Almost Optimal Inapproximability of Multidimensional Packing Problems
AAlmost Optimal Inapproximability of MultidimensionalPacking Problems
Sai Sandeep ∗ Computer Science DepartmentCarnegie Mellon UniversityPittsburgh, PA 15213
Abstract
Multidimensional packing problems generalize the classical packing problems suchas Bin Packing, Multiprocessor Scheduling by allowing the jobs to be d -dimensionalvectors. While the approximability of the scalar problems is well understood, therehas been a significant gap between the approximation algorithms and the hardnessresults for the multidimensional variants. In this paper, we close this gap by givingalmost tight hardness results for these problems.1. We show that Vector Bin Packing has no polynomial time Ω(log d ) factor asymp-totic approximation algorithm when d is a large constant, assuming P (cid:54) = NP .This matches the ln d + O (1) factor approximation algorithms (Chekuri, KhannaSICOMP 2004, Bansal, Caprara, Sviridenko SICOMP 2009, Bansal, Eliás, KhanSODA 2016) upto constants.2. We show that Vector Scheduling has no polynomial time algorithm with anapproximation ratio of Ω (cid:0) (log d ) − (cid:15) (cid:1) when d is part of the input, assuming NP (cid:42) ZPTIME (cid:16) n (log n ) O (1) (cid:17) . This almost matches the O (cid:16) log d log log d (cid:17) factor algo-rithms(Harris, Srinivasan JACM 2019, Im, Kell, Kulkarni, Panigrahi SICOMP2019). We also show that the problem is NP-hard to approximate within (log log d ) ω (1) .3. We show that Vector Bin Covering is NP-hard to approximate within Ω (cid:16) log d log log d (cid:17) when d is part of the input, almost matching the O (log d ) factor algorithm (Alon et al ., Algorithmica 1998).Previously, no hardness results that grow with d were known for Vector Schedulingand Vector Bin Covering when d is part of the input and for Vector Bin Packing when d is a fixed constant. ∗ [email protected] . Research supported in part by NSF grants CCF-1563742 and CCF-1908125. a r X i v : . [ c s . D S ] J a n Introduction
Bin Packing and Multiprocessor Scheduling (also known as Makespan Minimization) areone of the most fundamental problems in Combinatorial Optimization. They have beenstudied intensely from the early days of approximation algorithms and have had a greatimpact on the field. These two are packing problems where we have n jobs with certainsizes, and the objective is to pack them into bins efficiently. In Bin Packing, each binhas unit size and the objective is to minimize the number of bins, while in MultiprocessorScheduling, we are given a fixed number of bins and the objective is to minimize themaximum load in a bin. A relatively less studied but natural problem is Bin Covering,where the objective is to assign the jobs to the maximum number of bins such that in eachbin, the load is at least . These problems are well understood in terms of approximationalgorithms: all the three problems are NP-hard, and all of them have a Polynomial TimeApproximation Scheme (PTAS) [dlVL81, HS87, CJK01].In this paper, we study the approximability of the multidimensional generalizations ofthese three problems. The three corresponding problems are Vector Bin Packing, VectorScheduling, and Vector Bin Covering. Apart from their theoretical importance, theseproblems are widely applicable in practice [Spi94, ST12, PTUW11] where the jobs oftenhave multiple dimensions such as CPU, Hard disk, memory, etc.In the Vector Bin Packing problem, the input is a set of n vectors in [0 , d and thegoal is to partition the vectors into the minimum number of parts such that in each part,the sum of vectors is at most in every coordinate. The problem behaves differently fromBin Packing even when d = 2 : Woeginger [Woe97] proved that there is no asymptotic PTAS for -dimensional Vector Bin Packing, assuming P (cid:54) = NP . On the algorithmicfront, the PTAS for Bin Packing [dlVL81] easily implies a d + (cid:15) approximation for VectorBin Packing. When d is part of the input, this is almost tight: there is a lower boundof d − (cid:15) shown by [CK04] . When d is a fixed constant , much better algorithms areknown [CK04, BCS09, BEK16] that get ln d + O (1) approximation guarantee. However,the best hardness factor (for arbitrary constant d ) is still the APX-hardness result of the -dimensional problem due to Woeginger from 1997. Closing this gap, either by obtaininga O (1) factor algorithm or showing a hardness factor that is a function of d , has remaineda challenging open problem. It is one of the ten open problems in a recent survey onmultidimensional scheduling problems [CKPT17]. It also appeared in a recent report byBansal [Ban17] on open problems in scheduling.In the Vector Scheduling problem, given a set of n vector jobs in [0 , d , and m identicalmachines, the objective is to assign the jobs to machines to minimize the maximum (cid:96) ∞ norm of the load on the machines. Chekuri and Khanna [CK04] introduced the problemas a natural generalization of Multiprocessor Scheduling and obtained a PTAS for theproblem when d is a fixed constant. When d is part of the input, they obtained a O (log d ) factor approximation algorithm. They also showed that it is NP-hard to obtain a C factor approximation algorithm for the problem, for any constant C . Meyerson, Roytman,and Tagiku [MRT13] gave an improved O (log d ) factor algorithm while the current bestfactor is O (cid:16) log d log log d (cid:17) due to Harris and Srinivasan [HS19] and Im, Kell, Kulkarni, andPanigrahi [IKKP19]. The algorithm of Harris and Srinivasan [HS19] works for the moregeneral setting of unrelated machines where each job can have a different vector loadfor each machine. However, no super constant hardness is known even in this unrelated The asymptotic approximation ratio (formally defined in Section 2) of an algorithm is the ratio of itscost and the optimal cost when the optimal cost is large enough. All the approximation factors mentionedin this paper for Vector Bin Packing are asymptotic. [CK04] actually give d − (cid:15) hardness, but it has been shown later (see e.g., [CKPT17]) that a slightmodification of their reduction gives d − (cid:15) hardness. The algorithms are now allowed to run in time n f ( d ) , for some function f . n vectors in [0 , d . Theobjective is to partition these into the maximum number of parts such that in each part, thesum of vectors is at least in every coordinate. It is also referred to as “dual Vector Packing”in the literature. This problem is introduced by Alon et al . [AAC +
98] who gave a O (log d ) factor approximation algorithm. They also gave a d factor algorithm using a methodfrom the area of compact vector approximation that outperforms the above algorithm forsmall values of d . On the hardness front, Woeginger’s hardness result [Woe97] for VectorBin Packing can be easily modified to give APX-hardness for -dimensional Vector BinCovering as well. We prove almost optimal hardness results for the three multidimensional problems dis-cussed above.
For the Vector Bin Packing problem, we prove a
Ω(log d ) asymptotic hardness of ap-proximation when d is a large constant, matching the ln d + O (1) approximation algo-rithms [CK04, BCS09, BEK16], up to constants. Theorem 1.1.
There exists an integer d and a constant c > such that for all constants d ≥ d , d -dimensional Vector Bin Packing has no asymptotic c log d factor polynomial timeapproximation algorithm unless P = NP . We obtain our hardness result via a reduction from the set cover problem on certainstructured instances. In the set cover problem, we are given a set system
S ⊆ V on auniverse V , and the goal is to pick the minimum number of sets from S whose union is V . Observe that Vector Bin Packing is a special case of the set cover problem with thevectors being the elements and every maximal set of vectors whose sum is at most inevery coordinate (known as “configurations”) being the sets. In fact, in the elegant Round& Approx framework [BCS09, BEK16], the Vector Bin Packing problem is viewed as a setcover instance, and the algorithms proceed by rounding the standard set cover LP. Towardsproving the hardness of Vector Bin Packing, we ask the converse: Which families of setcover instances can be cast as d -dimensional Vector Bin Packing? We formalize this question using the notion of packing dimension of a set system S ona universe V : it is the smallest integer d such that there is an embedding f : V → [0 , d such that a set S ⊆ V is in S if and only if (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) v ∈ S f ( v ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ If a set system has packing dimension d , then the corresponding set cover problem canbe embedded as a d -dimensional Vector Bin Packing instance. However, it is not clearif the hard instances of the set cover problem have a low packing dimension. Indeed theinstances in the (1 − (cid:15) ) ln n set cover hardness [Fei98] have large packing dimension thatgrows with n , which we cannot afford as we are operating in the constant d regime. Weget around this by starting our reduction from highly structured yet hard instances of setcover. In particular, we study simple bounded set systems which satisfy the following threeproperties:1. The set system is simple i.e., every pair of sets intersect in at most one element. Simple set families are also known as linear set families.
3. The cardinality of each set is at most k , a fixed constant.3. Each element in the family is present in at most ∆ = k O (1) sets.Kumar, Arya, and Ramesh [KAR00] proved that simple set cover i.e., set cover with therestriction that every pair of sets intersect in at most one element, is hard to approximatewithin Ω(log n ) . We observe that by modifying the parameters slightly in their proof, wecan obtain the Ω(log k ) hardness of simple bounded set cover.We prove that simple bounded set systems have packing dimension at most k O (1) . Thus,the Ω(log k ) simple bounded set cover hardness translates to Ω(log d ) hardness of VectorBin Packing when d is a constant. Note that the optimal value of the set cover instancescan be made arbitrarily large in terms of k by starting with a Label Cover instance withan arbitrarily large number of edges. Thus, our Vector Bin Packing hardness holds forasymptotic approximation as well.Our upper bound on the packing dimension is obtained in two steps: First, we write thegiven simple bounded set system as an intersection of ( k ∆) O (1) structured simple boundedset systems on the same universe, and then we give an embedding using ( k ∆) O (1) dimen-sions bounding the packing dimension of these structured simple bounded set systems.This idea of decomposition into structured instances is inspired from a work of Chandran,Francis, and Sivadasan [CFS08] where an upper bound on the Boxicity of a graph isobtained in terms of its maximum degree. We believe that the packing dimension of setsystems is worth studying on its own, especially in light of its close connections to thenotions of dimension of graphs such as Boxicity. For the Vector Scheduling problem, we obtain a Ω (cid:0) (log d ) − (cid:15) (cid:1) hardness under NP (cid:42) ZPTIME (cid:16) n (log n ) O (1) (cid:17) , almost matching the O (cid:16) log d log log d (cid:17) algorithms [HS19, IKKP19]. Theorem 1.2.
For every constant (cid:15) > , assuming NP (cid:42) ZPTIME (cid:16) n (log n ) O (1) (cid:17) , d -dimensional Vector Scheduling has no polynomial time Ω (cid:0) (log d ) − (cid:15) (cid:1) -factor approximationalgorithm when d is part of the input. We obtain the hardness result via a reduction from the Monochromatic Clique problem.In the
Monochromatic-Clique (k,B) problem, given a graph G = ( V, E ) with | V | = n andparameters k ( n ) and B ( n ) , the goal is to distinguish between the case when G is k -colorableand the case when in any assignment of k -colors to vertices of G , there is a clique of size B all of whose vertices are assigned the same color. When B = 2 , this is the standard graphcoloring problem. Note that the problem gets easier as B increases. Indeed, when B > √ n ,we can solve the problem in polynomial time by computing the Lovász theta function ofthe complement graph. We are interested in proving the hardness of the problem for B aslarge a function of n as possible, for some k . For example, given a graph that is promisedto be k colorable, can we prove the hardness of assigning k colors to the vertices of thegraph in polynomial time where each color class has maximum clique at most B = log n ? The Monochromatic Clique problem was defined formally by Im, Kell, Kulkarni, andPanigrahi [IKKP19] in the context of proving lower bounds for online Vector Scheduling.It was also used implicitly in the ω (1) NP-hardness of Vector Scheduling by Chekuri andKhanna [CK04]. They proved (implicitly) that Monochromatic Clique is NP-hard when B is an arbitrary constant using a reduction from n − (cid:15) hardness of graph coloring. We observethat the same reduction combined with better hardness of graph chromatic number [Kho01]proves the hardness of Monochromatic Clique when B = (log n ) γ , for some constant γ > under the assumption that NP (cid:42) ZPTIME (cid:16) n (log n ) O (1) (cid:17) .4e then amplify this hardness to B = (log n ) C , for every constant C > . Our mainidea in this amplification procedure is the notion of a stronger form of MonochromaticClique where given a graph and parameters k, B, C , the goal is to distinguish between thecase that G is k colorable vs. in any k C coloring of G , there is a monochromatic cliqueof size B . It turns out that the graph coloring hardness of Khot [Kho01] already provesthe hardness of this stronger variant of Monochromatic clique when B = (log n ) γ for anyconstant C . We then use lexicographic product of graphs to amplify this result into thehardness of original Monochromatic Clique problem with B = (log n ) C for any constant C under the same assumption that NP (cid:42) ZPTIME (cid:16) n (log n ) O (1) (cid:17) . This directly gives therequired hardness of Vector Scheduling using the reduction in [CK04].The Vector Scheduling problem is also closely related to the Balanced HypergraphColoring problem where the input is a hypergraph H and a parameter k , and the objectiveis to color the vertices of H using k colors to minimize the maximum number of times acolor appears in an edge. We use this connection to improve upon the NP-hardness of theproblem: Theorem 1.3.
