On Hardness of Approximation of Parameterized Set Cover and Label Cover: Threshold Graphs from Error Correcting Codes
aa r X i v : . [ c s . CC ] S e p On Hardness of Approximation ofParameterized Set Cover and Label Cover:Threshold Graphs from Error Correcting Codes
Karthik C. S. ∗ Tel Aviv University [email protected]
Inbal Livni-Navon † Weizmann Institute of Science [email protected]
Abstract
In the ( k , h ) - SetCover problem, we are given a collection S of sets over a universe U , andthe goal is to distinguish between the case that S contains k sets which cover U , from thecase that at least h sets in S are needed to cover U . Lin (ICALP’19) recently showed a gapcreating reduction from the ( k , k + ) - SetCover problem on universe of size O k ( log |S| ) to the (cid:18) k , k r log |S| log log |S| · k (cid:19) - SetCover problem on universe of size |S| . In this paper, we prove a morescalable version of his result: given any error correcting code C over alphabet [ q ] , rate ρ , andrelative distance δ , we use C to create a reduction from the ( k , k + ) - SetCover problem on uni-verse U to the (cid:16) k , k q − δ (cid:17) - SetCover problem on universe of size log |S| ρ · |U | q k .Lin established his result by composing the input SetCover instance (that has no gap) witha special threshold graph constructed from extremal combinatorial object called universal sets,resulting in a final
SetCover instance with gap. Our reduction follows along the exact samelines, except that we generate the threshold graphs specified by Lin simply using the basicproperties of the error correcting code C . We further show that one can recover the precise re-sult of Lin by using a code which also achieves optimal parameters as a perfect hash function.We use the same threshold graphs mentioned above to prove inapproximability results,under W[1] = FPT and ETH, for the parameterized label cover problem called k - MaxCover in-troduced by Chalermsook et al. (FOCS’17; SICOMP’20). Our inapproximaiblity results matchthe bounds obtained by Karthik et al. (STOC’18; JACM’19), although their proof framework isvery different, and involves generalization of the ‘distributed PCP framework’. To the best ofour knowledge, prior to this work, it was not clear how to adopt the proof strategy of Lin toprove inapproximability results for k - MaxCover . ∗ This work was supported by the Israel Science Foundation (grant number 552/16) and the Len Blavatnik and theBlavatnik Family foundation. † This work was supported by Irit Dinur’s ERC-CoG grant 772839. Introduction
Many optimization problems that we care about are NP -Hard. Two typical ways to cope with NP -Hardness are to design approximation algorithms or fixed parameter algorithms. For example,consider the classic SetCover problem which was shown in the seminal work of Karp [Kar72]to be NP -Hard. Researchers coped with this hardness by designing approximation algorithms[Chv79, Sri95, Sla96] and studying its fixed parameter tractability.Nevertheless, in many cases we have that even finding a good approximate solution to NP -Hard optimization problems is still NP -Hard (and such results are proved using the celebratedPCP theorem [AS98, ALM +
98, Din07]). In similar spirit, it is also possible to show that manynatural parameterized variants of NP -Hard problems are W[1] -Hard and thus not fixed parametertractable. Case in point, it was shown that on one hand it is NP -Hard to approximate SetCover below logarithmic factors [Fei98, DS14] and on the other hand that the
SetCover problem param-eterized by the solution size is not fixed parameter tractable assuming
W[2] = FPT [DF95]. Thus,one may further try to cope with both hardness of approximation and fixed parameter intractabil-ity, simultaneously, by the design of fixed parameter approximation algorithms. In this paper, weare interested in the recently emerging theory of fixed parameter inapproximability , i.e., the subareaformed by the intersection of hardness of approximation and parameterized complexity.The results in fixed parameter inapproximability can be broadly divided into two parts. First,we have the results obtained under non-gap assumptions such as W [ ] = FPT , ETH , and
SETH [IP01, IPZ01]. The main difficulty addressed in these results is generating a gap, i.e., we focus onhow to start from a hard problem with no gap, say k - Clique , and reduce it to a problem of interestwhile generating a non-trivial gap in the process. We elaborate below on these set of results. Theother collection of results in fixed parameter inapproximability are under gap assumptions suchas the Gap Exponential Time Hypothesis [MR16, Din16] and Parameterized InapproximabilityHypothesis [LRSZ20]. In these results the gap is inherent in the assumption, and the challenge isto construct gap-preserving reductions. These results are not the focus of this paper and we shallnot elaborate further on them, and the interested reader may see the recent survey of Feldman etal. [FKLM20] for more details.There are two main techniques to generate the gap in the fixed parameter inapproximabilityliterature . Threshold Graph Composition.
The Threshold Graph Composition (
TGC ) technique was intro-duced in the breakthrough work of Lin [Lin18] to show the
W[1] -Hardness of the k - Biclique prob-lem via the inapproximability of the k - One - Sided - Biclique problem. This technique was later usedto prove the first non-trivial inapproximability result for the k - SetCover problem [CL19], and isalso the current technique used to prove the state-of-the-art inapproximability result for the same[Lin19]. Moreover, the result on the k - One - Sided - Biclique problem in Lin [Lin18] was used by Bhat-tacharyya et al. [BBE +
19] as the starting point to prove inapproximability results for problemsin coding theory such as the k - Minimum Distance problem (a.k.a. k - Even Set problem) and the k - Nearest Codeword problem, and also for lattice problems such as the k - Shortest Vector problem andthe k - Nearest Vector problem. One additional technique that we do not address in this paper is due to Wlodarczyk [Wlo20], who recently useda variant of the gap amplification via graph products technique to prove hardness of approximation for connectivityproblems, including the k - SteinerOrientation problem.
2t a very high level, in
TGC we compose an instance of the input problem that has no gap,with a threshold graph (that is constructed oblivious to the input instance; see Section 1.2 for thedefinition), to produce a gap instance of the desired problem. The main challenge here is to findthe right way to compose the input and the threshold graph, although we remark that even thetask of constructing the requisite threshold graphs is in many cases non-trivial.
Distributed PCP Framework.
The Distributed PCP Framework (
DPCPF ) was introduced in theseminal work of Abboud et al. [ARW17] and laid the foundation for a series of inapproximabilityresults in the area of fine-grained complexity [Rub18, Che18, AR18, CGL + k - SetCover problem. En route, they also provided inapproximability for k - MaxCover , a parameter-ized variant of the label cover problem which was introduced and identified by Chalermsook etal. [CCK +
20] as a key intermediate gap problem to be studied in parameterized complexity. At avery high level, in
DPCPF , given an instance of the input problem with no gap, one first designs aprotocol for a specific communication problem formulated based on the input problem, and thenextracts an instance of the gap k - MaxCover problem from the transcript of the protocol. Finally,one designs a gap preserving reduction from the gap k - MaxCover problem to the gap problem ofinterest.
Meeting Point of the Two Techniques.
While the aforementioned two techniques seems verydifferent, rather surprisingly, they both yield very similar inapproximability results for the sameproblem: k - SetCover [KLM19, Lin19]. This leads to the following natural question:
Is there a unified technique to yield all inapproximability resultsin parameterized complexity?
More concretely, one can ask if it is possible to recover using the
DPCPF all the inapproxima-bility results that are currently only obtained using the
TGC technique and vice-versa? In [KM19]the authors made the connection that if one could construct certain high dimensional extremalcombinatorial objects then it is possible to prove the inapproximability of k - One - Sided - Biclique through
DPCPF (specifically by using the result of [KLM19] on k - MaxCover ). However, the con-struction of the desired combinatorial objects seem far from reach using current techniques. Inthis paper, we look at the other direction of the question and address which results obtained in
DPCPF can now be obtained using the
TGC technique.
