)(N ( 1)Z) = ( 1)N ( 1) Z. Rerrnging gives Z ( r)n ( 1). As we sw erlier, by using I we hve r Z( )+N N Z, which is equivlent to Z( + r) N(r 1). Combining the lower bound nd upper bound on Z results in ( r)n( + r) ( 1) N(r 1), or equivlently r + r( ) ( + ) 0. Since 0 holds for nd + > 0 holds for, it is sufficient to find the (unique) positive root which is equl to + + ( ) +( +). The lst expression is lower bound on r nd thus the following holds.

9 J. Blogh, J. Béési, Gy. Dós, L. Epstein, nd A. Levin 10:9 Theorem 6. For ny, the symptotic competitive rtio for bin pcing with crdinlity constrints is t lest The lst lower bound is equl to pproximtely for =, 1.60 for =, for = 5, 1.71 for = 6, for = 7, for = 8, for = 9, nd for = 10. For = the resulting lower bound is nd the construction (for the cse = ) is indeed similr to tht of [7, ]. 5 Better lower bounds for CCBP for some smll vlues of In this section we prove the next theorem tht improves the resulting bounds of Theorem 6 for these vlues of. Theorem 7. The following pproximte vlues re lower bounds on the symptotic competitive rtio:the vlue for = (the exct vlue of this lower bound is 10 7 ), for =, for = 5, for = 6, 1.77 for = 7, for = 8, for = 9, nd for = Vector pcing As explined in the introduction, vector pcing is generliztion of CCBP, nd thus the results of the previous sections imply, in prticulr, lower bound of on the symptotic competitive rtio for VP in two or more dimensions. In this section we show tht VP is more generl, by improving the result, nd showing lower bound bove for VP with constnt dimensions. Our result is the first lower bound strictly bove for ny fixed dimension VP (recll tht currently, the best nown upper bound for d-dimensionl VP is d nd for -dimensionl VP.7 [1]). We prove the result for two dimensions (nd the result for higher dimensions follows since the symptotic competitive rtio is monotone in the dimension, s ny d-dimensionl vector cn be ugmented by d d zeroes to become d -dimensionl vector). Once gin we consider fixed deterministic online lgorithm ALG, but this time it is n lgorithm for VP. Let R be the symptotic competitive rtio. The min ide of the lower bound is t follows. First, there re items whose first component is 1 for n ppropritely chosen integer, while the second components re very smll. The items re such tht the second components re sufficiently lrger for items pced first into their bins by the lgorithm compred to those tht re not pced first. Afterwrds, one option is tht the following items hve very lrge second component nd their first component is zero (this is equivlent to the items in the construction for CCBP). Every such item could be pced with items tht rrived erlier, but never with the first item of bin of the lgorithm, nd thus the new items require new bins, while n optiml solution cn pc lmost everything densely. For this option it is most profitble for the lgorithm to pc items in ech bin. In the other cses, it turns out tht it is better to pc much less thn items per bin, s further items will hve first components of for n integer vlue of (which is selected bsed on the ction of the lgorithm). Those items will hve second components bove 1, nd there my be further items whose second components re bove 1. Let 10 be lrge integer. The set of inputs we define will consist of t most three phses (where phse is sub-input). The first phse of the input is bsed on the construction for CCBP s follows. For lrge integer N 1000, there re N items whose. E S A 0 1 7