For every constant
C > , d -dimensional Vector Scheduling is NP-hard toapproximate within Ω (cid:0) (log log d ) C (cid:1) when d is part of the input. Consider the case when each vector job is from { , } d . In this setting, we can vieweach coordinate as an edge in a hypergraph, and each vector corresponds to a vertex of thehypergraph. The goal is to find a m -coloring of vertices of the hypergraph i.e., an assign-ment of the vectors to m machines to minimize the maximum number of monochromaticvertices in an edge, which directly corresponds to the maximum load on a machine.This problem of coloring a hypergraph to ensure that no color appears too many timesin each edge is known as Balanced Hypergraph Coloring. Guruswami and Lee [GL18] ob-tained strong hardness results for this problem when k , the uniformity of the hypergraphis a constant, using the Label Cover Long Code framework combined with analytical tech-niques such as the invariance principle. However, when k is super constant, the invarianceprinciple based methods give weak bounds as the soundness of the Label Cover has to be atleast exponentially small in k . Recently, using combinatorial tools to analyze the gadgetsinstead of the standard analytical techniques, improvements have been obtained for vari-ous hypergraph coloring problems [Bha18,ABP19] in the super-constant inapproximabilityregime. We follow the same route and use combinatorial tools to analyze the gadgets in theLabel Cover Long Code framework and obtain better hardness of Balanced HypergraphColoring in the regime of super-constant uniformity k . The key combinatorial lemma usedin our analysis was proved recently by Austrin, Bhangale, and Potukuchi [ABP20] using ageneralization of the Borsuk-Ulam theorem.The NP-hardness of Vector Scheduling follows directly from the hardness of BalancedHypergraph Coloring using the above-described reduction. This NP-hardness result usesnear-linear size Label Cover hardness results [MR10, DS14]. By using the standard LabelCover hardness obtained by combining PCP Theorem and Parallel Repetition in the samereduction, we also prove an intermediate result bridging the above two hardness results forVector Scheduling. Theorem 1.4.
There exists a constant γ > such that assuming NP (cid:42) DTIME (cid:0) n O (log log n ) (cid:1) , d -dimensional Vector Scheduling is hard to approximate within Ω ((log d ) γ ) when d is partof the input. For the Vector Bin Covering problem, we show Ω (cid:16) log d log log d (cid:17) hardness, almost matching the O (log d ) factor algorithm [AAC + heorem 1.5. d -dimensional Vector Bin Covering is NP-hard to approximate within Ω (cid:16) log d log log d (cid:17) factor when d is part of the input. Similar to Vector Scheduling, the hard instances for the Vector Bin Covering are wheneach vector is in { , } d . Using the same connection as before, we view this problemas a hypergraph coloring problem where each edge of the hypergraph corresponds to acoordinate, and each vertex corresponds to a vector. Assigning the vectors to the binssuch that in each bin, the sum is at least in every coordinate corresponds to assigningcolors to vertices of the hypergraph such that in every edge, all the colors appear. Sucha coloring of the hypergraph with k colors where all the k colors appear in every edge isknown as a k -rainbow coloring of the hypergraph.Strong hardness results are known for approximate rainbow coloring [GL18, ABP20,GS20] when k is a constant. While these results give decent bounds in the super con-stant regime, they proceed via the hardness of Label Cover whose soundness is an inversepolynomial function of k . Because of this, in the NP-hard instances, the number of edgesis at least doubly exponential in k . Instead, by losing a factor of in the hardness, wegive a reduction to the approximate rainbow coloring problem from Label Cover with nogap i.e., just a “Label Coverized” -SAT instance. In these hard instances, the number ofedges is single exponential in k , proving that it is NP-hard to distinguish the case that ahypergraph with m edges has a rainbow coloring with Ω (cid:16) log m log log m (cid:17) colors vs. it cannot berainbow colored with colors . This hardness of approximate rainbow coloring gives therequired Vector Bin Covering hardness immediately using the earlier mentioned analogybetween rainbow coloring and Vector Bin Covering.We summarize our results in Table 1. Online Algorithms.
Multidimensional packing problems have been extensively stud-ied in the online setting. For the d -dimensional Vector Bin Packing, the classical First-Fit algorithm [GGJY76] gives O ( d ) competitive ratio, and Azar, Cohen, Kamara, andShepherd [ACKS13] recently gave an almost matching Ω (cid:0) d − (cid:15) (cid:1) lower bound. For the d -dimensional Vector Scheduling, Im, Kell, Kulkarni and Panigrahi [IKKP19] gave a O (cid:16) log d log log d (cid:17) competitive online algorithm and proved a matching lower bound. For the d -dimensional Vector Bin Covering problem, Alon et al. [AAC +
98] gave a d competitivealgorithm and proved a lower bound of d + . Geometric variants.
There are various natural geometric variants of Vector Bin Packingthat have been studied in the literature. A classical problem of this sort is the -dimensionalGeometric Bin Packing, where the input is a set of rectangles that need to be packed intothe minimum number of unit squares. After a long line of works, Bansal and Khan [BK14]gave a . factor asymptotic approximation algorithm for the problem. On the hardnessfront, Bansal and Sviridenko [BS04] showed that the problem does not admit an asymptoticPTAS, and this APX hardness result has been generalized to several related problems byChlebík and Chlebíková [CC06]. We refer the reader to the excellent survey [CKPT17]regarding the geometric problems. We first define the multidimensional problems and the Label Cover problem formally in Sec-tion 2. Next, we prove the hardness results for Vector Bin Packing, Vector Scheduling, Note that rainbow coloring with colors is the same as proper coloring (Property B) of the hyper-graphs. d = 1 PTAS [dlVL81] NP-Hard [GJ79]Fixed d ln d + O (1) [BEK16] Ω(log d ) Arbitrary d (cid:15)d + O (cid:0) ln (cid:15) (cid:1) [CK04] d − (cid:15) [CK04, CKPT17]VS d = 1 EPTAS [Jan10] No FPTAS [FKT89]Fixed d PTAS [CK04] No EPTAS [BOVvdZ16]Arbitrary d O (cid:16) log d log log d (cid:17) [HS19, IKKP19] Ω (cid:0) (log d ) − (cid:15) (cid:1)(cid:16) NP (cid:42) ZPTIME (cid:16) n (log n ) O (1) (cid:17)(cid:17) (log d ) Ω(1) (cid:0) NP (cid:42) DTIME (cid:0) n O (log log n ) (cid:1)(cid:1) (log log d ) ω (1) (NP-Hardness)VBC d = 1 FPTAS [JS03] NP-Hard [AJKL84]Arbitrary d O (log d ) [AAC + Ω (cid:16) log d log log d (cid:17) Table 1: Approximation algorithms for the multidimensional packing problems. VBP,VS and VBC stand for Vector Bin Packing, Vector Scheduling and Vector Bin Coveringrespectively. All the results for VBP and the FPTAS for VBC are asymptotic. The resultswithout citations are our new results.and Vector Bin Covering in Section 3, Section 4 and Section 5 respectively. Finally, weconclude by mentioning a few open problems in Section 6.
Notations.
We use [ n ] to denote the set { , , . . . , n } . We use d to denote the d -dimensional vector (1 , , . . . , . For two d -dimensional real vectors a and b , we say that a ≥ b if a i ≥ b i for all i ∈ [ d ] . For a graph G , we let ω ( G ) , α ( G ) , χ ( G ) be the largestclique size, largest independent size, and the chromatic number of G respectively. A setsystem or set family S on a universe V is a collection of subsets of V . Problem Statements.
We give formal definitions of the three problems that we study.
Definition 2.1. (Vector Bin Packing) In the Vector Bin Packing problem, the input isa set of n rational vectors v , v , . . . , v n ∈ [0 , d . The objective is to partition [ n ] intominimum number of parts A , A , . . . , A m such that (cid:13)(cid:13)(cid:13) (cid:88) j ∈ A i v j (cid:13)(cid:13)(cid:13) ∞ ≤ ∀ i ∈ [ m ] Definition 2.2. (Vector Scheduling) In the Vector Scheduling problem, the input is a setof n rational vector jobs v , v , . . . , v n ∈ [0 , d , and m identical machines. The objective isto assign the jobs to machines i.e. partition [ n ] into m parts A , A , . . . , A m to minimizethe makespan which is defined as the maximum (cid:96) ∞ load on a machine. max i ∈ [ m ] (cid:13)(cid:13)(cid:13) (cid:88) j ∈ A i v j (cid:13)(cid:13)(cid:13) ∞ efinition 2.3. (Vector Bin Covering) In the Vector Bin Covering problem, we are given n vectors v , v , . . . , v n ∈ [0 , d . The goal is to partition the input vectors into maximumnumber of parts A , A , . . . , A m such that (cid:88) j ∈ A i v j ≥ d ∀ i ∈ [ m ] Asymptotic Approximation.
For the Bin Packing problem, it is NP-Hard to identifyif all the vectors can be packed into bins or need bins. This already proves that theproblem is NP-hard to approximate within as per the usual notion of multiplicativeapproximation ratio. However, this is less interesting as there are much better asymptotic approximation algorithms for the problem which get (1 + (cid:15) ) -factor approximation whenthe optimal value is large enough, for every positive constant (cid:15) > .Even for the Vector Bin Packing problem, the performance of an algorithm is typicallymeasured in the asymptotic setting. We give the formal definition [CKPT17] of asymptoticapproximation ratio of an algorithm A for the Vector Bin Packing problem. Definition 2.4. (Asymptotic Approximation Ratio) The asymptotic approximation ratio ρ ∞A of an algorithm A for the Vector Bin Packing problem is ρ ∞A = lim sup n →∞ ρ n A , ρ n A = sup I ∈I (cid:26) A ( I ) OPT ( I ) : OPT ( I ) = n (cid:27) where I denotes the set of all possible Vector Bin Packing instances. All the results mentioned in this paper regarding Vector Bin Packing are with respectto the asymptotic approximation ratio.
Label Cover.
We define the Label Cover problem:
Definition 2.5. (Label Cover) In an instance of the Label Cover problem G = ( V = L ∪ R, E, Σ L , Σ R , Π) with | Σ L | ≥ | Σ R | , the input is a bipartite graph L ∪ R with constraintson every edge. The constraint on an edge e is a projection Π e : Σ L → Σ R . We say alabeling σ : V → Σ L ∪ Σ R satisfies the constraint on the edge e = ( u, v ) if Π e ( σ ( u )) = σ ( v ) .The objective is to find a labeling σ : V → Σ L ∪ Σ R that satisfies as many constraints aspossible. By a simple reduction from the -SAT problem, we can prove that Label Cover isNP-hard when Σ L and Σ R are constants (See e.g., Lemma . in [BG16]). Theorem 2.6.
Given a Label Cover instance when Σ L = Σ R = [6] , it is NP-hard toidentify if it has a labeling that satisfies all the constraints. The real use of Label Cover, however, lies in its strong hardness of approximation. PCPTheorem [ALM +
98] combined with Raz’s parallel repetition [Raz98] yields the followingstrong inapproximability of Label Cover problem.
Theorem 2.7.
There exists an absolute constant c > such that for every integer n and (cid:15) > , there is a reduction from -SAT instance I over n variables to Label Cover instance G = ( V = L ∪ R, E, Σ L , Σ R , Π) with | V | ≤ n O ( log ( (cid:15) )) , | Σ L | ≤ (cid:0) (cid:15) (cid:1) c satisfying the following:1. (Completeness.) If I is satisfiable, there exists a labeling to G that satisfies all theconstraints.2. (Soundness.) If I is not satisfiable, no labeling can satisfy an (cid:15) fraction of the con-straints of G . . (Biregularity.) The graph L ∪ R, E is biregular with degrees on either side boundedby poly (cid:0) (cid:15) (cid:1) .Furthermore, the running time of the reduction is poly (cid:0) n, (cid:15) (cid:1) . Moshkovitz-Raz [MR10] proved the following hardness of near linear size Label Cover.
Theorem 2.8.
There exist absolute constants c, c (cid:48) > such that for every n and (cid:15) > ,there is a reduction from -SAT instance I over n variables to Label Cover instance G =( V = L ∪ R, E, Σ L , Σ R , Π) with | V | ≤ n o (1) (cid:0) (cid:15) (cid:1) c , | Σ L | ≤ (cid:15) ) c (cid:48) satisfying the following:1. (Completeness.) If I is satisfiable, there exists a labeling to G that satisfies all theconstraints.2. (Soundness.) If I is not satisfiable, no labeling can satisfy an (cid:15) fraction of the con-straints of G .3. (Biregularity.) The graph L ∪ R, E is biregular with degrees on either side poly (cid:0) (cid:15) (cid:1) .Furthermore, when (cid:15) is a constant, the running time of the reduction is poly ( n ) . In this section, we prove the hardness of approximation of Vector Bin Packing. First, wedefine the packing dimension of a set family and bound the packing dimension of simpleset families. Next, we combine this upper bound with the hardness of set cover on simplebounded set systems to prove Theorem 1.1.