Towards answering the raised question we use the
TGC technique to prove the following gapcreating self reduction for the k - MaxCover problem. We start by describing the k - MaxCover problem(see Section 2.1 for a formal definition).In the k - MaxCover problem we are given a bipartite graph Γ = ( V ∪ W , E ) , where the vertexset is partitioned as follows: V = V ˙ ∪ · · · ˙ ∪ V k and W = W ˙ ∪ · · · ˙ ∪ W t . We denote by | Γ | = | V | + | W | + | E | . A labeling of V is a k -tuple of vertices ( v , . . . , v k ) ∈ V × · · · × V k , and we say that it covers W j (for some j ∈ [ t ] ) if ∃ w ∈ W j which is a joint neighbor of all of v , . . . , v k . We denote by3 axCover ( Γ ) the maximal fraction of W j that can be simultaneously covered, i.e., MaxCover ( Γ ) : = max ( v ,..., v k ) ∈ V ×···× V k (cid:18) Pr j ∼ [ t ] (cid:2) W j is covered by ( v , . . . , v k ) (cid:3)(cid:19) .It is easy to see that k - MaxCover is a parameterized variant of the classical label cover problem.Our first result is a reduction, by only using an arbitrary error correcting code, from the exact k - MaxCover problem with a certain projection property, (which we call pseduo-projection property,and is analogous to the standard projection property of label cover problem) to the gap k - MaxCover problem.
Theorem 1.1 ( MaxCover
Gap Creation using
TGC technique; Informal statement of Theorem 4.2) . Let Γ be a k- MaxCover instance with the “pseudo-projection” property. Let C be an error correcting codeover alphabet set [ q ] of block length ℓ and message length log q | Γ | . Then there exists a reduction in timeO ( | Γ | ℓ · q t ) to a k- MaxCover instance Γ of size | Γ | ℓ · q t . The new instance Γ satisfies, Completeness: If MaxCover ( Γ ) = , then MaxCover ( Γ ) = , Soundness: If MaxCover ( Γ ) < , then MaxCover ( Γ ) ≤ − ∆ ( C ) ,where ∆ ( C ) is the relative distance of C. It is clear that in the above theorem, by taking any code with constant relative distancebounded away from 0, we already obtain a k - MaxCover instance with constant gap. We elabo-rate more on the proof technique in the next subsection of the introduction, but for now discussthe context of the above result.We use the above gap creation theorem to show inapproximability results for k - MaxCover based on
ETH and W [ ] = FPT . In particular, we show in Theorem 4.3 (resp. Theorem 4.4) thatassuming
ETH (resp. W [ ] = FPT ), there is no algorithm running in time n o ( k ) (resp. F ( k ) · n O ( ) time, for some computable function F ), that can decide if a k - MaxCover instance has complete-ness 1 or soundness at most (cid:0) n (cid:1) / poly ( k ) , where n = | V | . The proofs of Theorems 4.3 and 4.4 fol-lows by first showing ETH -Hardness and
W[1] -Hardness of exact k - MaxCover having the pseudo-projection property, and then applying Theorem 1.1.
Comparison to [KLM19].
We remark that the proof technique in [KLM19] also gives us Theo-rem 1.1 with identical parameters (i.e., even using
DPCPF , we can create gap in k - MaxCover asin Theorem 1.1 using an arbitrary error correcting code). Therefore, our contribution is aboutproving the same result using
TGC technique. See Remark 4.5 for more details.Our next contribution is a more scalable version of Lin’s result on gap k - SetCover . Theorem 1.2 (Scalable version of [Lin19]; Informal statement of Theorem 5.1) . There is a polynomialtime algorithm taking an instance ( S , U ) of k- SetCover problem, and an error correcting code C overalphabet set [ q ] of block length ℓ and message length log q |S | , and outputs an instance U ′ , S ′ of k- SetCover problem, of size |S ′ | = |S | , |U ′ | = ℓ · |U | q k such that the following holds. Completeness: If ( S , U ) has a cover of size k, then so does ( S ′ , U ′ ) , Soundness: If ( S , U ) does not have a cover of size k then ( S ′ , U ′ ) does not have a cover of size k q − ∆ ( C ) , here ∆ ( C ) is the relative distance of C. Again, it’s clear that in the above theorem, starting from a set-system of n sets on O k ( log n ) size universe with no gap and by taking an arbitrary good code with relative distance greater than1 − / ( k ) k , we already obtain a k - SetCover instance with constant gap, and the universe size isblown up to merely ( log n ) O k ( ) .In fact our soundness result is stronger than as stated above. We show that the soundnessprobability in Theorem 1.2 is actually at most the ‘collision number’ of C (denoted by Col ( C ) )which informally is the smallest number of codewords needed to have collisions each coordinate(see Definition 2.5). We show that the soundness probability in the above theorem is actually Col ( C ) and that q − ∆ ( C ) is merely a lower bound on Col ( C ) . Thus using codes for which thevalue of Col ( C ) is optimal we get inapproximability result matching the parameters of Lin [Lin19](see Corollary 5.3). It is worth noting that the codes achieving optimality of Col ( C ) are simplyobjects called perfect hash functions but we view them as codes (Proposition 3.8). Comparison to [Lin19] and [KLM19].
We emphasize that our contribution in the above resultis in the construction of threshold graphs (using arbitrary error correcting codes) and not in thecomposition of the threshold graph with the input instance. In particular, Lin showed how toconstruct one particular threshold graph (using universal sets), and we show how to build themin a general way using any code. On the other hand, comparing Theorem 1.2 to [KLM19], we notethat it is possible to obtain time lower bounds using
DPCPF technique for k - SetCover instance withconstant gap, and the universe size blown up to merely ( log n ) O k ( ) , but this cannot be done usingarbitrary good codes as in Theorem 1.2. In [KLM19] the code used for generating gap is sensitiveto the starting hypothesis. Therefore even to get constant gap under SETH using [KLM19], wewould still need to use the highly non-trivial algebraic geometric codes.
Our main technical contribution is the construction of a class of threshold graphs. A specificthreshold graph was constructed in [Lin19] using extremal combinatorial objects called universalsets, and in this work, we show how to construct them in general by starting from just errorcorrecting codes.We note that there are several notions of threshold graphs in literature. A common aspectin all constructions is the threshold property: we want a base graph such that a certain subgraphappears many times in the base graph, whereas a slightly bigger (or different) version of thissubgraph does not appear (or appears very few times) in the base graph. We now formally definethe threshold graph that we use in this work.
Definition 1.3 (Thereshold Graph) . A bipartite graph G = ( A ˙ ∪ B , E ) , with A = A ˙ ∪ · · · ˙ ∪ A ℓ , B = B ˙ ∪ · · · ˙ ∪ B k has the threshold property with collision parameter h > k and soundness parameter δ if Completeness:
For every b , . . . b k ∈ B × · · · × B k and every i ∈ [ ℓ ] there exist a ∈ A i which is acommon neighbor of b , . . . b k . Collision Property:
Let X ⊆ B such that for every i ∈ [ ℓ ] we have that exists a ∈ A i which is a commonneighbor of (at least) k + vertices in X. Then | X | ≥ h. oundness: For every j ∈ [ t ] and every distinct b = b ′ ∈ B j , for all except ( − δ ) ℓ of the parts i ∈ [ ℓ ] ,we have that N ( b ) ∩ N ( b ′ ) ∩ A i = ∅ . Notice that a threshold graph G should contain many ( k , ℓ ) bicliques, one for each k -tuple in B × · · · × B k . The same G should not contain any ( k + ℓ ′ ) biclique (for some ℓ ′ ≪ ℓ ) when theright side has at most one vertex from each A i ( i ∈ [ ℓ ] ); this is exactly the Threshold property.The soundness property is also similar, we require that for every b j ∈ B j , b r ∈ B r , there is ajoint neighbor a i ∈ A i for every i ∈ [ ℓ ] . For b , b ′ ∈ B j , G should not contain a common neighbor inalmost all of A i ’s.We show a construction taking any error correcting code and transforming it into a thresholdgraph. The construction appears in Section 3. Theorem 1.4 (Threshold Property; Informal statement of Lemma 3.2) . Let C : Σ r → Σ ℓ be a codeof distance δ , then for every integer t ∈ N , there is polynomial time algorithm creating a graph G =( A ∪ B , E ) of size O ( t ℓ | Σ | t | Σ | r ) , which has the Threshold property with collision parameter q − δ andsoundness parameter δ . The parameters in the informal statement are not optimal, and in fact to match the bounds of[Lin19] we need the parameters in the formal statement.Using our construction when the code C is a random error correcting code (i.e. matchingeach string w ∈ Σ r to a random string in Σ ℓ ) gives optimal parameters for large enough ℓ (seeSection 3.1). On the other hand, taking a completely random graph does not give a good thresholdgraph matching our requirements, because the soundness property is very unlikely to happen (seeRemark 3.6). Composition Step of Threshold Graph with Input Graph.
