## Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring

Citations: | 60 - 8 self |

### BibTeX

@MISC{Khot_improvedinapproximability,

author = {Subhash Khot},

title = {Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring},

year = {}

}

### OpenURL

### Abstract

In this paper, we present improved inapproximability re-sults for three problems: the problem of finding the maximum clique size in a graph, the problem of finding the chro-matic number of a graph, and the problem of coloring a graph with a small chromatic number with a small numberof colors. H*astad's celebrated result [13] shows that the maximumclique size in a graph with n vertices is inapproximable inpolynomial time within a factor n1-ffl for arbitrarily smallconstant ffl> 0 unless NP=ZPP. In this paper, we aimat getting the best subconstant value of ffl in H*astad's re-sult. We prove that clique size is inapproximable within a factor n2(log n)1-fl (corresponding to ffl = 1(log n)fl) for some constant fl> 0 unless NP ` ZPTIME(2(log n) O(1)). This improves the previous best inapproximability factor of

### Citations

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Citation Context ... probability c. Otherwise, no proof is accepted with probability more than s. Feige et al. [12] showed that NP ⊆ PCP1,1/2(O(log n log log n), O(log n log log n)). Arora and Safra [3] and Arora et al. =-=[2]-=- improved this result to show that NP ⊆ PCP1,1/2(O(log n), O(1)), a result known as the PCP Theorem. Since then, many different PCP constructions for languages in NP have led to inapproximability resu... |

647 | Some optimal inapproximability results
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Citation Context ...ted so that the remaining system of equations has a satisfying assignment. We show a hardness factor of 2 Ω(√ log n) for this problem, improving upon the hardness factor of (log n) β shown by H˚astad =-=[18]-=-, for some small (unspecified) constant β > 0. The hardness results for clique and chromatic number are then obtained using the reduction from Min-3Lin-Deletion as given in [20]. 1 Introduction A cliq... |

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(Show Context)
Citation Context ...that is accepted with probability c. Otherwise, no proof is accepted with probability more than s. Feige et al. [12] showed that NP ⊆ PCP1,1/2(O(log n log log n), O(log n log log n)). Arora and Safra =-=[3]-=- and Arora et al. [2] improved this result to show that NP ⊆ PCP1,1/2(O(log n), O(1)), a result known as the PCP Theorem. Since then, many different PCP constructions for languages in NP have led to i... |

318 | A Parallel Repetition Theorem
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(Show Context)
Citation Context ...that the constant γ ′ in Khot’s hardness results for MaxClique and Chromatic Number is a non-explicit (possibly extremely tiny) constant that depends on the proof of Raz’s Parallel Repetition Theorem =-=[23]-=-. 2 Our Results and Techniques We show the following inapproximability results for MaxClique and chromatic n number, taking us closer to the goal of 2O(√log n) (or even n/polylog(n)). (log n)O(1) Theo... |

233 | On the power of unique 2-prover 1-round games
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Citation Context ... Theorem 4.15 and Lemma 4.12 of Arora [1]. We skip the details due to lack of space. � 4 Conclusion Recently, Samorodnitsky and Trevisan [25] showed that, assuming the Unique Games Conjecture of Khot =-=[21]-=-, it is hard to approximate MaxClique in degree d graphs better than d/polylog(d). This suggests that MaxClique on general graphs could be hard to approximate within n/polylog(n). We think it is a cha... |

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Citation Context ...e chromatic number of a graph G, denoted by χ(G), is the minimum number of colors required to color the vertices of G such that for any edge, its end-points receive different colors. Feige and Kilian =-=[14]-=- showed the connection between randomized PCPs and inapproximability of chromatic number. Using this result, they prove that it is hard to approximate chromatic number within a factor better than n1−ɛ... |

171 | Efficient probabilistically checkable proofs and applications to approximations
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Citation Context ...ness Factor Assumption Feige et al. [12] 2 log1−ɛ n , for any ɛ > 0 NP � DTIME(2 (log n) O(1) ) Arora and Safra [3] (log n)1/2−ɛ 2 P �= NP Arora et al. [2] n c , for some c > 0 P �= NP Bellare et al. =-=[5]-=- n 1/30 NP � BPP Bellare et al. [5] n 1/25 NEXP � BPEXP Feige and Kilian [13] n 1/15 NP � coRP Bellare and Sudan [6] n 1/4−ɛ NP � ZPP Bellare et al. [4] n 1/3−ɛ NP � ZPP H˚astad [17] n 1/2−ɛ NP � coRP... |