For a set family S on a universe V , we define the packing dimension pdim ( S ) below. For afunction f : V → [0 , K and a set S ⊆ V , we let f ( S ) denote the vector f ( S ) = (cid:80) v ∈ S f ( v ) . Definition 3.1.
For a set family S on a universe V , the packing dimension pdim ( S ) isdefined as the smallest positive integer K such that there exists an embedding f : V → [0 , K that satisfies the following property: For every set S ⊆ V , S is in the family S ifand only if (cid:107) f ( S ) (cid:107) ∞ ≤ . If no such embedding exists, we say that pdim ( S ) is infinite. For a set family S to have finite packing dimension i.e. for an embedding f : V → [0 , K realizing the above condition to exist requires two conditions:1. The set family is downward closed i.e. for every S ∈ S and T ⊆ S , T ∈ S as well.2. For every element v ∈ V , there is a set S ∈ S with v ∈ S . We call a set family S ona universe V non-trivial if for every v ∈ V , there is a set S ∈ S with v ∈ S .On the other hand, any set system that satisfies the above two conditions i.e. being down-ward closed and non-trivial has a finite packing dimension. Before proving this statement,we first prove the following simple but useful proposition. Proposition 3.2.
For a pair of set families S and S defined on the same universe V such that pdim ( S ) and pdim ( S ) are finite, pdim ( S ∩ S ) ≤ pdim ( S ) + pdim ( S ) roof. Let K = pdim ( S ) and K = pdim ( S ) . Suppose that f : V → [0 , K be suchthat for every set S ⊆ V , (cid:107) f ( S ) (cid:107) ∞ ≤ if and only if S ∈ S . Similarly, let f : V → [0 , K be such that for every set S ⊆ V , (cid:107) f ( S ) (cid:107) ∞ ≤ if and only if S ∈ S . Consider the function f : V → [0 , K + K defined as f ( v ) =( f ( v ) , f ( v )) . Then, for every set S ⊆ V , (cid:107) f ( S ) (cid:107) ∞ ≤ if and only if (cid:107) f ( S ) (cid:107) ∞ ≤ and (cid:107) f ( S ) (cid:107) ∞ ≤ . Thus, for every set S ⊆ V , (cid:107) f ( S ) (cid:107) ∞ ≤ if and only if S ∈ S and S ∈ S , or equivalently, if S ∈ S ∩ S . Hence, the packing dimension of S ∩ S is at most K + K .For a set S ⊆ V , let S ↑ be the family of sets T ⊆ V such that S ⊆ T . Similarly, let S ↓ be the family of sets T ⊆ V such that T ⊆ S . For a set system S , we let S ↑ (resp. S ↓ )denote the union of S ↑ (resp. S ↓ ) over all S ∈ S .Consider a set S ⊆ V with | S | > . For the set family V \ S ↑ , we have the embedding f : V → [0 , defined as f ( v ) = (cid:40) | S | + | S | , if v ∈ S otherwise.This shows that pdim (2 V \ S ↑ ) ≤ for all S ⊆ V with | S | > . Note that we have S = (cid:92) S / ∈S V \ S ↑ for every downward closed set system S . Combined with Proposition 3.2, we obtain thatfor every non-trivial downward closed family S on a universe V , pdim ( S ) ≤ | V | .We are interested in the classes of set families for which there is an efficient embeddingwith packing dimension being independent of | V | . In particular, the class of set familiesthat we study are bounded set families where each set has cardinality at most k , and eachelement appears in at most ∆ sets. We can show that such bounded set families thatare downward closed and non-trivial have packing dimension at most ( k ∆) O (∆) . Togetherwith the Ω(log k ) hardness [Tre01] of k -set cover where each set has cardinality at most k , a fixed constant and each element appearing in (log k ) O (1) sets, this packing dimensionbound gives the hardness of (log d ) Ω(1) for the Vector Bin Packing problem when d is a largeconstant. Unfortunately, the exponential dependence on ∆ is necessary for the packingdimension of bounded set systems, and thus, this approach does not yield the optimal Ω(log d ) hardness of Vector Bin Packing.Instead of using arbitrary bounded set families, we bypass this barrier by using simplebounded set families. Recall that a set family is called simple if any two distinct setsin the family intersect in at most one element. It turns out that for simple bounded setfamilies i.e. simple set families S where each set has cardinality at most k , and each elementappears in at most ∆ sets, the packing dimension of S ↓ can be upper bounded by ( k ∆) O (1) .Together with the Ω(log k ) hardness of simple k -set cover (proved in Appendix A), we getthe optimal Ω(log d ) hardness of Vector Bin Packing when d is a large constant. In the nextsubsection, we prove the packing dimension upper bound, and we use this upper bound toprove the hardness of Vector Bin Packing in Section 3.3. The main embedding result that we prove is that the downward closure of simple setsystems where each set has cardinality k and each element appears in at most ∆ sets haspacking dimension at most polynomial in k, ∆ .10 heorem 3.3. Suppose that S is a simple non-trivial set system on a universe V whereeach set has cardinality at most k ≥ and each element appears in at most ∆ sets. Then, pdim ( S ↓ ) ≤ ( k ∆) O (1) Furthermore, an embedding realizing the above can be found in time polynomial in | V | . We prove the embedding result by writing the set family S ↓ as an intersection of ( k ∆) O (1) structured set families each of which has packing dimension at most ( k ∆) O (1) . Wecan then upper bound the packing dimension of S ↓ using Proposition 3.2. The structuredset systems we study are sunflower-bouquets , which are a disjoint union of sunflowers thathave a single element as the kernel. The formal definition of the sunflower-bouquet setfamilies is below. See Figure 1 for an illustration. Definition 3.4. (Sunflower-bouquets) A simple set system S on a universe V is called asunflower-bouquet with core U ⊆ V, U (cid:54) = φ if the following hold.1. Every set S ∈ S satisfies | S ∩ U | = 1 . Furthermore, for every u ∈ U , there is a set S ∈ S with u ∈ S .2. For any pair of sets S , S ∈ S with S ∩ S (cid:54) = ∅ , we have S ∩ U = S ∩ U = S ∩ S . We now give an efficient embedding for a sunflower-bouquet S on a universe V withcore U ⊆ V, U (cid:54) = φ . The main motivation behind this lemma is to upper bound the packingdimension of the set system T ↓ = S ↓ ∪ { S ⊆ V \ U : | S | ≤ k } . Lemma 3.5.
Fix an integer k ≥ . Let S be a simple set family defined on a universe V that is a sunflower-bouquet with core U . Furthermore, each set in the family has cardinalityat most k and each element appears in at most ∆ sets. Then, there exists an embedding f : V → [0 , K that satisfies(A) For every set S ∈ S , (cid:107) f ( S ) (cid:107) ∞ ≤ . (B) For every set S / ∈ S ↓ with S ∩ U (cid:54) = ∅ , (cid:107) f ( S ) (cid:107) ∞ > . (C) For every set S ⊆ V with S ∩ U = ∅ and | S | ≤ k , (cid:107) f ( S ) (cid:107) ∞ ≤ . (D) For every set S ⊆ V with | S | > k , (cid:107) f ( S ) (cid:107) ∞ > . with K = ( k ∆) O (1) . Furthermore, such an embedding can be found in time polynomial in | V | given S .Proof. Let U = { u , u , . . . , u m } . We can partition V \ U into V , V , . . . , V m with V i = (cid:91) S ∈S : u i ∈ S S \ { u i } for all i ∈ [ m ] . As each set in S has cardinality at most k and each element appears in atmost ∆ sets, we get that | V i | ≤ k ∆ for all i ∈ [ m ] . For every i ∈ [ m ] , we order the elementsof V i as { v i, , v i, , . . . , v i,k ∆ } (with repetitions if needed).We construct the embedding f in two steps. Step-1. Eliminating the cross-sunflower sets . In the first step, our goal is to findan embedding g : V → [0 , K with K = 2 k ∆ + 2 such that11 u u Figure 1: An illustration of a sunflower-bouquet set family. Here, S is the family of all thegreen colored sets. It is a sunflower-bouquet with core U = { u , u , u } . In the embedding,we ensure that the (cid:96) ∞ norm of the left red set is greater than in the first step while theright side red set is handled in the second step.1. g satisfies the conditions ( A ) and ( C ) i.e. for every set S ∈ S , (cid:107) g ( S ) (cid:107) ∞ ≤ , and forevery set S ⊆ V with S ∩ U = φ and | S | ≤ k, (cid:107) g ( S ) (cid:107) ∞ ≤ .
2. For every “cross-sunflower” set S ⊆ V with u i ∈ S for some i ∈ [ m ] , and S ∩ V i (cid:48) (cid:54) = φ for i (cid:48) ∈ [ m ] , i (cid:48) (cid:54) = i , we have (cid:107) g ( S ) (cid:107) ∞ > .We achieve this by setting g = ( f , f , . . . , f k ∆ ) , where each f l : V → [0 , , l ∈ { , , . . . , k ∆ } satisfies the conditions ( A ) and ( C ) , and overall, the embedding g satisfies the second con-dition above.We define the embedding f : V → [0 , as f ( v ) = (cid:0) , k (cid:1) , if v ∈ U (cid:0) k , k (cid:1) if v ∈ V (cid:0) , k (cid:1) otherwise.We can verify that f satisfies the conditions ( A ) and ( C ) .We choose m distinct rational numbers α , . . . , α m with − k < α i < for all i ∈ [ m ] .We define the embeddings f l : V → [0 , , l ∈ [ k ∆] as follows. Consider an l ∈ [ k ∆] .1. For i ∈ [ m ] , we set f l ( u i ) = (cid:18) α i , − k − α i (cid:19)
2. For i ∈ [ m ] and v i,j ∈ V i , we set f l ( v i,j ) = (0 , if v i,j (cid:54) = v i,l . We set f l ( v i,l ) = (cid:18) − α i , α i + 1 k − (cid:19)
3. For v ∈ V , we set f l ( v ) = (0 , .We verify that these embeddings satisfy the conditions ( A ) and ( C ) . Fix an l ∈ [ k ∆] .12A) Consider a set S ∈ S . Let i ∈ [ m ] be such that { u i } = S ∩ U . We have f l ( S ) = (cid:88) v ∈ S f l ( v ) ≤ (cid:88) v ∈{ u i }∪ V i f ( v )= f l ( u i ) + f l ( v i,l )= (cid:18) α i , − k − α i (cid:19) + (cid:18) − α i , α i + 1 k − (cid:19) = (1 , . (C) This follows directly from the fact that (cid:107) f l ( v ) (cid:107) ≤ k for all l ∈ [ k ∆] and v ∈ V \ U .Let g : V → [0 , k ∆+2 be defined as g = ( f , f , . . . , f k ∆ ) . As each of the individualembeddings satisfies ( A ) and ( C ) , g also satisfies the conditions ( A ) and ( C ) .Let S ⊆ V be such that (cid:107) g ( S ) (cid:107) ∞ ≤ . i.e. (cid:107) f l ( S ) (cid:107) ∞ ≤ for all l ∈ { , , . . . , k ∆ } . We collect the following observations that willbe used later when we analyze the final embedding.1. As f ( v ) = 1 for all v ∈ U , | S ∩ U | ≤ . As f ( v ) = k for all v ∈ V , if S ∩ U (cid:54) = ∅ ,then S ∩ V = ∅ .2. As f ( v ) = k for all v ∈ V , | S | ≤ k .3. Suppose that S ∩ U = { u i } . Then, we claim that S ⊆ { u i } ∪ V i . Suppose forcontradiction that this is not the case, and there exists v i (cid:48) ,l ∈ V i (cid:48) with i (cid:48) (cid:54) = i, i (cid:48) ∈ [ m ] and l ∈ [ k ∆] such that v i (cid:48) ,l ∈ S . We have f l ( S ) = (cid:88) v ∈ S f l ( v ) ≥ f l ( u i ) + f l ( v i (cid:48) ,l )= (cid:18) α i , − k − α i (cid:19) + (cid:18) − α i (cid:48) , α i (cid:48) + 1 k − (cid:19) = (1 + α i − α i (cid:48) , α i (cid:48) − α i ) As α i (cid:54) = α i (cid:48) , (cid:107) f l ( S ) (cid:107) ∞ > , a contradiction. Step 2. Pinning down the intra-sunflower sets.
