On hardness of approximating the parameterized clique problem

Subhash Khot, Igor Shinkar

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

In the Gap-clique(k; k/2 ) problem, the input is an n-vertex graph G, and the goal is to decide whether G contains a clique of size k or contains no clique of size k/2 . It is an open question in the study of fixed parameterized tractability whether the Gap-clique(k; k/2 ) problem is fixed parameter tractable, i.e., whether it has an algorithm that runs in time f(k) n, where f(k) is an arbitrary function of the parameter k and the exponent is a constant independent of k. In this paper, we give some evidence that the problem Gap-clique(k; k/2 ) is not fixed parameter tractable. Speciff-cally, we define a constraint satisfaction problem, which we call Deg-2-sat, where the input is a system of k0 quadratic equations in k0 variables over a finite field F of size n0, and the goal is to decide whether there is a solution in F that satisfies all the equations simultaneously. The main result in this paper is an "FPT-reduction" from Deg-2-sat to the Gap-clique(k; k/2 ) problem. If one were to hypothesize that the Deg-2-sat problem is not fixed parameter tractable, then our reduction would imply that the Gap-clique(k; k/2 ) problem is not fixed parameter tractable either. The reduction relies on the algebraic techniques used in proof of the PCP theorem.

Original languageEnglish (US)
Title of host publicationITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science
PublisherAssociation for Computing Machinery, Inc
Pages37-45
Number of pages9
ISBN (Electronic)9781450340571
DOIs
StatePublished - Jan 14 2016
Event7th ACM Conference on Innovations in Theoretical Computer Science, ITCS 2016 - Cambridge, United States
Duration: Jan 14 2016Jan 16 2016

Publication series

NameITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science

Other

Other7th ACM Conference on Innovations in Theoretical Computer Science, ITCS 2016
CountryUnited States
CityCambridge
Period1/14/161/16/16

Keywords

  • Clique
  • Fixed parameter tractability
  • Hardness of approximation
  • Parameterized complexity

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Khot, S., & Shinkar, I. (2016). On hardness of approximating the parameterized clique problem. In ITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science (pp. 37-45). (ITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science). Association for Computing Machinery, Inc. https://doi.org/10.1145/2840728.2840733