We close this subsection by givingsome intuition on how the threshold graph given in Definition 1.3 is used in
TGC . First, we rewriteour initial problem (with no gap) as a problem on some bipartite graph G ( U ˙ ∪ W , E ) . This refor-mulaztion is explicit in the definition of the MaxCover problem and for the
SetCover problem weconsider the bipartite graph formed between universe and collection of input subsets (the edgesrepresenting the membership of a universe element in a subset). We then construct a thresholdgraph G ( A ˙ ∪ B , E ) where we have some canonical bijection between B and W . Therefore we nowhave the tripartite graph H ( U ˙ ∪ W ˙ ∪ A , E ′ ) , where the edge set between U and W is E and the edgeset between W (i.e., B ) and A is E . Given H , the goal is then to create a bipartite graph G between U and A , where the edges depend on H in some way. The resulting graph G has to be designedto be a gap instance of the starting problem, where the gap is obtained by using the thresholdproperties of G . This step of constructing G from H is the most non-trivial part of the TGC tech-nique. In Theorem 1.1, we indeed provide a novel way to combine the
MaxCover problem havingthe pseudo-projection property and no gap, with the threshold graph of Definition 1.3, to obtainan instance of
MaxCover problem with gap. For Theorem 1.2, we simply use the compositionprovided by Lin [Lin19].
The paper is organized as follows. In Section 2 we define the problems and hypotheses of rele-vance to this paper, and also recall some basic notions in coding theory. In Section 3, we show how6o construct threshold graphs from arbitrary error correcting codes. In Sections 4 and 5 resp., weshow how to compose these threshold graphs with k - MaxCover and k - SetCover instances resp., inorder to create a gap. Finally in Section 6, we discuss an important open problem stemming fromour work.
For a graph G = ( V , E ) and a vertex v ∈ V , we denote by N ( v ) ⊂ V the set of neighbors of v . -SAT. In the 3-SAT problem, we are given a
CNF formula ϕ over n variables x , . . . x n , such thateach clause contains at most 3 literals. Our goal is to decide if there exist an assignment to x , . . . x n which satisfies ϕ . ( k , h ) - SetCover problem.
For every k , h ∈ N , k < h , in the ( k , h ) - SetCover problem we receive auniverse U and k collection of sets S , . . . , S k over U . The goal is to distinguish between the twocases: • There exists ( S , . . . S k ) ∈ S × · · · × S k such that S ki = S i = U . • For every S ′ ⊂ S i ∈ [ k ] S i , if |S ′ | < h , then S S ∈S ′ S = U .Notice that the ( k , k + ) - SetCover problem has no gap, and we are interested in creating a gap forthe
SetCover problem, starting from a no gap instance. k - MaxCover problem.
We now recall the k - MaxCover problem introduced byChalermsook et al. [CCK + k - MaxCover super-node.The k - MaxCover instance Γ consists of a bipartite graph G = ( V ˙ ∪ W , E ) such that V is parti-tioned into V = V ˙ ∪ · · · ˙ ∪ V k and W is partitioned into W = W ˙ ∪ · · · ˙ ∪ W ℓ . We sometimes refer to V i ’s and W j ’s as left super-nodes and right super-nodes of Γ , respectively.A solution to k - MaxCover is called a labeling , which is a subset of vertices v ∈ V , . . . v k ∈ V k .We say that a labeling v , . . . . v k covers a right super-node W i , if there exists a vertex w i ∈ W i whichis a joint neighbor of all v , . . . v k , i.e. ( v j , w i ) ∈ E for every j ∈ [ k ] . We denote by MaxCover ( Γ ) themaximal fraction of right super-nodes that can be simultaneously covered, i.e. MaxCover ( Γ ) = ℓ (cid:18) max labeling v ,... v k |{ i ∈ [ ℓ ] | W i is covered by v , . . . v k }| (cid:19) .In exact k - MaxCover , the input is a maxcover instance Γ and the the goal is to decide whether MaxCover ( Γ ) = ε -gap k - MaxCover , on input Γ the goal is to distinguish between the two cases:7 MaxCover ( Γ ) = • MaxCover ( Γ ) < ε . k - Clique problem
In the k -clique problem we receive a graph H = ( V , E ) with | V | = n , and ourgoal is to decide if H contains a clique of size k , i.e. there exists v , . . . v k ∈ V such that for every i = j ∈ [ k ] , ( v i , v j ) ∈ E . Hypothesis 2.1 (W[1] = FPT) . For any computable function F : N → N , there is no F ( k ) poly ( n ) -timealgorithm which solves the k- Clique problem over n vertices.
Hypothesis 2.2 (Exponential Time Hypothesis (
ETH ) [IP01, IPZ01, Tov84]) . There exists ε > suchthat no algorithm can solve 3- SAT on n variables in time O ( ε n ) . Moreover, this holds even when restrictedto formulae in which each variable appears in at most three clauses. Note that the original version of the hypothesis from [IP01] does not enforce the requirementthat each variable appears in at most three clauses. To arrive at the above formulation, we firstapply the Sparsification Lemma of [IPZ01], which implies that we can assume without loss ofgenerality that the number of clauses m is O ( n ) . We then apply Tovey’s reduction [Tov84] whichproduces a 3- CNF instance with at most 3 m + n = O ( n ) variables and every variable occurs inat most three clauses. This means that the bounded occurrence restriction is also without loss ofgenerality. Definition 2.3 (Distance) . Let Σ be finite set and ℓ ∈ N , the distance between x , y ∈ Σ ℓ is ∆ ( x , y ) = ℓ |{ i ∈ [ ℓ ] | x i = y i }| . Definition 2.4 (Error Correcting Code) . Let Σ be finite set, for every ℓ ∈ N a subset C : Σ r → Σ ℓ isan error correcting code with message length r, block length ℓ and relative distance δ if for every x , y ∈ Σ r , ∆ ( C ( x ) , C ( y )) ≥ δ . We denote then ∆ ( C ) = δ . We sometimes abuse notations and treat an error correcting code as its image, i.e. C ⊂ Σ ℓ .For the purpose of this paper, we introduce a new notion on codes called collision number . Definition 2.5 (Collision Number) . Let C ⊂ Σ ℓ , we say that a subset S ⊂ C is colliding on coordinatei ∈ [ ℓ ] if there exists x , y ∈ S such that x i = y i . The collision number of a code C, s = Col ( C ) , is thesmallest integer s, for which there exists a set S ⊂ C , | S | = s which collides on every coordinate in [ ℓ ] . Proposition 2.6.
For every error correcting code C ⊆ Σ ℓ of relative distance δ we have, r − δ ≤ Col ( C ) ≤ | Σ | + Proof.