In the second step, our goal isto find an embedding g (cid:48) : V → [0 , K with K = ( k ∆) such that1. g (cid:48) satisfies the conditions ( A ) and ( C ) .2. For every i ∈ [ m ] and “intra-sunflower” set S ⊆ { u i } ∪ V i such that u i ∈ S and S / ∈ S ↓ , we have (cid:107) g (cid:48) ( S ) (cid:107) ∞ > .We achieve this by setting g (cid:48) = ( g , g , . . . , g ( k ∆) ) ) where each g l , l ∈ [( k ∆) ] satisfies theconditions ( A ) and ( C ) , and the overall function g (cid:48) satisfies the second condition above.For every i ∈ [ m ] , we order all the pairs of distinct elements x, y ∈ V i as { V i, , V i, , . . . , V i, ( k ∆) } (with repetitions if needed). The upper bound on the number of such pairs is obtainedusing the fact that | V i | ≤ k ∆ for all i ∈ [ m ] .We define the embeddings g l : V → [0 , , l ∈ [( k ∆) ] below. Fix an l ∈ [( k ∆) ] .1. Consider an i ∈ [ m ] . We have two different cases:13a) If V i,l ∪ { u i } ∈ S ↓ , we set g l ( u i ) = 0 and g l ( v ) = 0 for all v ∈ V i .(b) If V i,l ∪ { u i } / ∈ S ↓ , we set g l ( v ) = k for all v ∈ V i,l , and g l ( v ) = 0 for all v ∈ V i \ V i,l . We set g l ( u i ) = 1 − k + 1 k
2. For all v ∈ V , we set g l ( v ) = 0 .We now verify that these embeddings satisfy the conditions ( A ) and ( C ) . Fix an integer l ∈ [( k ∆) ] .(A) Consider a set S ∈ S . Let { u i } = S ∩ U . If { u i } ∪ V i,l ∈ S ↓ , g l ( v ) = 0 for all v ∈ S ,and thus we have | g l ( S ) | ≤ . Now suppose that { u i } ∪ V i,l / ∈ S ↓ . This implies that V i,l is not a subset of S . As | V i,l | = 2 , | V i,l ∩ S | ≤ . We get (cid:88) v ∈ S g l ( v ) = g l ( u i ) + (cid:88) v ∈ S ∩ V i g l ( v )= g l ( u i ) + (cid:88) v ∈ S ∩ V i,l g l ( v ) ≤ g l ( u i ) + 1 k = 1 − k + 1 k + 1 k ≤ (C) This follows from the fact that g l ( v ) ≤ k for all v ∈ V \ U . Final embedding.
We define the final embedding f : V → [0 , k ∆+2+( k ∆) as f =( g, g , g , . . . , g ( k ∆) ) . As each of these embeddings satisfies the conditions ( A ) and ( C ) ,the final embedding f also satisfies the conditions ( A ) and ( C ) .Suppose that (cid:107) f ( S ) (cid:107) ∞ ≤ for a set S ⊆ V . Then, (cid:107) g ( S ) (cid:107) ∞ ≤ and (cid:107) g l ( S ) (cid:107) ∞ ≤ forall l ∈ [( k ∆) ] . Condition ( D ) follows easily as (cid:107) g ( S ) (cid:107) ∞ ≤ implies that | S | ≤ k .We now return to condition ( B ) . Suppose that S ⊆ V with S ∩ U (cid:54) = ∅ satisfies (cid:107) f ( S ) (cid:107) ∞ ≤ . Our goal is to show that S ∈ S ↓ . We have already deduced from (cid:107) g ( S ) (cid:107) ∞ ≤ that | S ∩ U | = 1 and | S | ≤ k . Let S ∩ U = { u i } and we also have S ⊆ { u i } ∪ V i . Let S = { u i , s , s , . . . , s p } where s j ∈ V i for all j ∈ [ p ] . Note that for every v ∈ V i , there isexactly one set S ( v ) ∈ S such that v ∈ S ( v ) and this set S ( v ) satisfies u i ∈ S ( v ) . Thisfollows from the definition of V i and the fact that the set family S is a sunflower-bouquet.We now claim that S ( s j ) = S ( s j ) for all j , j ∈ [ p ] . Suppose for contradictionthat there exist j , j ∈ [ p ] with S ( s j ) (cid:54) = S ( s j ) . This implies that { u i , s j , s j } / ∈ S ↓ asotherwise, if there exists T ∈ S such that { u i , s j , s j } ⊆ T , we have S ( s j ) = S ( s j ) = T .Let l ∈ [( k ∆) ] be such that V i,l = { s j , s j } . As V i,l ∪ { u i } / ∈ S ↓ , we have g l ( v ) = k for all v ∈ V i,l and g l ( u i ) = 1 − k + 1 k Thus, we get that (cid:88) v ∈ S g l ( v ) = g l ( u i ) + (cid:88) v ∈ S \{ u i } g l ( v )= g l ( u i ) + (cid:88) v ∈ V i,l g l ( v )= 1 − k + 1 k + 2 k = 1 + 1 k g l ( S ) ≤ . This completes the proof that S ( s j ) = S ( s j ) forall j , j ∈ [ p ] . Thus, there exists a set S ( s ) ∈ S such that S ⊆ S ( s ) , which implies that S ∈ S ↓ . Thus, every set S ⊆ V with (cid:107) f ( S ) (cid:107) ∞ ≤ and S ∩ U (cid:54) = ∅ satisfies S ∈ S ↓ , whichproves the condition ( B ) .Note that our construction is explicit, and we have a polynomial time algorithm tooutput the required embedding. The dimension of the embedding is k ∆ + 2 + ( k ∆) ,which is at most ( k ∆) O (1) .As a corollary, we bound the packing dimension of the set family T ↓ = S ↓ ∪ { S ⊆ V \ U : | S | ≤ k } . Corollary 3.6.
Suppose that T is a set family defined on a universe V with T = S ∪ { S ⊆ V \ U : | S | ≤ k } where S ⊆ V is a sunflower-bouquet with core U . Furthermore, each set in S has cardi-nality at most k ≥ and each element appears in at most ∆ sets in S . Then, pdim ( T ↓ ) ≤ ( k ∆) O (1) Furthermore, an embedding realizing this packing dimension can be found in time polyno-mial in | V | given S .Proof. As S is a sunflower-bouquet, from Lemma 3.5, there exists an embedding f : V → [0 , K that satisfies the conditions ( A ) , ( B ) , ( C ) and ( D ) with K = ( k ∆) O (1) . Conditions ( A ) and ( C ) together imply that (cid:107) f ( S ) (cid:107) ∞ ≤ for all S ∈ T . Note that T ↓ = S ↓ ∪ { S ⊆ V \ U : | S | ≤ k } . Suppose that S ⊆ V is a subset of V with S / ∈ T ↓ . If S ∩ U = φ , then | S | > k , whichimplies that (cid:107) f ( S ) (cid:107) ∞ > using condition ( D ) . If S ∩ U (cid:54) = φ , then S / ∈ S ↓ which impliesthat (cid:107) f ( S ) (cid:107) ∞ > using condition ( B ) . Thus, (cid:107) f ( S ) (cid:107) ∞ ≤ if and only if S ∈ T ↓ .We are now ready to prove our main embedding result i.e. Theorem 3.3. Proof of Theorem 3.3.
We define a graph G = ( V, E ) as follows: two elements u, v ∈ V areadjacent in G if there exist sets S , S ∈ S (not necessarily distinct) such that u ∈ S , v ∈ S , S ∩ S (cid:54) = ∅ . As the cardinality of each set in S is at most k and each element of V ispresent in at most ∆ sets, the maximum degree of a vertex in G can be bounded above as ∆( G ) ≤ k ( k − Thus, the chromatic number of G is at most L = χ ( G ) ≤ k ( k − + 1 ≤ k ∆ . Usingthe greedy coloring algorithm, we can partition V into L non-empty parts U , U , . . . , U L such that each U j is a independent set in G . For every j ∈ [ L ] , as U j is an independentset in G , we have1. For every set S ∈ S , | S ∩ U j | ≤ .2. Any two sets S , S ∈ S with S ∩ U j (cid:54) = ∅ , S ∩ U j (cid:54) = ∅ and S ∩ S (cid:54) = ∅ satisfy S ∩ U j = S ∩ U j = S ∩ S . 15e now define the set families S , S , . . . , S L as follows: S j = { S ∈ S : S ∩ U j (cid:54) = ∅} ∪ { S ⊆ V \ U j : | S | ≤ k } We claim that (cid:84) j ∈ [ L ] S ↓ j = S ↓ . First, consider an arbitrary set S ∈ S ↓ and an integer j ∈ [ L ] . As | S | ≤ k , irrespective of S intersects U j or not, S ∈ S ↓ j . Thus, S ↓ ⊆ S ↓ j for all j ∈ [ L ] . Consider a non-empty set S / ∈ S ↓ . As U , U , . . . , U L is a partition of V , thereexists a j ∈ [ L ] such that S ∩ U j (cid:54) = ∅ . As S / ∈ S ↓ , S / ∈ S ↓ j . This implies that (cid:92) j ∈ [ L ] S ↓ j = S ↓ Using Proposition 3.2, in order to bound the packing dimension of S ↓ , it suffices to boundthe packing dimension of S ↓ j , j ∈ [ L ] .Fix an integer j ∈ [ L ] and consider the set family S ↓ j . It is defined on the universe V and there exists a non-empty subset U j ⊆ V such that S j = S (cid:48) j ∪ { S ⊆ V \ U j : | S | ≤ k } with S (cid:48) j = { S ∈ S : S ∩ U j (cid:54) = ∅} . Here, S (cid:48) j is a simple set system which satisfies the following properties:1. Each set in S (cid:48) j has cardinality at most k ≥ and each element appears in at most ∆ sets in S (cid:48) j .2. Every set S ∈ S (cid:48) j satisfies | S ∩ U j | = 1 . As S is non-trivial, for every u ∈ U j , thereexists a set S ∈ S (cid:48) j with u ∈ S .3. For every pair of sets S , S ∈ S (cid:48) j with S ∩ S (cid:54) = φ , S ∩ U j = S ∩ U j = S ∩ S .In other words, the set family S (cid:48) j is a sunflower-bouquet with core U j . Using Corol-lary 3.6, we get that pdim ( S ↓ j ) ≤ ( k ∆) O (1) for all j ∈ [ L ] , which completes the proof. We show that for large enough constant d , Vector Bin Packing is hard to approximatewithin Ω(log d ) . Our hardness is obtained via the hardness of set cover on simple boundedinstances.In the set cover problem, the input is a set family S on a universe V with | V | = n .The objective is to pick the minimum number of sets { S , S , . . . , S m } ⊆ S from the familysuch that their union is equal to V . The greedy algorithm where we repeatedly pick the setthat covers the maximum number of new elements achieves a ln n approximation factor.Fiege [Fei98] proved a matching hardness of (1 − (cid:15) )(ln n ) . On set systems where each pairof sets intersect in at most one element i.e. simple instances, Ω(log n ) hardness of setcover is proved by Kumar, Arya, and Ramesh [KAR00]. We observe that by changing theparameters slightly, their reduction also implies the same hardness on instances where themaximum set size is bounded: Theorem 3.7. (Set Cover on simple bounded instances) There exists an integer B suchthat for every constant B ≥ B , the Set Cover problem on simple set systems in which eachset has cardinality at most B is NP-hard to approximate within Ω(log B ) . Furthermore, inthe hard instances, each element occurs in at most O ( B ) sets. Proof of Theorem 1.1.
We prove the result by giving an approximation preserving reduc-tion from the NP-hard problem of set cover on simple bounded set systems. Let S be theset system from Theorem 3.7 defined on a universe V . Note that each set in the family hascardinality at most k = B and each element in the universe appears in at most ∆ = O ( B ) sets. We now output a set V of | V | vectors in [0 , d such that1. (Completeness.) If there is a set cover of size m in S , there is a packing of V using m bins.2. (Soundness.) If there is no set cover of size m (cid:48) in S , there is no packing of V using m (cid:48) bins.We use Theorem 3.3 to compute an embedding f : V → [0 , d in polynomial time suchthat (cid:107) f ( S ) (cid:107) ∞ ≤ if and only if S ∈ S ↓ , with d = ( k ∆) O (1) = B O (1) . Our output Vector Bin Packing instanceis the set of vectors f ( v ) , v ∈ V . V = { f ( v ) : v ∈ V } Completeness.
Suppose that there exist sets S , S , . . . , S m ∈ S whose union is V .Then, we use m bins with the vectors { f ( v j ) : j ∈ S i } in the i th bin. A vector mightappear in multiple bins, but we can arbitrarily pick one bin for each vector while stillmaintaining the property that in each bin, the (cid:96) ∞ norm of the sum of the vectors is atmost . Soundness.
