A New PCP Outer Verifier with Applications to Homogeneous Linear Equations and Max-Bisection

摘要

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 not is contained in ∩_(ε > 0) DTIME(2~(n~ε)). 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 un-correlated. 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.