Fix a code C ⊂ Σ ℓ , we prove the bounds. 8 pper Bound Let S ⊆ C be any set of cardinality | Σ | +
1, and let i ∈ [ ℓ ] be any coordinate. Sincethere are | Σ | possible values for the i th coordinate and | S | ≥ | Σ | +
1, by the pigeonhole principlethere must be two x , y ∈ S such that x i = y i . Lower bound
Let S ⊆ C be a set which has a collision on every coordinate i ∈ [ ℓ ] . Let x , y ∈ S and let L x , y ⊆ [ ℓ ] be defined as follows. L x , y : = { i ∈ [ ℓ ] | x i = y i } .Since x and y are codewords of C we have | L x , y | ≤ ( − δ ) · ℓ . On the other since S is a coveringsubset we have ∑ x , y ∈ Sx = y | L x , y | ≥ ℓ .This implies that ( | S | ) · | L x , y | ≥ ℓ or in other words, ( | S | ) · ( − δ ) ≥
1. After rearrangement, wehave that | S | ≥ q − δ .In this work we use Reed Solomon code, although in fact we only use the distance of the codeand not any other properties. Theorem 2.7 (Reed-Solomon Codes [RS60]) . For every prime power q, and every r ≤ q, there exists acode of message length r, block length q, and relative distance − rq . We define a threshold graph, essentially as in [Lin19] (with an additional soundness property)which is used to create gap instances in later sections.
Definition 3.1 (Threshold Graphs from Error Correcting Codes) . For every error correcting code C : Σ r → Σ ℓ and integer t ∈ N we define the threshold graph G C , t = ( A ∪ B , E ) as follows. The vertex setsare A = A ˙ ∪ A ˙ ∪ · · · ˙ ∪ A ℓ , ∀ i ∈ [ ℓ ] , | A i | = | Σ | t and B = B ˙ ∪ B ˙ ∪ · · · ˙ ∪ B t , ∀ j ∈ [ t ] , | B j | = | Σ | r . Forevery j ∈ [ t ] , we associate B j with the set of all codewords in C, i.e., each vertex in B j is a unique codewordin the image of C . Similarly, for every i ∈ [ ℓ ] , we associate A i with the set Σ t . We have an edge betweenb ∈ B j and a ∈ A i if and only if C ( b ) i = ( a ) j . Various variants of threshold graphs were studied in Theoretically Computer Science as earlyas in the works of Babai et al. [BGK + k - Biclique . We emphasize that the above definition of threshold graphs are a novel contributionof this paper and their properties below are as in [Lin19] albeit that he constructed one specificthreshold graph for a certain range of parameters using very non-trivial objects such as universalsets, whereas we provide a generic way to construct them using relatively basic objects such aserror correcting codes.
Lemma 3.2 (Threshold Property) . Let C : Σ r → Σ ℓ be a code, let t ∈ N , then the following holds for thegraph G C , t defined above. ompleteness: For every ( b , . . . , b t ) ∈ B × · · · × B t , and every i ∈ [ ℓ ] , there is a unique vertex a ∈ A i which is a common neighbor of b , . . . , b t . Collision Property:
Let X ⊆ B such that for every i ∈ [ ℓ ] we have that exists a ∈ A i which is a commonneighbor of (at least) t + vertices in X. Then | X | ≥ Col ( C ) . Soundness:
For every j ∈ [ t ] and every distinct b = b ′ ∈ B j , for all except ( − ∆ ( C )) ℓ of the partsi ∈ [ ℓ ] , we have that N ( b ) ∩ N ( b ′ ) ∩ A i = ∅ . The soundness means that two distinct vertices in B j don’t have any joint neighbor in almostall of the partitions A i (given that C is a code with large distance). Proof.
Let G C , t be the threshold graph defined above. Completeness:
Fix ( b , . . . b t ) ∈ B × · · · × B t and i ∈ [ ℓ ] . Let x , . . . x t be the codewords that areassociated to b , . . . b t (may have repetitions). Then the vertex a = (( x ) i , ( x ) i . . . ( x t ) i ) ∈ A i is connected to b , . . . b t by definition. Furthermore, it is the only common neighbor. Let a ′ = a ∈ A i , and let j ∈ [ t ] be a location in which ( a ) j = ( a ′ ) j , then a ′ is not connected to b j ,since ( a ′ ) j = ( x j ) i = ( a ) j . Collision Property:
Let X ⊂ B be a set such that for every i ∈ [ ℓ ] , there exist a i ∈ A i such that |N ( a i ) ∩ X | ≥ t +
1, we prove that | X | ≥ Col ( C ) .For every i ∈ [ ℓ ] , we know that |N ( a i ) ∩ X | ≥ t +
1, and B is divided to t parts, so there mustbe j ∈ [ t ] such that N ( a i ) contains at least two vertices in X ∩ B j , formally (cid:12)(cid:12) N ( a i ) ∩ X ∩ B j (cid:12)(cid:12) ≥
2. Denote these vertices by b , b ′ , and let x , x ′ be the codewords associated to b , b ′ . Since bothvertices are connected to a i , ( x ) i = ( x ′ ) i = ( a i ) j .Let S be the set of codewords which are the encoding of elements in X : S = { x ∈ C | x is an encoding of some b ∈ X } .From above, for every i ∈ [ ℓ ] there must be x , x ′ ∈ S such that x i = x ′ i , so the set S collideson every coordinate, and | S | ≥ Col ( C t ) .Every element b ∈ X contributed at most a single element to S , so | X | ≥ | S | ≥ Col ( C ) . Soundness:
Fix some j ∈ [ t ] and b = b ′ ∈ B j . Let x = x ′ be the codewords associated to b , b ′ respectively. Let i ∈ [ ℓ ] be a partition in A in which b , b ′ has a joint neighbor a . By definition,this means that ( x ) i = ( x ′ ) i = ( a ) j . As x , x ′ are different codewords in C , this can happen forat most ( − ∆ ( C )) ℓ of the indices in [ ℓ ] . We show that a random code C : Σ r → Σ ℓ has good distance and collision number when ℓ is largeenough. A random error correcting code C : Σ r → Σ ℓ is a mapping in which for every x ∈ Σ r , C ( x ) is chosen uniformly at random in Σ ℓ . Corollary 3.3.
For every t , r , q ∈ N and ℓ ≥ r q , let C : [ q ] r → [ q ] ℓ be a random code, then the graphG C , t has the threshold property with collision property Col ( C ) ≥ q and soundness ∆ ( C ) ≥ − q . Claim 3.4.
Fix ℓ ≥ r | Σ | , and let C : Σ r → Σ ℓ be a random code, then with high probability, Col ( C ) ≥ | Σ | .Proof. Denote | Σ | = q . Fix a set S ⊂ Σ r , | S | ≤ q . We say that S has no collision on coordinate i ∈ [ ℓ ] , if there are no x , y ∈ S such that C ( x ) i = C ( y ) i .Fix an arbitrary coordinate i , we lower bound the probability for S to have no collision at i .We enumerate over all elements in S , S = x , x , . . . x | S | , and for each j ∈ | S | take the probabilitythat ( x j ) i = ( x t ) i for all t < j . Assuming that there was no collision on t < j , and that ( x j ) i isdistributed uniformly in [ q ] , this equals exactly q − j + q , hence:Pr [ S has no collisions at i ] = · q − q q − q · · · q − | S | + q ≥ (cid:18) (cid:19) q .The probability that there is some i ∈ [ ℓ ] such that S has collisions at i is at most ( − (cid:0) (cid:1) q ) ℓ .By union bound over all S , there are at most ( q r ) q sets S :Pr h Col ( C ) > q i ≤ ( q r ) q − (cid:18) (cid:19) q ! ℓ ≤ e rq log q e − ℓ q .In our case ℓ ≥ r q , so ℓ q ≫ rq log q , so it holds with high probability. Claim 3.5.
Let ℓ ≥ r | Σ | ln | Σ | and C : Σ r → Σ ℓ be a random error correcting code, then with highprobability ∆ ( C ) ≥ − | Σ | .Proof. Denote | Σ | = q , in this proof we treat C as the image of the code, i.e. C ⊂ Σ ℓ . For every x , y ∈ C , E [ ∆ ( x , y )] = − q . By a Chernoff bound:Pr [ ∆ ( x , y ) ≤ − q ] ≤ e − q ℓ .Preforming union bound over all pairs x , y ∈ C (there are q r such pairs):Pr (cid:20) min x = y ∈ C { ∆ ( x , y ) } < − q (cid:21) ≤ q r e − q ℓ ,with ℓ ≥ rq ln q , with high probability the distance is at least 1 − q . Remark 3.6.