Suppose that the minimum set cover in S has cardinality at least m (cid:48) + 1 .Then, we claim that the set of vectors V needs m (cid:48) + 1 bins to be packed. Suppose forcontradiction that there is a vector packing with m (cid:48) bins. In other words, there exists apartition of V into B , B , . . . , B m (cid:48) such that (cid:107) f ( B i ) (cid:107) ∞ ≤ for all i ∈ [ m (cid:48) ] . As (cid:107) f ( B i ) (cid:107) ∞ ≤ , B i ∈ S ↓ for all i ∈ [ m (cid:48) ] . That is, for every i ∈ [ m (cid:48) ] , there exists a set S i ∈ S such that B i ⊆ S i . This implies that { S , S , . . . , S m (cid:48) } is a set cover of V , a contradiction.As the original bounded simple set cover problem is hard to approximate within Ω(log B ) = Ω(log d ) , the resulting Vector Bin Packing is hard to approximate within Ω(log d ) . Furthermore, in the hard instances, the optimal value i.e. the minimum numberof bins needed to pack the vectors can be made arbitrarily large, and thus, the hardnessapplies to the asymptotic approximation ratio. In the Monochromatic Clique problem, given a graph G = ([ n ] , E ) and a parameter k ( n ) ,the objective is to assign k colors to the vertices of G so as to minimize the largestmonochromatic clique. More formally, we study the following decision version of the prob-lem. 17 efinition 4.1. ( Monochromatic-Clique ( k, B ) ) In the Monochromatic-Clique ( k, B ) prob-lem, given a graph G = ( V, E ) with | V | = n and parameters k ( n ) , B ( n ) , the goal is todistinguish between the following:1. (YES case) The chromatic number of G is at most k .2. (NO case) In any assignment of k colors to the vertices of G , there is a clique of size B , all of whose vertices are assigned the same color. It generalizes the standard k -Coloring problem, which corresponds to the case when B = 2 . Note that the problem gets easier as B increases. Indeed, when B > √ n , we cansolve the problem in polynomial time using the canonical SDP relaxation. We present thisalgorithm and an almost matching integrality gap in Appendix B.On the hardness front, we now prove that Monochromatic-Clique ( k, B ) is hard when B = (log n ) C , for any constant C . We achieve this in two steps: First, we observe that theexisting chromatic number hardness results already imply the hardness of monochromaticclique when B = (log n ) γ for some constant γ > . Next, we amplify this hardness byusing lexicographic graph product. We start with a couple of basic Ramsey theoretic lemmas from [CK04].
Lemma 4.2.
For a graph G = ( V, E ) with | V | = n , if ω ( G ) ≤ B , then α ( G ) ≥ n B . Lemma 4.3.
For a graph G = ( V, E ) with | V | = n , if ω ( G ) ≤ B , then χ ( G ) ≤ O ( n − B log n ) . We can use the above lemmas to prove that if the chromatic number of a graph is largeenough, then in any assignment of k colors to the vertices of the graph, there is a largemonochromatic clique. Lemma 4.4.
For every constant (cid:15) > , if a graph G = ( V, E ) with | V | = n satisfies χ ( G ) ≥ k n (log n ) α for some integer k and < α < , then in any assignment of k colors to V , there is a monochromatic clique of size B = Ω (cid:0) (log n ) − α − (cid:15) (cid:1) .Proof. Suppose for contradiction that there is an assignment of k colors V without amonochromatic clique of size B . Using Lemma 4.3, the subgraphs corresponding to eachof the k color classes has chromatic number at most O ( n − B log n ) = n Ω((log n ) α + (cid:15) ) log n < n (log n ) α colors. Thus, the whole graph has chromatic number at most k n (log n ) α colors, a contradic-tion.Khot [Kho01] proved that assuming NP (cid:42) ZPTIME (cid:16) n (log n ) O (1) (cid:17) , the chromatic numberof graphs is hard to approximate within a factor of n (log n )1 − γ for an absolute constant γ > .More formally, he proved the following: Theorem 4.5. ( [Kho01]) There exists a constant γ > , a function k = k ( n ) , and arandomized reduction that takes as input a -SAT instance I on n variables and outputs agraph G = ( V, E ) with | V | = N = 2 log n O (1) such that1. (Completeness) If I is satisfiable, χ ( G ) ≤ k .2. (Soundness) If I is not satisfiable, with probability at least , χ ( G ) > k N (log N )1 − γ . uthermore, the reduction runs in time poly ( N ) = 2 (log n ) O (1) . We observe that Khot’s chromatic number hardness immediately gives (log n ) Ω(1) hard-ness of Monochromatic Clique.
Lemma 4.6.
There exists a constant γ > , a function k = k ( n ) such that the followingholds. Assuming NP (cid:42) ZPTIME (cid:16) n (log n ) O (1) (cid:17) , given a graph G = ([ n ] , E ) , there is no n (log n ) O (1) time algorithm for Monochromatic-Clique ( k, B ) when B = Ω ((log n ) γ ) .Proof. Using Khot’s reduction, we get that there exists an absolute constant γ > suchthat assuming NP (cid:42) ZPTIME (cid:16) n (log n ) O (1) (cid:17) , given a graph G = ([ n ] , E ) and a parameter k ( n ) , there is no n (log n ) O (1) time algorithm to distinguish between the following:1. (Completeness) χ ( G ) ≤ k .2. (Soundness) χ ( G ) > k n (log n )1 − γ .Using Lemma 4.4, the Soundness condition implies that in any assignment of k colors to G , there is a monochromatic clique of size Ω ((log n ) γ − (cid:15) ) , for any constant (cid:15) > . Thus,given a graph G and a parameter k , assuming NP (cid:42) ZPTIME (cid:16) n (log n ) O (1) (cid:17) , there is no n (log n ) O (1) time algorithm to distinguish between the following:1. (Completeness) χ ( G ) ≤ k .2. (Soundness) In any assignment of k colors to the vertices of G , there is a monochro-matic clique of size Ω((log n ) γ (cid:48) ) .for any constant γ (cid:48) < γ . We cannot directly amplify the hardness of the Monochromatic-Clique problem by takinggraph products as we cannot preserve the chromatic number and also amplify the largestclique in an assignment of k colors at the same time. We get around this issue by defininga harder variant of Monochromatic Clique called Strong Monochromatic Clique and thenamplifying it. Definition 4.7. ( Strong Monochromatic-Clique ( k, B, C ) ) In the Strong Monochromatic-Clique ( k, B, C ) , given a graph G and parameters k ( n ) , B ( n ) , C , the goal is to distinguishbetween the following two cases:1. (YES case) The chromatic number of G is at most k .2. (NO case) In any assignment of k C colors to the vertices of G , there is a monochro-matic clique of size B . We now observe that the chromatic number hardness of Khot [Kho01] implies the samehardness as Lemma 4.6 for Strong Monochromatic Clique as well.
Lemma 4.8.
There exists a constant γ > and a function k = k ( n ) such that for everyconstant C ≥ , the following holds. Assuming NP (cid:42) ZPTIME (cid:16) n (log n ) O (1) (cid:17) , there is no n (log n ) O (1) time algorithm for Strong Monochromatic-Clique ( k, B, C ) when B = Ω((log n ) γ ) . roof. Note that the function k in Theorem 4.5 satisfies k = o (cid:16) (log N ) − γ (cid:17) . Thus, wecan replace the soundness condition in Theorem 4.5 with χ ( G ) ≥ k C N C (log N )1 − γ . UsingLemma 4.4, this implies that in any assignment of k C colors to the vertices of G , there isa monochromatic clique of size Ω((log N ) γ − (cid:15) ) , where (cid:15) > is an absolute constant. Thehardness of Strong Monochromatic Clique then follows along the same lines as Lemma 4.6.We amplify the hardness of Strong Monochromatic-Clique ( k, B, C ) to Monochromatic-Clique ( k C , B C ) using the lexicographic product of graphs. First, we define lexicographicproduct and prove some properties of it. Definition 4.9. (Lexicographic product of graphs) Given two graphs G and H , the Lexico-graphic graph product G · H has vertex set V ( G ) × V ( H ) , and two vertices ( u , v ) , ( u , v ) are adjacent if either ( u , u ) ∈ E ( G ) or u = u and ( v , v ) ∈ E ( H ) . The lexicographic product can be visualized as replacing each vertex of G with a copyof H and forming complete bipartite graphs between copies of vertices adjacent in G . Forease of notation, we let G = G · G . More generally, for an integer n that is a power of ,we define G n as taking the above lexicographic product of G with itself recursively log n times. Lemma 4.10.
Let n ≥ be a power of . If χ ( G ) ≤ k , then χ ( G n ) ≤ k n .Proof. We prove that χ ( G ) ≤ k , and the statement follows by induction on n . If f : G → [ k ] is a proper k -coloring of G , then the coloring f (cid:48) ( u, v ) = ( f ( u ) , f ( v )) is a proper k -coloring of G × G . Lemma 4.11.
Let n ≥ be a power of . Suppose that in any assignment of k colors tothe vertices of G , there is a monochromatic clique of size B . Then, in any assignment of k colors to the vertices of G n , there is a monochromatic clique of size B n .Proof. We prove the statement for n = 2 and the lemma follows by induction on n . Let f : V ( G ) → [ k ] be a given assignment. For a vertex v ∈ G , consider the assignment g v : V ( G ) → [ k ] defined as g v ( u ) = f ( v, u ) . As every assignment of k colors to thevertices of G has a monochromatic clique of size B , there is a color α ( v ) ∈ [ k ] and a clique S ( v ) ⊆ V ( G ) with | S ( v ) | ≥ B such that g v ( u ) = α ( v ) for all u ∈ S ( v ) , or in other words, f ( v, u ) = α ( v ) for all u ∈ S ( v ) . Note that such a set S ( v ) and α ( v ) exist for v ∈ V ( G ) .The function α : V ( G ) → [ k ] can also be visualized as an assignment of k colors to thevertices of G , and thus there is a monochromatic clique T of size at least B with respectto this assignment. The set { S ( v ) : v ∈ T } is a monochromatic clique of size B with respect to f in G .By using the lexicographic product, we can get a polynomial time reduction from StrongMonochromatic Clique to Monochromatic Clique. Lemma 4.12.
For every constant C ≥ that is a power of , there exists a polynomial timereduction from Strong Monochromatic-Clique ( k, B, C ) to Monochromatic-Clique ( k C , B C ) .Proof. Given a graph G as an instance of Strong Monochromatic-Clique ( k, B, C ) , we com-pute the graph G (cid:48) = G C . We claim that solving Monochromatic-Clique ( k C , B C ) on G (cid:48) solves the original Strong Monochromatic Clique problem.1. (Completeness.) Suppose that χ ( G ) ≤ k . Then, by Lemma 4.10, χ ( G (cid:48) ) ≤ k C .20. (Soundness.) Suppose that in any assignment of k C colors to the vertices of G , thereis a monochromatic clique of size B . Then, by Lemma 4.11, in any assignment of k C colors to the vertices of G (cid:48) , there is a monochromatic clique of size B C .Putting everything together, we obtain the following hardness of Monochromatic Clique. Theorem 4.13.
For every constant
C > , there exists a function k = k ( n ) such thatthe following holds. Assuming NP (cid:42) ZPTIME (cid:16) n (log n ) O (1) (cid:17) , there is no n (log n ) O (1) timealgorithm for Monochromatic-Clique ( k, B ) when B = Ω (cid:0) (log n ) C (cid:1) .Proof. The proof follows directly by combining Lemma 4.8 and Lemma 4.12.
We now prove Theorem 1.2 using the above hardness of Monochromatic Clique.
Proof of Theorem 1.2.
The reduction from
Monochromatic-Clique ( k, B ) to Vector Schedul-ing is (implicitly) proved in [CK04]. We present it here for the sake of completeness. Givena graph G = ( V = [ n ] , E ) , parameters k and B , we order all the B -sized cliques of G as T , T , . . . , T d with d ≤ n B . We define a set of n vectors v , v , . . . , v n of dimension d with ( v i ) j = (cid:40) if i ∈ T j otherwise.The instance of the Vector Scheduling has these n vectors as the input and the number ofmachines is equal to k .We analyze the reduction.1. (Completeness.) Suppose that there exists a proper k -coloring of G , c : V → [ k ] . Weassign the vector v i to the machine c ( i ) . For every j ∈ [ d ] , all the B vectors thathave in the j th dimension are assigned to distinct machines. Thus, the makespanof the scheduling is at most .2. (Soundness.) Suppose that in any assignment of k colors to the vertices of G , thereis a monochromatic clique of size B . In this case, the makespan of the scheduling isat least B .We set B = (log n ) C for a large constant C to be set later. We choose k from Theorem 4.13such that assuming NP (cid:42) ZPTIME (cid:16) n (log n ) O (1) (cid:17) , there is no n (log n ) O (1) time algorithm for Monochromatic-Clique ( k, B ) . By the above reduction, we can conclude that there is no poly-nomial time algorithm that approximates the resulting Vector Scheduling instances withina factor of B = (log n ) C . As d ≤ n B , we get that log d ≤ (log n ) C +1 , and B ≥ (log d ) − C +1 .Setting C = (cid:15) − , we get that d -dimensional Vector Scheduling has no polynomial time Ω((log d ) − (cid:15) ) approximation algorithm assuming NP (cid:42) ZPTIME (cid:16) n (log n ) O (1) (cid:17) , for everyconstant (cid:15) > . Remark 4.14.
