A new PCP outer verifier with applications to homogeneous linear equations and Max-Bisection

Jonas Holmerin, Subhash Khot

Research output: Contribution to journalConference articlepeer-review

Abstract

We show an optimal hardness result for the following problem : Given a system of homogeneous linear equations over GF(2) with 3 variables per equation, find a balanced assignment that satisfies maximum number of equations. For arbitrarily small constant ζ > 0, we show that it is hard to determine (in polynomial time) whether such a system has a balanced assignment that satisfies 1 - ζ fraction of equations or there is no balanced assignment that satisfies more than 1/2 + ζ fraction of equations, As a corollary, we show that it is hard to approximate (in polynomial time) the Max-Bisection problem within factor 16/15 - ζ These hardness results hold under the assumption NP ⊈ > 0 DTIME(2). Our results are obtained via a construction of a new PCP outer verifier that has a mixing property and a smoothness property. These properties are crucial in the analysis of the inner verifier. No previous outer verifier can achieve both these properties simultaneously. An outer verifier is essentially a 2-query PCP over a large alphabet. Loosely speaking, the mixing property says that the locations of the two queries read by the verifier are uncorrelated. The smoothness property says that the verifier's acceptance predicate is close to being a bijective predicate. Our construction relies on the algebraic techniques used to prove the PCP Theorem. This is in contrast with all earlier constructions that use the PCP Theorem as a black-box. The progress in inapproximability theory seems to require new ideas for building outer verifiers and our construction takes a first step in that direction.

Original languageEnglish (US)
Pages (from-to)11-20
Number of pages10
JournalConference Proceedings of the Annual ACM Symposium on Theory of Computing
DOIs
StatePublished - 2004
EventProceedings of the 36th Annual ACM Symposium on Theory of Computing - Chicago, IL, United States
Duration: Jun 13 2004Jun 15 2004

Keywords

  • Hardness of Approximation
  • Linear Equations
  • Max-Bisection
  • PCPs

ASJC Scopus subject areas

  • Software

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