On the approximability of clique and related maximization problems

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Abstract

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.

Keywords

Inapproximability
Approximation algorithms
Clique
Independent set
Packing integer programs
Random sampling

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A preliminary version of this work appears as “The value of strong inapproximability results for clique” in the Proceedings of the ACM Symposium on Theory of Computing, 2000, pp. 144–152.

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Most of this work was done while at Bell Laboratories, Lucent Technologies; part of this work was supported by NSF Award CCR-0208005.