10 10:10 Online Bin Pcing with Crdinlity Constrints Resolved first component is 1. The second components of items re constructed using procedure SMALL with ε = N+, such tht SMALL is pplied for the construction of N elements (i.e., condition C never hppens). The condition C is tht the item is pced s the first item of some bin (i.e., it is pced into n empty bin), nd the condition C 1 is tht the item is pced into non-empty bin. The N (two-dimensionl) items presented so fr will be clled the first phse items. The second components of the first phse items re no lrger thn N+ 1. Due to the vlue of the first component, in ny pcing every bin hs t most first phse items. A first phse item pced s the first item of bin will be clled lrge nd ny other first phse item will be clled smll. The multiplictive gp between the smllest second component of ny lrge item nd the lrgest second component of ny smll item is greter thn. Let δ > 0 denote the lrgest second component of ny smll first phse item. Let α = δ. Any lrge first phse item hs second component strictly bove α. Let < 1 be the lrgest second component of ny first phse item. Obviously, 1 α = 1 δ > 1 1 > Let X i denote the number of bins pced with i first phse items nd let Θ = ( Xi) i=1 N, where Θ 1 s every bin hs t lest one item out of the N items. Let the input of first phse items be denoted by I. At this time, ny items cn be pced into bin, nd thus OP T (I) N. If ALG(I) = ΘN N, we get R ALG(I) OP T (I) 1. Thus, we ssume in wht follows tht Θ <. Since every bin of the lgorithm contins exctly one lrge item nd the remining items re smll, there re ΘN < N lrge items nd t lest N ΘN > ( )N 7N 10 smll items. The first option for the second prt of the input is similr to the construction for CCBP (the second prt of the input will lso be the lst prt of the input in this specific cse). The next phse of items will consist of N Θ N items clled second phse items, whose first component is zero nd the second component is 1 α = 1 δ. This input (consisting of the first phse items nd the second phse items) is clled I. By the following lemm we hve 1 + ( 1)Θ R. Lemm 8. We hve ALG(I ) = ΘN + N ΘN OP T (I ) N ΘN + ΘN = N. ΘN + N ΘN = N+( 1)ΘN Proof. It is possible to crete fesible solution for I where ech bin (except for possibly two bins) hs first phse items. This solution hs N ΘN bins, ech with one second phse item nd smll first phse items, nd ΘN bins with lrge first phse items (for ech one of these two bin types, the lst bin my hve smller number of first phse items). Indeed the lst solution is n optiml solution (since second phse items cnnot be pced with lrge first phse items), though we will only use tht it is fesible solution. We conclude tht OP T (I ) N ΘN + ΘN = N. The lgorithm uses different new bin for ech second phse item, since every such item hs second component lrger thn 1, nd every bin with first phse items hs totl size bove α in its second component. Thus, we get ALG(I ) = ΘN + N ΘN ΘN + N ΘN = N+( 1)ΘN. nd Let b be n integer such tht b <. For ny integer such tht 1 b, there will be two possible inputs I 1 nd I. All inputs strt with the first phse items defined bove. The second phse of items is identicl for the two inputs I 1 nd I (but it is different for different vlues of ). Let Γ = N NΘ. Intuitively, when considering n optiml pcing of the smll first phse items in I 1 nd I, most of the bins will contin smll first phse items, nd thus Γ is pproximtely their number. The second phse items re constructed