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Citation Context ...or arbitrarily small δ > 0. This implies a hardness factor of n1/4−ɛ for MaxClique. The result was improved by Bellare et al. [4] by constructing PCPs with 2 + δ amortized free bits. Finally, H˚astad =-=[16]-=- gave a construction that achieved an amortized free bit complexity of δ for any constant δ > 0, proving n1−ɛ hardness for MaxClique. Simpler proofs of H˚astad’s result were given by Samorodnitsky and... |

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Citation Context ... showing that the Lovász θ-function n does not approximate MaxClique better than 2 √ c log n , where c > 0 is a constant. The first inapproximability result for MaxClique was obtained by Feige et al. =-=[12]-=- who discovered the connection between hardness of approximation and Probabilistically Checkable Proofs(PCPs). We summarize the progress on showing hardness results for MaxClique in Table 1. Let PCPc,... |

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(Show Context)
Citation Context ...known as the PCP Theorem. Since then, many different PCP constructions for languages in NP have led to inapproximability results for several other problems in addition to MaxClique. Bellare and Sudan =-=[6]-=- defined a parameter called amortized free bits for PCPs. They showed that if problems in NP have PCPs that use logarithmic randomness and ¯ f amortized free bits, then MaxClique is hard to approximat... |

86 |
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(Show Context)
Citation Context ...nstruction that achieved an amortized free bit complexity of δ for any constant δ > 0, proving n1−ɛ hardness for MaxClique. Simpler proofs of H˚astad’s result were given by Samorodnitsky and Trevisan =-=[24]-=- and H˚astad and Wigderson [19]. Both these results achieved amortized free bit complexity δ and amortized query complexity 1 + δ for any constant δ > 0 (both parameters are optimal). Khot [20] showed... |

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Citation Context ... approximation ratio as bad as 2c√log n for some constant c > 0. It would be interesting to prove the same lower bound for any polynomial time algorithm. It would also fit in nicely with Trevisan’s d =-=[27]-=- lower bound of 2O(√log d) for MaxClique on degree d graphs (d thought of n as a large constant). Blum [7] showed that if there exists a factor 2 √ b log n quasipolynomial time approximation algorithm... |

74 |
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Citation Context ...t. We instead use a technique based on the Sum-Check Protocol. Reducing the Size of Equations in a Min-Lin-Deletion Instance We use the Sum-Check Protocol combined with the Low-degree Test (see Arora =-=[1]-=-) to construct a PCP verifier for Min-Lin-Deletion. A typical constraint of the Min-Lin-Deletion instance looks like: x1 ⊕ x2 ⊕ . . . ⊕ xn = a (1) The verifier tries to verify that this constraint is ... |

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Citation Context ...ved factor for Min-3Lin-Deletion Hardness of Approximation Result for Min-Lin-Deletion The tensoring operation we use on a Min-Lin-Deletion instance is similar to an operation defined by Dumer et al. =-=[8]-=- on linear codes. Tensoring involves taking all possible pairs of linear equations, computing their “product”, and then replacing the terms of the form xixj with xij and xi with xii to get back a line... |

51 | Gowers uniformity, influence of variables, and PCPs
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Citation Context ...analysis of Sum-Check Protocol and Low-degree Test are based on Theorem 4.15 and Lemma 4.12 of Arora [1]. We skip the details due to lack of space. � 4 Conclusion Recently, Samorodnitsky and Trevisan =-=[25]-=- showed that, assuming the Unique Games Conjecture of Khot [21], it is hard to approximate MaxClique in degree d graphs better than d/polylog(d). This suggests that MaxClique on general graphs could b... |

47 |
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Citation Context ...ME(2 (log n) O(1) ) Arora and Safra [3] (log n)1/2−ɛ 2 P �= NP Arora et al. [2] n c , for some c > 0 P �= NP Bellare et al. [5] n 1/30 NP � BPP Bellare et al. [5] n 1/25 NEXP � BPEXP Feige and Kilian =-=[13]-=- n 1/15 NP � coRP Bellare and Sudan [6] n 1/4−ɛ NP � ZPP Bellare et al. [4] n 1/3−ɛ NP � ZPP H˚astad [17] n 1/2−ɛ NP � coRP H˚astad [16] n 1−ɛ NP � ZPP Engebretsen and n 2 O(log n/√ log log n) NP � Ho... |

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Citation Context ...als the size of the largest clique. For random graphs, the gap between these two values can be as bad as Ω( √ n/ log n). The conjecture says that this may essentially be the worst possible gap. Feige =-=[10]-=- disproved the conjecture by showing that the Lovász θ-function n does not approximate MaxClique better than 2 √ c log n , where c > 0 is a constant. The first inapproximability result for MaxClique w... |