In [IKKP19], Im, Kell, Kulkarni, and Panigrahi also study the (cid:96) r -normminimization of Vector Scheduling where the objective is to minimize max k ∈ [ d ] (cid:32) m (cid:88) i =1 ( L ki ) r (cid:33) r here L ki denotes the load on the machine i on the k th dimension. They gave an al-gorithm with an approximation ratio O (cid:18)(cid:16) log d log log d (cid:17) − r (cid:19) . Our reduction from Monochro-matic Clique gives almost optimal hardness for this variant as well: we get the hardness of Ω (cid:16) (log d ) − r − (cid:15) (cid:17) assuming NP (cid:42) ZPTIME (cid:16) n (log n ) O (1) (cid:17) , for every constant (cid:15) > . Observe that the resulting Vector Scheduling instances in the above reduction satisfy astronger property: the vectors are from { , } d . In the setting where the vectors arefrom { , } d , the Vector Scheduling problem is closely related to the Balanced HypergraphColoring problem. In this problem, given a hypergraph H and an integer k , the objective isto assign k colors to the vertices of H minimizing the maximum number of monochromaticvertices in an edge. More formally, we study the following decision version of the problem. Definition 4.15. (Balanced Hypergraph Coloring.) In the Balanced Hypergraph Coloringproblem, given a s -uniform hypergraph H and parameters k and c < s , the objective is todistinguish between the following:1. There is an assignment of k colors to the vertices of H such that in every edge, eachcolor appears at most c times.2. The hypergraph H has no proper coloring with k colors i.e., in any assignment of k colors to the vertices of H , there is an edge all of whose s vertices are assigned thesame color. We give a simple reduction from Balanced Hypergraph Coloring to Vector Scheduling.
Lemma 4.16.
Given a s -uniform hypergraph H = ( V (cid:48) = [ n (cid:48) ] , E (cid:48) ) and parameters k, c ,there is a polynomial time reduction that outputs a Vector Scheduling instance I over n (cid:48) vectors v , v , . . . , v n (cid:48) ∈ { , } d on m (cid:48) machines with m (cid:48) = k, d = | E (cid:48) | such that1. (Completeness.) If there is an assignment of k colors to the vertices of H such thateach color appears at most c times in every edge, then there is a scheduling of I withmakespan at most c .2. (Soundness.) If H has no proper coloring with k colors, then in any scheduling of I ,the makespan is at least s .Proof. Let d = | E (cid:48) | . Order the edges of the hypergraph H as e , e , . . . , e d . We define theset of vectors v , v , . . . , v n (cid:48) ∈ { , } d as follows: ( v i ) j = (cid:40) if i ∈ e j otherwise.We set the number of machines m (cid:48) to be equal to the number of colors k . There is a naturalcorrespondence between the assignment of k -colors to the vertices of H f : V (cid:48) → [ k ] , andthe scheduling where we assign the vector v i to the machine f ( i ) . We now analyze ourreduction.1. (Completeness.) If there exists an assignment of k colors f : V (cid:48) → [ k ] where eachcolor appears at most c times in each edge, we assign the vector v i , i ∈ [ n (cid:48) ] to themachine f ( i ) . In any dimension j ∈ [ d ] , at most c vectors v i with ( v i ) j = 1 arescheduled on any machine. Thus, in any machine, the total load in each dimensionis at most c . 22. (Soundness.) If there exists a vector scheduling f : [ n ] → [ m (cid:48) ] with makespan strictlysmaller than s , assign the color f ( i ) to the i th vertex of the hypergraph. In any edgeof the hypergraph, each color appears fewer than s times as the makespan is smallerthan s . Thus, f : V (cid:48) → [ k ] is a proper k -coloring of the hypergraph H .We prove the hardness results for Vector Scheduling, namely Theorem 1.3 and Theo-rem 1.4 by combining this reduction with the hardness of Balanced Hypergraph Coloring.Note that the dimension of the resulting instances in the above reduction is equal to m , thenumber of edges in the hypergraph H , and the ratio of the makespans in the completenessand soundness is equal to sc . Thus, our goal is to prove the hardness of the Balanced Hy-pergraph Coloring problem where sc is as large as possible, as a function of m , the numberof edges in the underlying hypergraph.Towards this, we first give a reduction from the Label Cover problem to the BalancedHypergraph Coloring problem. Lemma 4.17.
Fix an odd prime number k ≥ and let (cid:15) = k . Given a Label Coverinstance G = ( V = L ∪ R, E, Σ L , Σ R , Π) , there is a polynomial time reduction that outputsa k uniform hypergraph H = ( V (cid:48) , E (cid:48) ) with | V (cid:48) | ≤ | L | k | Σ L | such that1. (Completeness) If G is satisfiable, there is an assignment of k colors to the verticesof H such that in every edge, each color occurs at most k times.2. (Soundness) If no labeling to G can satisfy an (cid:15) fraction of the constraints, then H has no proper k -coloring, that is, in any assignment of k colors to the vertices of H ,there is an edge all of whose vertices are assigned the same color.Furthermore, | E (cid:48) | is at most | R | ∆ k k | Σ L | k where ∆ is the maximum degree of a vertex v ∈ R . We defer the proof of Lemma 4.17 to Section 4.4.Using Lemma 4.17, we can prove the hardness of Balanced Hypergraph Coloring viaLabel Cover hardness results. We obtain two different hardness results for the BalancedHypergraph Coloring problem, one under NP (cid:42) DTIME (cid:0) n O (log log n ) (cid:1) and another NP-hardness result, by using two different hardness results for the Label Cover problem. Thesetwo hardness results prove Theorem 1.4 and Theorem 1.3 respectively, using Lemma 4.16.First, using the standard Label Cover hardness obtained using PCP Theorem [ALM + Theorem 4.18.
Assuming NP (cid:42) DTIME (cid:0) n O (log log n ) (cid:1) , there is no polynomial time al-gorithm for the following problem. Given a k -uniform hypergraph H = ( V (cid:48) , E (cid:48) ) with m = | E (cid:48) | and k = (log m ) Ω(1) , distinguish between the following:1. There is an assignment of k colors to the vertices of H such that in any edge of thehypergraph, each color appears at most k times.2. The hypergraph H has no proper k coloring.Proof. By setting (cid:15) = k in Theorem 2.7, we have a reduction from the -SAT problem on n variables to the Label Cover problem G = ( V = L ∪ R, E, Σ L , Σ R , Π) with soundness (cid:15) and | V | ≤ n O (log k ) , | Σ L | ≤ k O (1) and ∆ ≤ k O (1) . Using Lemma 4.17, we can reducethis Label Cover instance to a Balanced Hypergraph Coloring instance H = ( V (cid:48) , E (cid:48) ) with | V (cid:48) | ≤ n O (log k ) k O (1) and | E (cid:48) | ≤ n O (log k ) k O (1) . We set k = (log n ) Ω(1) such that | V (cid:48) | = n O (log log n ) and | E (cid:48) | = n O (log log n ) to obtain the required hardness of Balanced HypergraphColoring. 23he proof of Theorem 1.4 follows immediately from Theorem 4.18 and Lemma 4.16.Next, using the hardness of near linear sized Label Cover due to Moshkovitz andRaz [MR10], we obtain the following NP-hardness of Balanced Hypergraph Coloring. Theorem 4.19.
For any constant C ≥ , given a k uniform hypergraph H = ( V (cid:48) , E (cid:48) ) with m = | E (cid:48) | and k = (log log m ) C , it is NP-hard to distinguish between the following:1. There is an assignment of k colors to the vertices of H such that in any edge of thehypergraph, each color appears at most k times.2. The hypergraph H has no proper k coloring.Proof. By setting (cid:15) = k in Theorem 2.8, we can reduce a -SAT instance over n variablesto a Label Cover instance G = ( V = L ∪ R, E, Σ L , Σ R , Π) with soundness (cid:15) and | V | ≤ n o (1) k O (1) , | Σ L | ≤ k O (1) , ∆ = k O (1) . By using Lemma 4.17, we can reduce the LabelCover instance to a Balanced Hypergraph Coloring instance H = ( V (cid:48) , E (cid:48) ) with | V (cid:48) | ≤ n o (1) kO (1) and | E (cid:48) | at most n o (1) kO (1) . We set k = (log log n ) Ω(1) to obtain | V (cid:48) | = O ( n ) , | E (cid:48) | = O ( n ) .Dinur and Steurer [DS14] gave an improvement to [MR10]–in the new Label Coverhardness, the alphabet size | Σ L | can be taken to be (cid:15) ) γ for every constant γ > . Usingthis improved Label Cover hardness, we can set k = (log log n ) C for any constant C ≥ in the hardness of Balanced Hypergraph Coloring.The proof of Theorem 1.3 follows immediately from Theorem 4.19 and Lemma 4.16.Finally, we remark that if the structured graph version of the Projection Games Con-jecture [Mos15] holds, Lemma 4.17 and Lemma 4.16 together prove that d -dimensionalVector Scheduling is NP-hard to approximate within a factor of (log d ) Ω(1) . We follow the standard Label Cover-Long Code framework–see e.g., [ABP20].
Reduction.
For ease of notation, let n = | Σ L | . For every node v ∈ L of the Label Coverinstance, we have a set of k n vertices denoted by f v = { v } × [ k ] n . The vertex set of thehypergraph is V (cid:48) = (cid:83) v ∈ L f v .For every u ∈ R , and k distinct neighbors of u , v , v , . . . , v k ∈ L with projectionconstraints π i : [Σ L ] → [Σ R ] , i ∈ [ k ] , consider the set of k vectors x i,j for i ∈ [ k ] , j ∈ [ k ] which satisfy the following: For every β ∈ Σ R , and for all α , α , . . . , α k ∈ Σ L such that π i ( α i ) = β for all i ∈ [ k ] , we have (cid:12)(cid:12) { ( i, j ) | x i,jα i = p } (cid:12)(cid:12) ≤ k ∀ p ∈ [ k ] (1)For every such set of k vectors, we add the edge { ( v i , x i,j ) : 1 ≤ i, j ≤ k } to E (cid:48) . We canobserve that | V (cid:48) | ≤ | L | k | Σ L | and | E (cid:48) | ≤ | R | (cid:18) ∆ k (cid:19)(cid:18) k | Σ L | k (cid:19) k ≤ | R | ∆ k k | Σ L | k . Completeness.
Suppose that there exists an assignment σ : V → Σ that satisfies all theconstraints of the Label Cover instance G . We color the set of vertices f v in the long codecorresponding to the vertex v ∈ L with the dictator function on the coordinate σ ( v ) i.e.for every x ∈ f v , we assign the color c ( { v, x } ) = x σ ( v )
24e can observe that this coloring satisfies the property that in every edge e ∈ E (cid:48) , eachcolor appears at most k times. Soundness.
Suppose that there is a proper k -coloring c : V (cid:48) → [ k ] of the hypergraph H i.e. in every edge e = { v , v , . . . , v k } , we have |{ c ( v ) , c ( v ) , . . . , c ( v k ) }| > Our goal is to prove that there is a labeling to the Label Cover instance that satisfies atleast (cid:15) = k fraction of constraints.We need the following lemma proved by Austrin, Bhangale, Potukuchi [ABP20] usinga generalization of Borsuk-Ulam theorem. Lemma 4.20. (Theorem . of [ABP20]) For every odd prime k and n ≥ k , in any k -coloring of [ k ] n , c : [ k ] n → [ k ] , there is a set of k vectors x , x , . . . , x k that are all assignedthe same color such that { x i , x i , . . . , x ki } = [ k ] for at least n − k distinct coordinates i ∈ [ n ] . Using this lemma, for every v ∈ L , we can identify a set of vectors x v, , x v, , . . . , x v,k ∈ f v such that all these vectors have the same color i.e. c ( { v, x v,i } ) = c (cid:48) ( v ) for all v ∈ L, i ∈ [ k ] for some function c (cid:48) : L → [ k ] . Furthermore, there are a set of coordinates S ( v ) ⊆ [ n ] with | S ( v ) | ≤ k such that { x v, i , x v, i , . . . , x v,ki } = [ k ] for every i ∈ [ n ] \ S ( v ) .For a set S ⊆ Σ L and a function π : Σ L → Σ R , we use π ( S ) to denote the set { π ( i ) : i ∈ S } . We now prove a key lemma that helps in the decoding procedure. Lemma 4.21.