We remark that if instead of taking a random error correcting code C, we would choose Gto be a random graph in the Erd¨os-R´enyi model (i.e. each edge appears with probability p) it would notbe possible to get the soundness property. This is because for graphs sampled from the Erd¨os-R´enyi model,there is no distinction between vertices in the same B j and vertices in different ones. It is unlikely that foreach b , . . . , b k ∈ B × · · · × B k there is a full bipartite graph with some a , . . . , a ℓ , but for two b , b ′ ∈ B j there is no full bipartite graph with some a , . . . , a ℓ . On the other hand, it is possible to get the collisionproperty for a random graph in the Erd¨os-R´enyi model, albeit with slightly worse parameters than randomcodes.
11e now show that there are deterministic codes for which we can obtain improvement aboveProposition 2.6 on the collision number and are in fact optimal. We start by defining perfect hashfamilies which have received considerable attention in literature (for example see [FK84, FKS84,AAB +
92, Nil94, AYZ95]).
Definition 3.7 (Perfect Hash Family) . For every N , ℓ , q ∈ N , we say that H : = { h i : [ N ] → [ q ] | i ∈ [ ℓ ] } is a [ N , ℓ ] q -Perfect hash family if for every subset T of [ N ] , where | T | ≤ q, there exists some i ∈ [ ℓ ] such that: ∀ x , y ∈ T , x = y , h i ( x ) = h i ( y ) . (1) Moreover, the computation time of H is defined to be the time needed to output the ℓ × N matrix withentries in [ q ] whose ( i , x ) th entry is simply h i ( x ) (for h i ∈ H).
In other words, H is a [ N , ℓ ] q -Perfect hash family if for every T ⊂ [ N ] , | T | ≤ q , there exists ahash function h ∈ H such that h on inputs in T gets | T | distinct values. Proposition 3.8 (Collision number of Perfect hash family) . Let N , ℓ , q ∈ N , and let H be a [ N , ℓ ] q -Perfect hash family. Then H can be interpreted as a code over alphabet [ q ] of message length log q N, blocklength ℓ and collision number q + .Proof. Label the hash functions in H using [ ℓ ] . We think of H as a code as follows. For every x ∈ [ N ] and i ∈ [ ℓ ] , the x th codeword’s i th coordinate is the image of the i th hash function in H onthe input x .To see the claim on the collision number of the aforementioned code, suppose for the contraryassume that there exists T ⊆ [ N ] of cardinality q such that for every i ∈ [ ℓ ] there exists x , y ∈ T such that x i = y i . This contradicts (1). k - MaxCover : Gap Creation by Threhosld Graph Composition
In this section we show a gap creation technique for k - MaxCover with a projection property we call pseudo projection . This property is an analog of the projection property of label cover.
Definition 4.1 (Pseudo Projection) . A k-
MaxCover instance Γ = ( V ˙ ∪ W , E ) , V = V ˙ ∪ · · · ˙ ∪ V k andW = W ˙ ∪ · · · ˙ ∪ W t has the pseudo projection property if for every i ∈ [ k ] , j ∈ [ t ] , one of the two holds: • Every v ∈ V i has exactly one neighbor w ∈ W j . • There is a full bipartite graph between V i and W j . Below is the main result of this section on gap creation in k - MaxCover . Theorem 4.2.
Let Γ = ( V ˙ ∪ W , E ) be a k- MaxCover instance with the pseudo projection property, withV = V , . . . V k , W = W , . . . W t . Let C : Σ r → Σ ℓ be an error correcting code such that | Σ | r ≥ (cid:12)(cid:12) W j (cid:12)(cid:12) forevery j ∈ [ t ] . Then there exists a reduction in time O ( | Γ | ℓ | Σ | t ) to a k- MaxCover instance Γ ( V ˙ ∪ A , E ) ofsize | V | ℓ | Σ | t with V divided into k parts, and A into ℓ parts. The new instance Γ satisfies • If MaxCover ( Γ ) = , then MaxCover ( Γ ) = . If MaxCover ( Γ ) < , then MaxCover ( Γ ) ≤ − ∆ ( C ) .Proof. Let G C , t be the threshold graph from Definition 3.1, with the error correcting code C andinteger t . We compose Γ with G C , t to create our new instance Γ .For every j ∈ [ t ] , we arbitrarily match every vertex in w j ∈ W j to a vertex b j ∈ B j withoutrepetitions. This can be done since (cid:12)(cid:12) W j (cid:12)(cid:12) ≤ (cid:12)(cid:12) B j (cid:12)(cid:12) . The new instance Γ is defined as follows: • The vertex sets are V from Γ , and A from G C , t . • A vertex v ∈ V i is connected to a ∈ A j if there exists w ∈ W , . . . w t ∈ W t such that v isconnected to w , . . . w t in Γ , and a is connected to the matching b , . . . b t in G C , t .We prove the reduction parameters and correctness. Runtime and Size
The size of Γ is bounded by | V | | A | = | V | ℓ | Σ | t . For the runtime, to create theedges in Γ , for each v ∈ V and a ∈ A we go over all their neighbors in Γ , G C , t and check if every W j is covered by a joint neighbor. This can be done in a linear time in | W | . Therefore, the runtimeof the reduction is bounded by | V | | A | | W | = O ( | Γ | ℓ | Σ | t ) . Completeness
Assume
MaxCover ( Γ ) =
1, let v , . . . v k ∈ V × · · · V k be a covering set, and let w , . . . w t ∈ W × · · · × W t be the vertices covered by v , . . . v k , i.e. there is a full bipartite graphbetween v , . . . v k and w , . . . w t .Let b , . . . b t the matching vertices to w , . . . w t . By the definition of the threshold graph, forevery l ∈ [ ℓ ] there exists a vertex a l ∈ A l which is a common neighbor of b , . . . b t . From thecomposition definition, for every i ∈ [ k ] , l ∈ [ ℓ ] , a l is a neighbor of v i . Therefore, for every l ∈ [ ℓ ] , A l is covered by v , . . . v k . Soundness
Assume
MaxCover ( Γ ) <
1. Fix any labeling v ∈ V , . . . v k ∈ V k of Γ . Since Γ is notsatisfiable, v , . . . v k do not cover all of W . Let j ∈ [ t ] be a super-node not covered by v , . . . v k .Define S ⊂ [ k ] to be all indices i such that there is a function from the set V i to W j . For every i ∈ S , denoted this function by f i : V i → W j . The instance Γ has the pseudo projection property,so for every i ′ / ∈ S there is a full bipartite graph between V i ′ and W j . Since W j is not covered by v , . . . v k , there must be i , i ∈ S such that f i ( v i ) = f i ( v i ) . Denote w = f i ( v i ) and w ′ = f i ( v i ) .Let b = b ′ ∈ B j be the vertices matched to w , w ′ ∈ W j . By our composition, any neighbor a ∈ N ( v i ) in Γ has to be a neighbor of b in G C , t (since w = b is the only neighbor of v i in W j ).Similarly, every neighbor of v i in Γ has to be a neighbor of b ′ in G C , t .By Lemma 3.2, the threshold graph G C , t is such that for all except ( − ∆ ( C )) ℓ of the indices l ∈ ℓ , N ( b ) ∩ N ( b ′ ) ∩ A l = ∅ . From above, for all these l ’s, v i , v i don’t have a common neighborin A l , and A l is uncovered by v , . . . v k .Using the above theorem we can prove strong inapproximability results for k - MaxCover basedon
ETH and W [ ] = FPT . The proofs of both the theorems essentially follow from the ideas givenin [KLM19] and we defer them to Appendix A. 13 heorem 4.3.