11 J. Blogh, J. Béési, Gy. Dós, L. Epstein, nd A. Levin 10:11 using SMALL with the sme vlue of s for the first phse items s follows. The number of items is N = Γ (nd once gin C never hppens nd ll items re presented). The sizes re built using ε = 1, nd the conditions C 1 nd C re s follows. We let C be the condition tht the item is pced into bin tht does not hve second phse item, nd C 1 is the condition tht the item is pced into bin tht lredy hs second phse item. The first component of ech item is. Given the ith output of SMALL denoted by z, for the ith item, the second component is defined s 1 + z. If z is defined when C holds, we sy tht the item whose vector is (, z) is lrge, nd otherwise it is smll. Since 0 < z for ny item, the items stisfy tht their second components re strictly lrger thn 1, nd they re not lrger thn < 0.5. Furthermore, we conclude tht the difference between the smllest second component of lrge second phse item nd the lrgest second component of smll second phse item is t lest N+. Obviously, since second phse items hve second components bove 1, no bin cn hve more thn two such items. Let Y1 nd Y denote the numbers of bins with one second phse item nd two second phse items, respectively (note tht there my be such bin tht contin first phse items nd bins tht do not, nd both inds re included in these two vlues ccording to their numbers of second phse items, while bins with only first phse items re not included). There re Y smll second phse items nd Y1 + Y lrge second phse items (nd Y1 + Y = Γ ). Note tht since the first component of second phse items is, they could not hve been pced into bins with t lest + 1 first phse items. Input I 1 continues with Γ items, ech of the form (, 0.6). Let 1 + δ be the lrgest second component of smll second phse item (such tht for ny lrge second phse item, its second component is lrger thn 1 + δ ), nd observe tht since δ N+, the totl sum of second component of set of t most first phse items is t most δ. Input I continues with the third phse items s follows. Γ+Y items, ech of the form (, δ ), nd N items, ech of the form (0, 1 α). Let c = i=c X i. Lemm 9. The costs of the lgorithm stisfy nd ALG(I 1 ) +1 + Y + Γ ALG(I) +1 + Y1 + Y + Γ + Y + N. Proof. For I, 1 the lgorithm cnnot use ny bin with t lest + 1 first phse items to pc ny other items (s second phse nd third phse items fterwrds hve first component of vlue ), nd the lgorithm cnnot pc n item of the form (, 0.6) into bin with two second phse items. Thus, using c = i=c X i, the totl number of bins of the lgorithm is t lest +1 + Y + Γ. For I, the lgorithm cnnot use ny bin with t lest + 1 first phse items to pc items whose first component is, nd it cnnot use ny bins with first phse items to pc items whose second component is 1 α. Moreover, since every bin with second phse items hs lrge second phse item, the lgorithm cnnot pc ny third phse item into bin contining t lest one second phse item (nd ech bin with third phse item will contin exctly one third phse item). The only bins tht cn possibly be used for third phse items re those with t most first phse items nd no other items. Thus, the number of bins is t lest +1 + Y1 + Y + Γ+Y + N. We next nlyze optiml solutions for I 1 nd I. E S A 0 1 7

12 10:1 Online Bin Pcing with Crdinlity Constrints Resolved Lemm 10. The cost of the optiml solutions for I 1 nd I stisfy nd OP T (I 1 ) Γ + NΘ OP T (I) N + Y 1 + Y + Γ + Y + 9N + = N + Γ + 9N +. Proof. For I 1 consider the following fesible solution. There re Γ bins, ech with second phse item (whose first component is nd its second component is in ( 1, 0.5)), one item of the form (, 0.6), nd first phse items where ech such item hs first component of 1 1 nd its second component is no lrger thn (the lst bin my contin smller number of first phse items). The first component of the sum of the vectors of these items is 1, nd the second component is t most < 1. The remining first phse items (there re t most NΘ such items) re pced in bin. We find tht OP T (I1 ) Γ + NΘ. For I, there re N bins, ech with one item of the form (0, 1 α) nd smll first phse items (recll tht the number of smll first phse items is lrger thn N + ), Y 1 +Y bins with t most two lrge second phse items nd t most first phse items, Γ+Y bins with one item of the form (, δ ), nd t most one smll second phse item, nd t most first phse items. The remining first phse items re pced into dditionl bins, such tht every bin hs such items. All items re pced since the number of smll second phse items, Y, is no lrger thn Γ Γ+Y, so Y. The totl spce for first phse items in the first three inds of bins is t lest N ( Y + ( ) 1 + Y + Γ + Y ) = N + Γ N + N(1 Θ) = N NΘ, so the number of bins of the lst ind is t most NΘ 9N + 1 since Θ. We find tht OP T (I) N + Y 1 + Y + Γ + Y + 9N + = N + Γ + 9N +. We get R ALG(I 1 ) OP T (I1 ) +1 + Y N NΘ + N NΘ + NΘ, R ALG(I ) OP T (I ) Y1 + Y + N NΘ +Y + N. N + N NΘ + 9N We let β = b/ nd let grows to infinity. Choosing β nd using the inequlities we showed, we find R.07119, nd thus we conclude the following theorem. Theorem 11. The symptotic competitive rtio of ny online lgorithm for vector pcing with d is t lest

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