Let u ∈ R be a node on the right side of the Label Cover instance. Thereare a set of labels S ( u ) ⊆ Σ R such that | S ( u ) | ≤ k , and for every v ∈ L that is a neighborof u with projection constraint π : Σ L → Σ R , we have S ( u ) ∩ π ( S ( v )) (cid:54) = φ .Proof. Fix a node u ∈ R on the right side of the Label Cover instance. Let v , v , . . . , v l ∈ L be the neighbors of u in the Label Cover instance corresponding to the projection con-straints π , π , . . . , π l respectively. As | S ( v i ) | ≤ k for all i ∈ [ l ] , and the constraints π i are projections, we have | π i ( S ( v i )) | ≤ k for all i ∈ [ l ] . Among these l subsets π i ( S ( v i )) of Σ R , let the maximum number of pairwise disjoint subsets be denoted by l (cid:48) . Without lossof generality, we can assume that S = { π i ( S ( v i )) : i ∈ [ l (cid:48) ] } is a pairwise disjoint family ofsubsets.We define the set S ( u ) as follows: S ( u ) = (cid:91) i ∈ [ l (cid:48) ] π i ( S ( v i )) As S is a family of maximum pairwise disjoint subsets, we have S ( u ) ∩ π i ( S ( v i )) (cid:54) = φ forall i ∈ [ l ] . Our goal is to bound the size of S ( u ) , which we achieve by bounding l (cid:48) .We claim that l (cid:48) ≤ k ( k − . Suppose for contradiction that l (cid:48) > k ( k − . This impliesthat there are l (cid:48) > k ( k − nodes v , v , . . . , v l (cid:48) all adjacent to u such that π i ( S ( v i )) , i ∈ [ l (cid:48) ] are all pairwise disjoint. Thus, there exists a color (cid:96) ∈ [ k ] and a set of k nodes w , w , . . . , w k adjacent to u corresponding to the projection constraints π (cid:48) , π (cid:48) , . . . , π (cid:48) k such that c (cid:48) ( w i ) = (cid:96) for all i ∈ [ k ] , and the k sets π (cid:48) i ( S ( w i )) are pairwise disjoint.Using this, we can construct a set of vectors x i,j , ≤ i, j ≤ k defined as x i,j = x w i ,j which satisfy the following properties: 25. All these vectors are colored the same: c ( { w i , x i,j } ) = (cid:96) ∀ ≤ i, j ≤ k
2. For every i ∈ [ k ] , { x i, i (cid:48) , x i, i (cid:48) , . . . , x i,ki (cid:48) } = [ k ] for every i (cid:48) ∈ [ n ] \ S ( w i ) .We claim that these set of vectors satisfy the condition in Equation (1). Fix a β ∈ Σ R , and α , α , . . . , α k ∈ Σ L such that π (cid:48) i ( α i ) = β for all i ∈ [ k ] . As the family of subsets π (cid:48) i ( S ( w i )) is a pairwise disjoint family, we can infer that there exists at most one i ∈ [ k ] such that α i ∈ S ( w i ) . Note that if α i / ∈ S ( w i ) , then { x i,jα i : j ∈ [ k ] } = [ k ] . Thus, we have (cid:12)(cid:12) { ( i, j ) | x i,jα i = p } (cid:12)(cid:12) ≤ k ∀ p ∈ [ k ] . Thus, the set of vectors { ( w i , x i,j ) : 1 ≤ i, j ≤ k } is indeed an edge of E (cid:48) . As all thesevectors are colored the same color (cid:96) , we have arrived at a contradiction to the fact that c is a proper k -coloring of H .Hence, we can conclude that l (cid:48) ≤ k ( k − , and thus, | S ( u ) | ≤ k ( k − k < k .Now, consider the labeling σ : L → Σ L , where σ ( v ) , v ∈ L is chosen uniformly atrandom from S ( v ) . Similarly, let σ : R → Σ R is chosen uniformly at random from S ( u ) , u ∈ R . Using Lemma 4.21, we can infer that for every edge e = ( v, u ) in the Label Cover,this labeling satisfies the edge e with probability at least | S ( v ) || S ( u ) | ≥ k . By linearityof expectation, this labeling satisfies at least k fraction of the constraints in expectation.Hence, with positive probability, the labeling satisfies at least k fraction of the constraints.This concludes the proof of soundness that if H has a proper k coloring, then there existsa labeling to G that satisfies at least k fraction of the constraints. As is the case with the Vector Scheduling problem, the hard instances for Vector BinCovering are when the vectors are from { , } d . In this setting, the Vector Bin Coveringproblem is closely related to the hypergraph rainbow coloring problem. A hypergraph H = ( V, E ) is said to be k -rainbow colorable if there is an assignment of k colors to thevertices of H such that in every edge, all the k colors appear. When k = 2 , it is equivalentto the standard -coloring of hypergraphs. Unlike the usual (hyper)graph coloring, rainbowcoloring gets harder with larger number of colors.In the approximate rainbow coloring problem, given a hypergraph that is promised tohave a rainbow coloring with a large number of colors, the goal is to find a coloring inpolynomial time using fewer number of colors. More formally, the computational problemwe study is the following. Definition 5.1. (Approximate Rainbow Coloring) In the approximate rainbow coloringproblem, the input is a hypergraph H = ( V, E ) and a parameter k ≥ . The objective is todistinguish between the following:1. The hypergraph H is k -rainbow colorable. . The hypergraph H has no valid -coloring. We now give a simple reduction from approximate rainbow coloring to Vector BinCovering.
Lemma 5.2.
Given a hypergraph H = ( V, E ) and a parameter k , there is a polynomialtime reduction that outputs a Vector Bin Covering instance v , v , . . . , v n ∈ { , } d with n = | V | , d = | E | such that1. (Completeness.) If H is k -rainbow colorable, there is a partition of [ n ] into k parts A , A , . . . , A k such that (cid:88) j ∈ A i v j ≥ d ∀ i ∈ [ k ]
2. (Soundness.) If H is not -colorable, there is no partition of [ n ] into A , A suchthat (cid:88) j ∈ A i v j ≥ d ∀ i ∈ [2] Proof.
Let n = | V | , d = | E | . We order the edges E as E = { e , e , . . . , e d } . We output aset of vectors V = { v , v , . . . , v n } where each v i ∈ { , } d is defined as follows: ( v i ) j = (cid:40) if i ∈ e j otherwise.We analyze this reduction:1. (Completeness.) Suppose that the hypergraph H has a rainbow coloring with k colors f : V → [ k ] . We partition [ n ] into k parts A , A , . . . , A k such that A i = { j ∈ [ n ] : f ( j ) = i } Consider an arbitrary integer i ∈ [ k ] . Note that for every edge e in H , e ∩ A i (cid:54) = φ .Thus, (cid:88) j ∈ A i v j ≥ d
2. (Soundness.) Suppose that the hypergraph H has no proper coloring with colors.Then, we claim that there is no partition of [ n ] into two parts A , A such that (cid:88) j ∈ A i v j ≥ d ∀ i ∈ [2] Suppose for contradiction that there exists A , A with the above property. Considerthe coloring of the hypergraph f : V → [2] as f ( v ) = (cid:40) if v ∈ A if v ∈ A Consider an arbitrary edge e l , l ∈ [ d ] of the hypergraph H . As (cid:80) j ∈ A i ( v j ) l ≥ forall i ∈ [2] , there exist v , v ∈ e l such that v ∈ A , v ∈ A . Thus, the coloring f isa proper coloring of the hypergraph H , a contradiction.27e combine this reduction with the hardness of approximate rainbow coloring to provethe hardness of Vector Bin Covering, namely Theorem 1.5. Note that the dimension of theresulting vectors in the Vector Bin Covering instance V is equal to the number of edges m = | E | of the hypergraph H , and the gap in the optimal Bin Covering value of V is equalto k , the number of colors. Hence, to obtain better inapproximability results for VectorBin Covering that grow with d , our goal is to show the hardness of approximate rainbowcoloring on hypergraphs with m edges where the number of colors k is as large a functionof m as possible. Towards this, we prove that it is NP-hard to -color a hypergraph with m edges that is promised to be rainbow colorable with k = Ω (cid:16) log m log log m (cid:17) colors. Theorem 5.3.
Given a hypergraph H with m edges, it is NP-hard to distinguish betweenthe following:1. (Completeness) H is k -rainbow colorable.2. (Soundness) H is not -colorable.where k = Ω (cid:16) log m log log m (cid:17) . We defer the proof of Theorem 5.3 to Section 5.2.We now prove the hardness of Vector Bin Covering using Theorem 5.3.
Proof of Theorem 1.5.
Using Theorem 5.3 combined with the reduction in Lemma 5.2, weget that the following problem is NP-hard. Given a set of n vectors v , v , . . . , v n ∈ { , } d ,distinguish between1. V can be partitioned into k = Ω (cid:16) log d log log d (cid:17) parts such that in each part, the sum ofvectors is at least in every coordinate.2. V cannot be partitioned into parts such that in each part, the sum of vectors isat least in every coordinate. In other words, the maximum number of parts intowhich V can be partitioned such that in each part, the sum of vectors is at least inevery coordinate is equal to .Thus, it is NP-hard to approximate d -dimensional Vector Bin Covering within k = Ω (cid:16) log d log log d (cid:17) . Our proof follows by viewing the hypergraph rainbow coloring problem as a promiseconstraint satisfaction problem(PCSP) and analyzing its polymorphisms [AGH17, BG16,BKO19]. The idea is to prove that the polymorphisms have a small number of “important”coordinates which can then be decoded in the Label Cover-Long Code framework. For thecase of the above rainbow coloring PCSP, we prove that the polymorphisms are -fixingin that there is a single coordinate which when set to a certain value fixes the value ofthe function. This characterization then implies the hardness of the approximate rainbowcoloring.For ease of readability, we skip defining polymorphisms formally, and instead presentthe proof as a simple gadget reduction from Label Cover. We first need a definition. Definition 5.4. ( -fixing [BG16, GS20]) A function f : [ k ] n → { , } is said to be -fixingif there exists an index (cid:96) ∈ [ n ] and values α, β ∈ [ k ] such that f ( x ) = 0 ∀ x : x (cid:96) = α and f ( x ) = 1 ∀ x : x (cid:96) = β
28n the analysis of the gadget, we need a definition and a lemma from [ABP20].
Definition 5.5. (The hypergraph H nr [ k ] ) The hypergraph H nr [ k ] = ( V, E ) is a k -uniformhypergraph with vertex set as the set of n -dimensional vectors over [ k ] i.e. V = [ k ] n . A setof k vectors v , v , . . . , v k form an edge of the hypergraph if n (cid:88) i =1 (cid:12)(cid:12)(cid:12) [ k ] \ { v ji : j ∈ [ k ] } (cid:12)(cid:12)(cid:12) ≤ r Lemma 5.6.
For every k ≥ , the hypergraph H n (cid:98) k (cid:99) [ k ] is not -colorable. We analyze the gadget used in our reduction.
Lemma 5.7.
Fix k ≥ . Suppose f : [ k ] n → { , } satisfies the below two-coloring property:For every k vectors v , v , . . . , v k ∈ [ k ] n with { v ji : j ∈ [2 k ] } = [ k ] ∀ i ∈ [ n ] , we have { f ( v j ) : j ∈ [2 k ] } = { , } . Then, f is -fixing.Proof. We first prove that there exist (cid:96) ∈ [ n ] , α ∈ [ k ] , b ∈ { , } such that f ( x ) = b for all x ∈ [ k ] n with x (cid:96) = α . Suppose for contradiction that this is not the case. Then, for every i ∈ [ n ] , j ∈ [ k ] there exist vectors x i,j , y i,j ∈ [ k ] n such that x i,ji = y i,ji = j , and f ( x i,j ) = 0 where as f ( y i,j ) = 1 .Let r = (cid:98) k (cid:99) . We view f : [ k ] n → { , } as an assignment of two colors to the verticesof the hypergraph H nr [ k ] . As the hypergraph is not two colorable (Lemma 5.6), we caninfer that there is an edge of H nr [ k ] all of whose vertices are assigned the same color. Inother words, there exist k vectors v , v , . . . , v k ∈ [ k ] n and b ∈ { , } such that f ( v j ) = b for all j ∈ [ k ] . Furthermore, there are at most r missing values in these vectors i.e. n (cid:88) i =1 (cid:12)(cid:12)(cid:12) [ k ] \ { v ji : j ∈ [ k ] } (cid:12)(cid:12)(cid:12) ≤ r Now, we pick r vectors u , u , . . . , u r (with repetitions if needed) by filling the missingvalues using x i,j , y i,j vectors such that1. f ( u j ) = b for all j ∈ [ r ] .2. For every i ∈ [ n ] , { v ji : j ∈ [ k ] } ∪ { u ji : j ∈ [ r ] } = [ k ] By taking the union of { v , v , . . . , v k } and { u , u , . . . , u r } , and repeating some vectors,we obtain k vectors w , w , . . . , w k with f ( w j ) = b for all j ∈ [2 k ] , and { w ji : j ∈ [2 k ] } = [ k ] ∀ i ∈ [ n ] However, this contradicts the two-coloring property of f . Thus, there exist (cid:96) ∈ [ n ] , α ∈ [ k ] , b ∈ { , } such that f ( x ) = b for all x ∈ [ k ] n with x (cid:96) = α .We now claim that there exists β ∈ [ k ] such that f ( x ) = 1 − b for all x ∈ [ k ] n with x (cid:96) = β . Suppose for contradiction that this is not the case. Then, there exist k vectors v , v , . . . , v k such that v j(cid:96) = j for all j ∈ [ k ] , and f ( v j ) = b for all j ∈ [ k ] . We now pick v k +1 , v k +2 , . . . , v k ∈ [ k ] n such that v j(cid:96) = α for all j ∈ { k + 1 , k + 2 , . . . , k } , and v ji = j − k for all i ∈ [ n ] with i (cid:54) = (cid:96) , and j ∈ { k + 1 , k + 2 , . . . , k } . These k vectors v , v , . . . , v k satisfy 29. f ( v j ) = b for all j ∈ [2 k ] .2. For every i ∈ [ n ] , { v ji : j ∈ [2 k ] } = [ k ] contradicting the two-coloring property of f . Thus, there exists β ∈ [ k ] such that f ( x ) =1 − b for all x ∈ [ k ] n with x (cid:96) = β , completing the proof that f is -fixing.We are now ready to prove Theorem 5.3. Our hardness result is obtained using a reduc-tion from the Label Cover problem. This reduction is standard in the PCSP literature.(Seee.g., [BKO19]) Reduction.