Assuming
ETH , there is no n o ( k ) time algorithm that given a k- MaxCover instance Γ ( G =( V ˙ ∪ W , E )) , where V is divided into k parts, can decide between the following two cases: Completeness:
MaxCover ( Γ ) = . Soundness:
MaxCover ( Γ ) ≤ n − O ( k ) . Theorem 4.4.
Assuming W [ ] = FPT , for every computable function F : N → N , there is no F ( k ) poly ( n ) time algorithm that given a k- MaxCover instance Γ ( G = ( V ˙ ∪ W , E )) , where V is divided into k parts, candecide between the following two cases: Completeness:
MaxCover ( Γ ) = . Soundness:
MaxCover ( Γ ) ≤ n − √ k . Remark 4.5 (Comparison to [KLM19]) . The proof technique in [KLM19] gives us the exact same state-ment as in Theorem 4.2 and with identical parameters. The difference being that in [KLM19], the
DPCPF isused, whereas we demonstrate that the result can be established using the
TGC technique as well. Addition-ally, it is easy to see that the k-
MaxCover problem with pseudo-projection property can be reduced withoutany loss in parameters to a product space problem
PSP ( f ) over the multi-equality Boolean function f (see[KLM19] for the definition of the two terms). It is also possible to reverse the direction of this reduction.Since [KLM19] show time lower bounds under ETH and W [ ] = FPT for
PSP ( f ) , it is easy to see why thetwo techniques yield the same result. In this section, we prove Theorem 1.2 formally.
Theorem 5.1.
For every integer n and every code C : Σ r → Σ ℓ of relative distance δ , such that | Σ | r ≥ n,there is an algorithm running in poly ( n ) time that takes as input an instance ( U , S = S ∪ · · · ∪ S k ) of ( k , k + ) - SetCover problem (where |S | = n) and outputs an instance ( U ′ , S ′ = S ′ ∪ · · · ∪ S ′ k ) of ( k , h ) - SetCover problem such that the following holds.
Size: |S | = |S ′ | and |U ′ | = ℓ · |U | | Σ | k . Completeness:
If there exists ( S , . . . S k ) ∈ S × · · · × S k such that S ki = S i = U then there exists ( S ′ , . . . S ′ k ) ∈ S ′ × · · · × S ′ k such that S ki = S ′ i = U ′ . Soundness:
If there is no cover for U of size k in S then there is no cover for U ′ of size h in S ′ whereh : = Col ( C ) ≥ q − δ . The proof of the above theorem follows immediately from combining the below lemma provedin [Lin19] with our lower bound on covering number given in Proposition 2.6.
Lemma 5.2 ([Lin19]) . There is an algorithm which, given an integer k, an instance ( U , S = S ∪ · · · ∪ S k ) of ( k , k + ) - SetCover problem (where |S | = n), and a threshold graph G C , k as described in Definition 3.1,outputs a ( k , h ) - SetCover instance ( U ′ , S ′ = S ′ ∪ · · · ∪ S ′ k ) with |S ′ | = |S | and |U ′ | = ℓ · |U | | Σ | k in |U | | Σ | k · n O ( ) time such that If there exists ( S , . . . S k ) ∈ S × · · · × S k such that S ki = S i = U then there exists ( S ′ , . . . S ′ k ) ∈S ′ × · · · × S ′ k such that S ki = S ′ i = U ′ . • If there is no cover for U of size k in S then there is no cover for U ′ of size Col ( C ) in S ′ (follows fromLemma 3.2). Next, we show that for a specific choice of code C , we can achieve the following parametersfor k - SetCover problem.
Corollary 5.3.
For every integer n, there is an algorithm running in poly ( n ) time that takes as input aninstance ( U , S = S ∪ · · · ∪ S k ) of ( k , k + ) - SetCover problem (where |S | = n) and outputs an instance ( U ′ , S ′ = S ′ ∪ · · · ∪ S ′ k ) of (cid:16) k , k q log |U ′ | log log |U ′ | (cid:17) - SetCover problem such that the following holds.
Size: |S | = |S ′ | and |U ′ | = n. Completeness:
If there exists ( S , . . . S k ) ∈ S × · · · × S k such that S ki = S i = U then there exists ( S ′ , . . . S ′ k ) ∈ S ′ × · · · × S ′ k such that S ki = S ′ i = U ′ . Soundness:
If there is no cover for U of size k in S then there is no cover for U ′ of size k q log |U ′ | log log |U ′ | in S ′ . Notice that the parameters obtained here match the parameters obtained by Lin [Lin19] byusing universal sets. The proof of the above corollary follows by combining Theorem 5.1 with thetheorem below by setting q = k q log |U ′ | log log |U ′ | and then applying Proposition 3.8. Theorem 5.4 (Alon et al. [AYZ95]) . For every N , q ∈ N there exists a [ N , 2 O ( q ) · log N ] q -Perfect hashfamily that can be computed in time e O q ( N ) . Remark 5.5.
Also, notice that in Lemma 5.2, starting from a universe of size O k ( log n ) , in order to obtaingood time lower bounds based on various assumptions such as SETH , ETH , and W [ ] = FPT , we wouldlike that the new universe size is n O ( ) . This implies that the alphabet of the code used in gap creation canbe at most O (cid:16) k q log n log log n (cid:17) . Since the collision number of a code is by Proposition 2.6 at most the alphabetsize (plus one), we have that it is not possible to obtain better gaps using Lin’s scheme of gap creation for SetCover problem.
The main open question that stems from our work is if we could prove Theorem 4.2 when the k - MaxCover instance does not have the pseudo-projection property but instead is obtained throughthe standard
SETH -hardness reduction from k - SAT to exact k - MaxCover . Assuming
SETH , can we show there is no n k − ε time algorithm (for some ε > )for gap k- MaxCover (of size n) using the Threshold Graph Composition technique?
A positive answer to the above question in particular for the case k = TGC technique to enter the world of inap-proximability in subquadratic time, and might provide a lot of new insights, including the poten-tial resolution of many open problems (for example, the subquadratic hardness of the gap closestpair problem [KM19]). Notice that one advantage of
TGC over
DPCPF is that, in theory, it canhandle monochromatic k - MaxCover (where instead of picking ( v , . . . , v k ) ∈ V × · · · × V k , we areallowed to pick any k distinct vertices in V ). We direct the reader to Section B for an attempt at try-ing to circumvent the need for pseudo-projection property for gap creation in Theorem 4.2, when k = Acknowledgements
We would like to thank Lijie Chen for his detailed comments on an earlier version of the paper.
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A Inapproximability of k - MaxCover
In this section we show hardness of k - MaxCover with pseudo projection property under W [ ] = FPT and
ETH , and use Lemma 4.2 to show gap k - MaxCover hardness under these hypothesis.
A.1 W[1]-Hardness of Approximation
Lemma A.1.
