On the approximability of clique and related maximization problems

We consider approximations of the form n1-o(1) for the Maximum Clique problem, where n is the number of vertices in the input graph and where the "o(1)" term goes to zero as n increases. We show that sufficiently strong negative results for such problems, which we call strong inapproximability results, have interesting consequences for exact computation. A simple sampling method underlies most of our results.

[1]  Lance Fortnow,et al.  Time-Space Tradeoffs for Satisfiability , 2000, J. Comput. Syst. Sci..

[2]  Magnús M. Halldórsson,et al.  Journal of Graph Algorithms and Applications Approximations of Weighted Independent Set and Hereditary Subset Problems , 2022 .

[3]  Uriel Feige,et al.  Randomized graph products, chromatic numbers, and the Lovász ϑ-function , 1997, Comb..

[4]  Jonas Holmerin,et al.  Clique Is Hard to Approximate within n1-o(1) , 2000, ICALP.

[5]  Aravind Srinivasan,et al.  Chernoff-Hoeffding bounds for applications with limited independence , 1995, SODA '93.

[6]  László Lovász,et al.  Interactive proofs and the hardness of approximating cliques , 1996, JACM.

[7]  Aravind Srinivasan,et al.  Improved Approximation Guarantees for Packing and Covering Integer Programs , 1999, SIAM J. Comput..

[8]  Magnús M. Halldórsson,et al.  Approximations of Independent Sets in Graphs , 1998, APPROX.

[9]  Michael Krivelevich,et al.  Approximating the Independence Number and the Chromatic Number in Expected Polynominal Time , 2000, ICALP.

[10]  Subhash Khot,et al.  Improved inapproximability results for MaxClique, chromatic number and approximate graph coloring , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[11]  J. Håstad Clique is hard to approximate withinn1−ε , 1999 .

[12]  Uriel Feige,et al.  Randomized graph products, chromatic numbers, and Lovasz j-function , 1995, STOC '95.

[13]  Prabhakar Raghavan,et al.  Probabilistic construction of deterministic algorithms: Approximating packing integer programs , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[14]  A. Blum ALGORITHMS FOR APPROXIMATE GRAPH COLORING , 1991 .

[15]  Panos M. Pardalos,et al.  On maximum clique problems in very large graphs , 1999, External Memory Algorithms.

[16]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[17]  Ravi B. Boppana,et al.  Approximating maximum independent sets by excluding subgraphs , 1992, BIT Comput. Sci. Sect..

[18]  AroraSanjeev,et al.  Probabilistic checking of proofs , 1998 .

[19]  Luca Trevisan,et al.  A PCP characterization of NP with optimal amortized query complexity , 2000, STOC '00.

[20]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.