We start with the Label Cover instance G = ( V = L ∪ R ) , E, Σ = Σ L =Σ R , Π) from Theorem 2.6 and output a hypergraph H = ( V (cid:48) , E (cid:48) ) . Let n denote the labelsize n = | Σ | . For each vertex v ∈ L ∪ R , we have a long code containing a set of nodes K v of size [ k ] n , indexed by n length vectors.1. The vertex set of the hypergraph V (cid:48) is the union of all the long code nodes. V (cid:48) = (cid:91) v ∈ V K v
2. Edges of the hypergraph: For every vertex v ∈ V of the Label Cover instance, weadd an edge in E (cid:48) for each set of k vectors { v , v , . . . , v k } in K v , if { v ji : j ∈ [2 k ] } = [ k ] ∀ i ∈ [ n ] . (2)The number of edges in H is at most | E (cid:48) | ≤ | V | (cid:18) k | Σ | k (cid:19) ≤ | V | k O ( k )
3. Equality constraints: For every constraint Π e : u → v of the Label Cover, we adda set of equality constraints between nodes x ∈ K u , y ∈ K v if for all i ∈ [ n ] , x i = y Π e ( i ) . By adding an equality constraint between two nodes, we identify thetwo nodes together and treat it as a single node. That is, we compute the connectedcomponents of the equality constraints graph and identify a single master node foreach component. We then obtain a multi-hypergraph H from H by replacing eachnode with the corresponding master node. However, a vertex could appear multipletimes in an edge in H . We delete such occurrences from H by setting each edge tobe a simple set of the vertices contained in it, and obtain the final hypergraph H .We note the following:(a) There exists a k -rainbow coloring of H , f : V (cid:48) → [ k ] that respects the equalityconstraints i.e. f ( x ) = f ( y ) for all pairs of nodes x , y with equality constraintsbetween them if and only if H is k -rainbow colorable.(b) Similarly, there exists a -coloring of H that respects equality constraints if andonly if H is -colorable.Finally, the number of edges in H is at most the number of edges in H . Completeness.
Suppose that there is a labeling σ : V → Σ that satisfies all the con-straints. We define the coloring f : V (cid:48) → [ k ] of H as follows. For every node x ∈ K v , weset the dictatorship function f ( x ) = x σ ( v )
30y the constraints added in Equation (2), the function f is a valid k -rainbow coloring of H . As σ satisfies all the constraints of the Label Cover, the coloring f satisfies all theequality constraints. Soundness.
Suppose that there is no labeling σ : V → Σ that satisfies all the constraints in G . Then we claim that there is no -coloring of H that respects all the equality constraints.Suppose for contradiction that there is a -coloring f : V (cid:48) → { , } that respects all theequality constraints.Consider a vertex v ∈ V . The function f v : [ k ] n → { , } , defined as f on K v satisfiesthe conditions in Lemma 5.7. Thus, f v is -fixing for every v ∈ V . Hence, there is afunction L : V → Σ such that for every v ∈ V , f v is -fixing on the coordinate L ( v ) . Wenow claim that the labeling σ : V → Σ defined as σ ( v ) = L ( v ) satisfies all the constraintsin G .Consider an edge e = ( u, v ) , u ∈ L, v ∈ R with the projection constraint Π e : Σ → Σ .Our goal is to show that Π e ( L ( u )) = L ( v ) . Suppose for contradiction that Π e ( L ( u )) (cid:54) = L ( v ) . By the -fixing property of f u , we have α u , β u ∈ [ k ] such that f u ( x ) = 0 ∀ x ∈ [ k ] n : x L ( u ) = α u and f u ( x ) = 1 ∀ x ∈ [ k ] n : x L ( u ) = β u Similarly, we have α v , β v ∈ [ k ] such that f v ( y ) = 0 ∀ y ∈ [ k ] n : y L ( v ) = α v and f v ( y ) = 1 ∀ y ∈ [ k ] n : y L ( v ) = β v By the equality constraints, f u ( x ) = f v ( y ) for all x , y ∈ [ k ] n such that x i = y Π e ( i ) ∀ i ∈ [ n ] .Let y (cid:48) ∈ [ k ] n be an arbitrary vector with y (cid:48) Π e ( L ( u )) = α u , y (cid:48) L ( v ) = β v . We choose x (cid:48) ∈ [ k ] n such that for all i ∈ [ n ] , x (cid:48) i = y (cid:48) Π e ( i ) . Note that x (cid:48) L ( u ) = α u . Thus, f u ( x (cid:48) ) = 0 where as f v ( y (cid:48) ) = 1 . However, this contradicts the equality constraints. We conclude by mentioning a few open problems.1. A drawback of our hardness result for Vector Bin Packing is that our lower bound isonly applicable when d is large enough. In particular, for the -dimensional VectorBin Packing, we still do not have an explicit constant hardness. The reason our hard-ness result needs d to be large enough is that the k -set cover hardness itself [Tre01]needs k to be large enough. By starting our reduction with alternate set cover hard-ness results such as the -dimensional matching problem, we can obtain improvedhardness for smaller values of d . However, this approach will still not help for d = 2 as the upper bound on the packing dimension of these set families is greater than .It is an interesting open problem to characterize set families with packing dimension . We believe that such a characterization could help in proving the hardness of ap-proximation results for the set cover problem on set families with packing dimension , which directly gives the hardness of -dimensional Vector Bin Packing.2. d -dimensional Geometric Bin Packing is another open problem where packing dimen-sion based ideas could help. The best hardness result for the d -dimensional GeometricBin Packing is still the (tiny) hardness factor from the -dimensional setting. A pos-sible avenue to obtain improved inapproximability for this problem is by a reductionfrom a suitable set cover variant using a notion of packing dimension. However, thisis easier said than done as the geometric packings are significantly harder to tame–forexample, it is NP-hard to decide if a given set of n rectangles can fit in a unit square,while the corresponding problem for Vector Bin Packing is trivial.31. Another interesting open problem is to close the gap between O (log d ) algorithmand Ω (cid:16) log d log log d (cid:17) hardness for Vector Bin Covering. For the rainbow coloring, thequestion is: Given a hypergraph H with m edges that is promised to be f ( m ) -rainbowcolorable, can we -color it in polynomial time? The answer to this question when f ( m ) is equal to O (log m ) is yes, by a simple random -coloring. By Theorem 5.3,the answer is no when f ( m ) = Ω (cid:16) log m log log m (cid:17) . Which of these is tight? Bhattiprolu,Guruswami, and Lee [BGL15] proved that in certain settings, the simple randomcoloring is optimal for rainbow coloring. This suggests that perhaps even here, theproblem is hard when f ( m ) = o (log m ) . On the other hand, obtaining any hardnessresult beyond Ω (cid:16) log m log log m (cid:17) seems to be impossible with the Label Cover-Long Codeframework. Acknowledgements
I am greatly indebted to Venkatesan Guruswami for helpful discussions, for his detailedfeedback on the manuscript which significantly improved the presentation, and for hisencouragement. I also thank Varun Gupta, Ravishankar Krishnaswamy, and Janani Sun-daresan for helpful discussions. I am especially grateful to Ravishankar Krishnaswamy forpointing me to [KAR00].
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A Hardness of simple k -set cover The hardness result of Kumar, Arya, and Ramesh [KAR00] is obtained from the LabelCover problem using a partition gadget along the lines of the reduction of Lund andYannakakis [LY94]. The set families in the reduction in [LY94] have large intersections.[KAR00] get around this by using two main ideas:1. They use a different partition system wherein each partition is a disjoint union of alarge (super constant) number of sets instead of just sets in [LY94].2. They use multiple sets for each label assignment to a vertex of the Label Cover,unlike a single set corresponding to each label of each vertex in [LY94].As [KAR00] were proving a Ω(log n ) hardness of the set cover, the universe size ofthe partition system is chosen to be the same as the number of vertices in the LabelCover instance. This forces the set sizes to be very large. We can get around this issueby simply defining the partition system on a set of size B , where B is a large constant.This also has an added benefit that we no longer require sub-constant hardness from theLabel Cover instances, thus giving us NP-hardness directly. This observation is used by35revisan [Tre01] to obtain ln B − O (ln ln B ) NP-hardness of set cover on instances whereeach set has cardinality at most B , from Feige’s (1 − (cid:15) ) ln n set cover hardness [Fei98].We now describe the parameter modifications in full detail. Let B be a large constant.We start our reduction from Label Cover instances with soundness γ = β log B where β is an absolute constant to be fixed later. Theorem A.1. ( [ALM +
98, Raz98]) Given a Label Cover instance defined on a bipartitegraph G = ( V, E ) with left alphabet Σ L and right alphabet Σ R , it is NP-hard to distinguishbetween the following:1. (Completeness). There exists a labeling σ : V → Σ L ∪ Σ R that satisfies all theconstraints.2. (Soundness). No labeling to V can satisfy more than γ fraction of the constraints.Furthermore the instances satisfy the following properties:1. The alphabet sizes d = | Σ L | and d (cid:48) = | Σ R | are both upper bounded by (log B ) O (1) .2. The maximum degree deg of G is upper bounded by (log B ) O (1) . Following the convention in [KAR00], we assume that the number of vertices on theleft side in G is equal to that on the right side of G , and we denote this number by n (cid:48) .We now construct a partition system P on a universe N of size B . The system P has d (cid:48) × ( deg +1) × d partitions. Each partition has m = B − (cid:15) parts, where (cid:15) is a small constantto be fixed later. The partition system is divided into d (cid:48) groups each containing ( deg +1) × d partitions. Each group is further organized into deg + 1 subgroups each of which contains d partitions. Let P g,s,p denote the p th partition in the s th subgroup of the g th group and P g,s,p,k denote the k th set in P g,s,p where g ∈ [ d (cid:48) ] , s ∈ [ deg + 1] , p ∈ [ d ] , k ∈ [ m ] . Thepartition system satisfies the four properties in Section of [KAR00], the only differencebeing that the universe N now has size B instead of n (cid:48) . Thus, the covering property(Property in [KAR00]) now states that any covering of N with βm log B sets shouldcontain at least m sets from the same partition. Such a partition system is shown to existfor large enough B in [KAR00] using a randomized construction. They also derandomizethe construction. But for our setting, as B is a constant, we just need to show the existenceof such a partition system.We reduce the Label Cover instance in Theorem A.1 to a set cover instance SC bythe same construction as in [KAR00]: we have a partition system corresponding to eachedge of the Label Cover instance, and the union of the elements in the partition systemsis the element set of SC . The sets in SC , C k ( v, a ) are defined exactly as in [KAR00]. Thecardinality of each set is at most B (cid:48) = deg × B ≤ B . Each element is present in atmost md = O ( B ) sets. The fact that SC is a simple set system follows from Lemma 1of [KAR00]. By Lemma 2 in [KAR00], if there is a labeling of the Label Cover instance,then there is a set cover of size n (cid:48) m in SC . If there is a set cover of size β n (cid:48) m log B in SC , then there is a labeling of G that satisfies γ fraction of constraints. The proof of thissoundness follows along the same lines as Lemma of [KAR00], with the only differencebeing that we now define the good edges as edges having e ) ≤ βm log B . B SDP Relaxation of Monochromatic-Clique
We consider the following SDP relaxation of the graph coloring problem on G = ( V, E ) :Minimize k (cid:104) u i , u i (cid:105) = 1 ∀ i ∈ V (cid:104) u i , u j (cid:105) ≤ − k − ∀ ( i, j ) ∈ E χ v ( G ) ofthe graph G . It is equivalent to the Lovasz theta function of the complement of G . Wehave the following sandwich property due to [Knu94]: ω ( G ) ≤ χ v ( G ) ≤ χ ( G ) B.1 Algorithm when
B > √ n There is a simple algorithm for the
Monochromatic-Clique ( k, B ) problem when B > k : Wecompute χ v ( G ) in polynomial time, and we check if χ v ( G ) ≤ k . In this case, there is noclique of size B in G , and we output YES. If χ v ( G ) > k , then the graph cannot be coloredwith k colors, and in this case, we output NO.Note that if k ( B − ≥ n , there is always an assignment of k colors to the vertices ofthe graph without a clique of size B , thus the problem is trivial. B.2 Integrality gap
The above algorithm proves that in any graph with vector chromatic number at most k ,there is an assignment of k colors to the vertices that has monochromatic clique of size atmost √ n . We now prove that this cannot be significantly improved: Theorem B.1.
For n large enough, there exists a graph G = ( V, E ) with n vertices, anda parameter k such that1. χ v ( G ) ≤ k .2. In any assignment of k colors to the vertices of G , there is a monochromatic cliqueof size n Ω(1) .Proof.