For every integer t ∈ N and every graph H = ( V ′ , E ′ ) over m vertices, where V ′ = V ′ ˙ ∪ V ′ ˙ ∪ · · · ˙ ∪ V ′ t , such that for all i ∈ [ t ] , V ′ i is an independent set of size mt , there is an O ( m ) - timereduction which outputs a k- MaxCover instance Γ = ( V ∪ W , E ) of size O ( m ) with V divided intok = ( t ) parts ans W into t parts. The k- MaxCover instance Γ satisfies • If H contains a t-clique, then
MaxCover ( Γ ) = . • If H does not contain a t-clique, then
MaxCover ( Γ ) < .Furthermore, Γ has the pseudo projection property.Proof. Given a graph H = ( V ′ , E ′ ) over m vertices V ′ = V ′ ˙ ∪ V ′ ˙ ∪ · · · ˙ ∪ V ′ t , such that each V ′ i is anindependent set, we denote by E ′ i , j the set of edges between V ′ i and V ′ j .We now construct the k - MaxCover instance, set k : = ( t ) , and let Γ = ( V ˙ ∪ W , E ) be a bipartitegraph with vertex sets V = V ˙ ∪ V · · · V k and W = W ˙ ∪ W · · · W t . We abuse notation a little andrefer to a super node V l for l ∈ [ k ] as V i , j , for i , j ∈ [ t ] .For all distinct i , j ∈ [ t ] , we define V i , j to be the set of edges E ′ i , j , V i , j ∆ = E ′ i , j . For every i ∈ [ t ] ,we set W i to be V ′ i , W i ∆ = V ′ i . The edges in Γ : • Between W i and V i , j , we connect w ∈ W i and e = ( v , v ) ∈ V i , j if w = v or w = v . • Between W i and V j , l , when i = j , l , we create a full bipartite graph.We now prove the properties of the reduction. Reduction Runtime and Size:
The size of each V i , j is at most m t and the size of each W j is mt .Since checking edge incidence is a trivial task, the total runtime is at most | V | · | W | ≤ m . Pseudo Projection:
By definition, every edge e = E ′ , e ∈ V i , j has exactly one neighbor in V ′ i andone in V ′ j , which corresponds in Γ to one neighbor on W i and one on W j . For every other W l , thereis a full bipartite graph in Γ between V i , j and W l , so Γ has the pseudo projection property.19 ompleteness: Suppose H contains a t -clique. Each V ′ j is an independent set, so the t clique hasto contain one vertex in each V ′ j , say { v ′ , . . . , v ′ t } ∈ V ′ × · · · × V ′ t . We show that there exist a coverin Γ with value 1. For every distinct i , j ∈ [ t ] , we pick the label e = ( v ′ i , v ′ j ) ∈ V i , j (it exists because { v ′ , . . . , v ′ t } is a clique). We show that this set covers all of W : for any W j , the vertex v ′ j ∈ W j isconnected to all of the edges: it is connected to its adjacent edges ( v ′ i , v ′ j ) ∈ V i , j , and because thereis a full bipartite graph, it is connected also to ( v ′ i , v ′ l ) ∈ V i , l when j = i , l . Soundness:
Suppose H does not contain a t -clique, then consider any labeling S of G . If S coversall the super nodes W j , then there exist { v i , j ∈ V i , j } i , j ∈ [ t ] such that they have one common neighborin each W j , say w j . Consider the set of t vertices: { w , . . . , w t } . For every i , j ∈ [ t ] we have that v i , j is an edge in H (as v i , j is a neighbor of w i and w j in G ). Therefore, { w , . . . , w t } is t -clique of H leading to a contradiction. Proof of Theorem 4.4.
We prove the theorem by a reduction from k - Clique to k - MaxCover and then togap k - MaxCover , in a similar way to Theorem 4.3. Assume towards contradiction that there is analgorithm A running in time F ( k ) poly ( n ) and solves the gap k - MaxCover problem described in thetheorem. We show an algorithm running in time F ′ ( t ) poly ( m ) for solving the t -clique problem on m vertices.The input is a graph H = ( V ′ , E ′ ) , where V ′ is divided into t parts. Denote m = | V ′ | .1. Use the reduction from Lemma A.1 on the graph H to get a k - MaxCover instance Γ = ( V ∪ W , E ) , such that V is divided into k = ( t ) parts, and W into t parts.2. Let C : F tq → F qq be the Reed Solomon code, for q a large prime power such that m t ≤ q ≤ m t . Run the reduction from Theorem 4.2 on Γ and C and receive a k - MaxCover instance Γ .3. Run algorithm A on Γ , answer like A .We prove the correctness of the algorithm. Runtime:
The reduction in Item 1 takes O ( m ) time and outputs a k - MaxCover instance Γ of sizeat most O ( m ) , with the pseudo projection property. The gap generating algorithm fromTheorem 4.2 runs in time at most | Γ | qq t ≤ m m + t , and outputs Γ which is of size at most | Γ | qq t ≤ m . Denote | Γ | = n , by our assumption, the runtime of A is F ( k ) poly ( n ) for acomputable function F . As n = poly ( m ) and k = ( t ) , this is at most F ( t ) poly ( m ) . Thus, thetotal runtime runtime of the algorithm in total is F ′ ( t ) poly ( m ) for a computable function F ′ . Correctness: If H contains a t-clique, then by Lemma A.1, Γ is a k - MaxCover instance with thepseudo projection property, such that
MaxCover ( Γ ) =
1. By Theorem 4.2
MaxCover ( Γ ) = A answers YES.If H does not have a t-clique, then by Lemma A.1, Γ satisfies MaxCover ( Γ ) <
1. Therefore,by Theorem 4.2
MaxCover ( Γ ) < − ∆ ( C ) . The distance of C is 1 − tq ≥ − tm t , in terms of | Γ | = n , MaxCover ( Γ ) ≤ − ∆ ( C ) ≤ n − t , so the algorithm A answers no.20 .2 ETH Hardness of Approximation Lemma A.2.
For every 3-
CNF formula ϕ over n variables such that each variable appears in at most clauses, and every integer k ∈ N , there is a n nk - time reduction which outputs a k- MaxCover instance Γ = ( V ∪ W , E ) of size nk poly ( k ) with V divided into k parts ans W into O ( k ) parts. The k- MaxCover instance Γ satisfies • If ϕ is satisfiable, then MaxCover ( Γ ) = . • If ϕ is not satisfiable, then MaxCover ( Γ ) < .Furthermore, Γ has the pseudo projection property.Proof. Given a 3-
CNF formula ϕ over n variables and m ≤ n clauses. Let C . . . C k be a partitionof the clauses into k approximately equal sets.Let G ( V ˙ ∪ W , E ) be a bipartite graph with vertex sets V = V ˙ ∪ V · · · V k and W = W ˙ ∪ W · · · W t ,where t = ( k ) + ( k ) + ( k ) . Each j ∈ [ t ] is associated to a subset J ⊂ [ k ] , 1 ≤ | J | ≤
3, in the proof weabuse notation a bit and use J ∈ [ t ] while treating J as a subset.Each V i contains the set of all partial assignments satisfying the clauses in C i . Note that | V i | ≤ n / k . For each J ∈ [ t ] , if J = { i , i , i } , let S J be the set of variables appears both in C i in C i andin C i . For smaller J , S J contains the set of variables appears exactly on all C i for i ∈ J and not inother partitions (such that each variable in [ n ] belongs exactly to a single S J ). The vertex set W J contains all assignments to S J . Again note that | W J | ≤ n / k .For every i ∈ [ k ] , J ∈ [ t ] , the edges are as follows: • If i ∈ J , then connect every v ∈ V i with it’s consistent assignment w ∈ W J . • If i / ∈ J , connect every v ∈ V i to every w ∈ W J .We now prove the properties of the reduction. Reduction Runtime and Size:
The size of each V i is at most 2 nk because its an assignment overat most nk variables. Similarly the size of each W j is at most 2 nk . The number of partition is clearfrom construction.It takes linear time in n to check if every partial assignment to a partition C i is satisfying. Foreach vertex v ∈ V i it takes at most time | W | to create all its edges. Therefore the total runtime is atmost n | W | | V | = n nk nk kt ≤ n nk . Partial Projection:
Each vertex v ∈ V i is an assignment to all variables in the clauses in C i . Forevery J ∋ i , a vertex w ∈ W J is an assignment to the variables in S J , which is a subset of thevariables in C i . Therefore, there is exactly a single w ∈ W J which is consistent with u .For every J such that i / ∈ J , there is a full bipartite graph between V i to W J , which matches thedefinition of the pseudo projection property. 21 ompleteness: Suppose ϕ is a satisfiable formula, and let x = x , . . . x n be an assignment whichsatisfies ϕ . Let v ∈ V , . . . v k ∈ V k be the vertices which represents the restriction of x to eachpart C i . Similarly, let w ∈ W , . . . w t ∈ W t be the restriction of x to each S J . We claim that W isfully covered by v , . . . v k , with the nodes w , . . . w t . Fix an arbitrary J ∈ [ t ] , i ∈ [ k ] , if i / ∈ J thenby definition v i is connected to all of W J , so w J is covered by v i . For i ∈ J , since both v i , w J are arestriction of x , they are consistent and are connected by an edge. Therefore v , . . . . v k covers W . Soundness:
Suppose ϕ is not satisfiable, and let v ∈ V , . . . v k ∈ V k to be some labeling of Γ . By definition, each v i satisfies all clauses in C i . Since ϕ is not satisfiable, v , . . . v k can’t be arestriction of a single assignment, and there must be some r ∈ [ n ] , i , i ∈ [ k ] such that v i , v i assign different values to x r . Assume towards contradiction that v , . . . v k fully covers W , andlet w ∈ W , . . . w t ∈ W t be the vertices which are the joint neighbors. Let J ∈ [ t ] be the subsetcontaining all indices from k in which x r appears, it must be that i , i ∈ J (there might be a thirdindex). The vertex w J assigns some value to x r , and it’s not possible that both v i , v i are consistentwith it (as they assign different values to x r ). Therefore w J is not a joint neighbor of v , . . . v k andwe have a contradiction. Proof of Theorem 4.3.