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Citation Context ...roximation factor of O( n(log log n)2 log3 ), where n is n the number of vertices in the input graph. It was conjectured that the Lovász θ-function might be a O( √ n) approximation for MaxClique (see =-=[22]-=- for details). Since the Lovász θ-function can be computed to any desired degree of accuracy in polynomial time, the conjecture implies a O( √ n) approximation algorithm for ⋆ The research is partly s... |

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Citation Context ... NP Bellare et al. [5] n 1/30 NP � BPP Bellare et al. [5] n 1/25 NEXP � BPEXP Feige and Kilian [13] n 1/15 NP � coRP Bellare and Sudan [6] n 1/4−ɛ NP � ZPP Bellare et al. [4] n 1/3−ɛ NP � ZPP H˚astad =-=[17]-=- n 1/2−ɛ NP � coRP H˚astad [16] n 1−ɛ NP � ZPP Engebretsen and n 2 O(log n/√ log log n) NP � Holmerin [9] ZPTIME(2 O(log n(log log n)3/2 ) ) Khot [20] n 2 (log n)1−γ′ , for some γ′ (log n)O(1) > 0 NP ... |

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Citation Context ...h. It has been a pivotal problem in the field of inapproximability, leading to the development of many important tools in this field. The best approximation algorithm for MaxClique was given by Feige =-=[11]-=-. The algorithm achieves an approximation factor of O( n(log log n)2 log3 ), where n is n the number of vertices in the input graph. It was conjectured that the Lovász θ-function might be a O( √ n) ap... |

22 | Algorithms for Approximate Graph Coloring
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Citation Context ...lower bound for any polynomial time algorithm. It would also fit in nicely with Trevisan’s d [27] lower bound of 2O(√log d) for MaxClique on degree d graphs (d thought of n as a large constant). Blum =-=[7]-=- showed that if there exists a factor 2 √ b log n quasipolynomial time approximation algorithm for MaxClique, then there exists a quasi-polynomial time algorithm to color a 3-colorable graph with nɛ c... |

19 | Hardness of Approximation of the Balanced Complete Bipartite Subgraph Problem
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Citation Context ...e graph with nɛ colors, where ɛ = O(1/b). Therefore, strong lower bounds for MaxClique give evidence that the graph coloring problem is hard. Another motivation comes from a result of Feige and Kogan =-=[15]-=- who showed that if the balanced bipartite clique problemsn can be approximated within a constant factor, then there is a 2O(√log n) approximation for MaxClique. We refer to Srinivasan’s paper [26] fo... |

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Citation Context ...rtized free bit complexity of δ for any constant δ > 0, proving n1−ɛ hardness for MaxClique. Simpler proofs of H˚astad’s result were given by Samorodnitsky and Trevisan [24] and H˚astad and Wigderson =-=[19]-=-. Both these results achieved amortized free bit complexity δ and amortized query complexity 1 + δ for any constant δ > 0 (both parameters are optimal). Khot [20] showed that MaxClique cannot be appro... |

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Citation Context ... NP � coRP Bellare and Sudan [6] n 1/4−ɛ NP � ZPP Bellare et al. [4] n 1/3−ɛ NP � ZPP H˚astad [17] n 1/2−ɛ NP � coRP H˚astad [16] n 1−ɛ NP � ZPP Engebretsen and n 2 O(log n/√ log log n) NP � Holmerin =-=[9]-=- ZPTIME(2 O(log n(log log n)3/2 ) ) Khot [20] n 2 (log n)1−γ′ , for some γ′ (log n)O(1) > 0 NP � ZPTIME(2 ) Table 1. Hardness Results for MaxClique The chromatic number of a graph G, denoted by χ(G), ... |

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Citation Context ...−ɛ unless NP ⊆ ZPP. They constructed PCPs with 3 + δ amortized free bits for arbitrarily small δ > 0. This implies a hardness factor of n1/4−ɛ for MaxClique. The result was improved by Bellare et al. =-=[4]-=- by constructing PCPs with 2 + δ amortized free bits. Finally, H˚astad [16] gave a construction that achieved an amortized free bit complexity of δ for any constant δ > 0, proving n1−ɛ hardness for Ma... |

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Citation Context ...an [15] who showed that if the balanced bipartite clique problemsn can be approximated within a constant factor, then there is a 2O(√log n) approximation for MaxClique. We refer to Srinivasan’s paper =-=[26]-=- for several other interesting consequences of proving strong hardness results for MaxClique. Hardness Factor Assumption Feige et al. [12] 2 log1−ɛ n , for any ɛ > 0 NP � DTIME(2 (log n) O(1) ) Arora ... |

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