Interactive proofs and the hardness of approximating cliques

Feige, Uriel; Goldwasser, Shafi; Lovász, László; Safra, Shmuel; Szegedy, Márió

doi:10.1145/226643.226652

Cited by 431 publications

(222 citation statements)

References 28 publications

(27 reference statements)

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“…Garg et al showed that the problem is APX-Hard [7]. More recently, Mastrolilli et al [18] showed that the problem is NP-hard to approximate within 6 5 − ε, and that it is hard to approximate within 4 3 − ε assuming the UGC. In this work we give a tight hardness result assuming the UGC and show that the problem is hard to approximate within 2 − ε.…”

Section: Concurrent Open Shopmentioning

confidence: 99%

“…The techniques developed were also used to show a tight 2 − ε inapproximability result (assuming a variant of the UGC) for the classic problem of minimizing the weighted completion time with precedence constraints. They designed a PCP with one free bit, near-perfect completeness, and arbitrarily low soundness, and used the equivalence between hardness of vertex cover problem and existence of PCPs with zero free bits [6,2].…”

Section: Introductionmentioning

confidence: 99%

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Inapproximability of Hypergraph Vertex Cover and Applications to Scheduling Problems

Bansal

Khot

2010

Automata, Languages and Programming

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Assuming the Unique Games Conjecture (UGC), we show optimal inapproximability results for two classic scheduling problems. We obtain a hardness of 2 − ε for the problem of minimizing the total weighted completion time in concurrent open shops. We also obtain a hardness of 2 − ε for minimizing the makespan in the assembly line problem.These results follow from a new inapproximability result for the Vertex Cover problem on k-uniform hypergraphs that is stronger and simpler than previous results. We show that assuming the UGC, for every k ≥ 2, the problem is inapproximable within k − ε even when the hypergraph is almost k-partite.

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Section: Concurrent Open Shopmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Inapproximability of Hypergraph Vertex Cover and Applications to Scheduling Problems

Bansal

Khot

2010

Automata, Languages and Programming

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“…The first stage is algebraic and gives local tests for algebraic codes, usually based on multivariate polynomials. This is based on a rich collection of results on "linearity testing" or "low-degree testing" [1,3,4,5,6,7,8,9,13,14,16,17,18,20,23,25]. This first stage either yielded codes of poor rate (mapping k information symbols to codewords of length exp(k)) as in [14], or yielded codes over large alphabets as in [25].…”

Section: Introductionmentioning

confidence: 99%

Robust locally testable codes and products of codes

Ben–Sasson¹,

Sudan²

2006

Random Struct Algorithms

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We continue the investigation of locally testable codes, i.e., error-correcting codes for whom membership of a given word in the code can be tested probabilistically by examining it in very few locations. We give two general results on local testability: First, motivated by the recently proposed notion of robust probabilistically checkable proofs, we introduce the notion of robust local testability of codes. We relate this notion to a product of codes introduced by Tanner, and show a very simple composition lemma for this notion. Next, we show that codes built by tensor products can be tested robustly and somewhat locally, by applying a variant of a test and proof technique introduced by Raz and Safra in the context of testing low-degree multivariate polynomials (which are a special case of tensor codes).Combining these two results gives us a generic construction of codes of inverse polynomial rate, that are testable with poly-logarithmically many queries. We note these locally testable tensor codes can be obtained from any linear error correcting code with good distance. Previous results on local testability, albeit much stronger quantitatively, rely heavily on algebraic properties of the underlying codes.

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“…Babai, Fortnow, Levin and Szegedy [5] showed that there exist PCPs (called holographic proofs in their result) for NP in which it is possible to verify the correctness of the proof in poly-logarithmic time. The seminal result indicating the intricate relationship between PCP systems and hardness of approximations was made by Feige, Goldwasser, LovAsz, Safra and Szegedy [12]. The definition of PCPs is implicit in their result and they show that NP C O(lognloglogn)].…”

Section: Pcps -A Brief Historymentioning

confidence: 99%

Small PCPs with low query complexity

Harsha¹,

Sudan²

2000

Comput. complex.

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Most known constructions of probabilistically checkable proofs (PCPs) either blow up the proofsize by a large polynomial, or have a high (though constant) query complexity. In this thesis we give a transformation with slightly-super-cubic blowup in proof-size and a low query complexity.Specifically, the verifier probes the proof in 16 bits and rejects every proof of a false assertion with probability arbitrarily close to 1, while accepting corrects proofs of theorems with probability one.The proof is obtained by revisiting known constructions and improving numerous components therein. In the process we abstract a number of new modules that may be of use in other PCP constructions.

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Interactive proofs and the hardness of approximating cliques

Cited by 431 publications

References 28 publications

Inapproximability of Hypergraph Vertex Cover and Applications to Scheduling Problems

Inapproximability of Hypergraph Vertex Cover and Applications to Scheduling Problems

Robust locally testable codes and products of codes

Small PCPs with low query complexity

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