We prove the theorem by a reduction from 3-
SAT to k - MaxCover , and then togap k - MaxCover .Assume towards contradiction that there exists an algorithm A which solves the gap k - MaxCover problem described in the theorem in time n o ( k ) . In particular, A runs in time less than n ε k , where ε is the constant from the ETH hypothesis (see Hypothesis 2.2). We show an algorithm for solving3-
SAT on m variables in time less than 2 ε m , refuting ETH .The input is a 3-
CNF formula ϕ on m variables, such that each variable appears in at most 3clauses.1. Set k = ε and run the reduction from Lemma A.2 on ϕ with parameter k , receiving a k - MaxCover instance Γ = ( V ∪ W , E ) with V divided into k parts and W into at most k parts.2. Let C : F k q → F qq be the Reed Solomon code, for q a large prime power such that mk ≤ q ≤ mk . Run the reduction from Theorem 4.2 on Γ and C and receive a k - MaxCover instance Γ .3. Run algorithm A on Γ , answer like A .We show that the described algorithm solves 3- SAT and runs in time O ( ε m ) . Runtime: ϕ is a 3- CNF formula on m variables such that each variable appears in at most 3 clauses.From Theorem A.2, the size of Γ is 2 mk poly ( k ) , and the reduction runtime is at most 2 mk m .By Theorem 4.2, Γ has size at most | Γ | qq k = mk poly ( k ) mk ≤ mk , and the runtime of Item2 is at most | Γ | qq k ≤ mk . Denote | Γ | = n , the runtime of A is at most n ε k , which is lessthan 2 mk ε k . So the total run time of the algorithm is less than 2 ε m . Correctness:
Suppose ϕ is a satisfiable 3- CNF , then according to Theorem A.2, Γ is a k - MaxCover instance with the pseudo projection property, and
MaxCover ( Γ ) =
1. From Theorem 4.2,
MaxCover ( Γ ) = A should answer YES in this case.22n the case ϕ is not satisfiable, then by Theorem A.2, Γ has the pseudo projection propertyand MaxCover ( Γ ) <
1. Therefore, by Theorem 4.2
MaxCover ( Γ ) < − ∆ ( C ) . The distanceof Reed Solomon code C : F k q → F qq is 1 − rq ≥ − − mk . Calculating the gap with respect to | Γ | , 1 − ∆ ( C ) ≤ − mk = n − O ( k ) , as | Γ | = n = mk . Therefore, A should answer NO. B Gap creation in
MaxCover for k = In the special case of k =
2, and where the out degree of vertices in V is small, we can show a gapcreating reduction for MaxCover without requiring the partial projection property.
Lemma B.1.
Let Γ = ( V ∪ W , E ) be a MaxCover instance with V = V ˙ ∪ V , W = W , ˙ ∪ · · · ˙ ∪ W t , andfor every v ∈ V and j ∈ t, (cid:12)(cid:12) N ( u ) ∩ W j (cid:12)(cid:12) ≤ d. Let C : Σ r → Σ ℓ be an error correcting code such that | Σ | r ≥ (cid:12)(cid:12) W j (cid:12)(cid:12) for every j ∈ [ t ] . Then there exists a reduction in time O ( | Γ | ℓ | Σ | t ) to a MaxCover instance Γ ( V ∪ A , E ) of size | V | ℓ | Σ | t with V divided into parts, and A into ℓ parts. The new instance Γ satisfies • If MaxCover ( Γ ) = , then MaxCover ( Γ ) = . • If MaxCover ( Γ ) < , then MaxCover ( Γ ) ≤ d ( − ∆ ( C )) . Notice that in order to use the reduction we must use an error correcting code with largedistance, else the soundness guarantee in the above lemma is meaningless.
Proof.
The proof is essentially the same as the proof of Theorem 4.2, with slight modifications. Let G C , t be the threshold graph from Definition 3.1, with the error correcting code C and integer t . Wecompose Γ with G C , t to create our new instance Γ .For every j ∈ [ t ] , we arbitrarily match every vertex in w j ∈ W j to a vertex b j ∈ B j withoutrepetitions. This can be done since (cid:12)(cid:12) W j (cid:12)(cid:12) ≤ (cid:12)(cid:12) B j (cid:12)(cid:12) . The new instance Γ is defined as follows: • The vertex sets of Γ are V from Γ , and A from G C , t . • A vertex v ∈ V i is connected to a ∈ A j if there exists w ∈ W , . . . w t ∈ W t such that v isconnected to w , . . . w t in Γ , and a is connected to the matching b , . . . b t in G C , t .We prove: Runtime and Size
The size Γ is | V | | A | , where | A | = ℓ | A i | = ℓ | Σ | t .To create the new instance, for each v ∈ V and a ∈ A we go over all their neighbors and checkif every W j is covered by a joint neighbor. This can be done in a linear time in | W | . Therefore, theruntime of the reduction is bounded by | V | | A | | W | = O ( | G | ℓ | Σ | t ) . Completeness
Suppose
MaxCover ( Γ ) =
1, let v ∈ V , v ∈ V be a covering set, and let w , . . . w t ∈ W × · · · × W t be the joint neighbors covered by v , v .Let b , . . . b t the matching vertices to w , . . . w t . By the definition of the threshold graph, forevery l ∈ [ ℓ ] there exists a vertex a l ∈ A l which is a common neighbor of b , . . . b t in G C , t . Fromthe composition definition, this a l is a neighbor of v and v in Γ , so A l is covered by v , v .23 oundness MaxCover ( Γ ) <
1. Let v ∈ V , v ∈ V to be some labeling of Γ . Since Γ isnot satisfiable, v , v does not cover all of W . Suppose W j is not covered by v , v , denote S = N ( v ) ∩ W j and S = N ( v ) ∩ W j . Because W j is uncovered by v , v , S ∩ S = ∅ , and the degreebounds promises that | S | , | S | ≤ d .Let S ′ , S ′ ⊂ B j be the vertices in G C , t which are matched to S , S respectively. The matchingis one to one, so S ′ ∩ S ′ = ∅ . By our composition, any neighbor a of v in Γ has to be a neighborof some vertex b ∈ S ′ in G C , t , and the same for v and S ′ .By Lemma 3.2, the threshold graph G C , t is such that for every j and for every b = b ′ ∈ B j , allexcept ( − ∆ ( C )) ℓ of the indices l ∈ ℓ , N ( b ) ∩ N ( b ′ ) ∩ A l = ∅ . For every b ∈ S ′ , b ′ ∈ S ′ , thereare at most ( − ∆ ( C )) ℓ indices such that N ( b ) ∩ N ( b ′ ) ∩ A l = ∅ . There are at most d pairs of b ∈ S ′ , b ′ ∈ S ′ , so for all except d ( − ∆ ( C )) ℓ of the indices l ∈ ℓ , there is no a ∈ A l which is acommon neighbor of v , v2