From Weak to Strong LP Gaps for all CSPs

TL;DR

This paper demonstrates that basic LP relaxations for all CSPs are essentially as powerful as high-level Sherali-Adams relaxations, establishing a fundamental limit on LP-based approximations and simplifying previous resistance results.

Contribution

It proves that basic LP relaxations match the power of \\Omega(\\frac{\\log n}{\\log \\log n}) levels of Sherali-Adams for all CSPs, and simplifies and strengthens existing approximation resistance bounds.

Findings

Basic LP relaxations are as strong as high-level Sherali-Adams relaxations.

Polynomial size LPs are no stronger than basic LP relaxations.

Strengthened bounds on approximation resistance for LPs.

Abstract

We study the approximability of constraint satisfaction problems (CSPs) by linear programming (LP) relaxations. We show that for every CSP, the approximation obtained by a basic LP relaxation, is no weaker than the approximation obtained using relaxations given by $\Omega\left(\frac{\log n}{\log \log n}\right)$ levels of the Sherali-Adams hierarchy on instances of size $n$. It was proved by Chan et al. [FOCS 2013] that any polynomial size LP extended formulation is no stronger than relaxations obtained by a super-constant levels of the Sherali-Adams hierarchy.. Combining this with our result also implies that any polynomial size LP extended formulation is no stronger than the basic LP. Using our techniques, we also simplify and strengthen the result by Khot et al. [STOC 2014] on (strong) approximation resistance for LPs. They provided a necessary and sufficient condition under which $\Omega(\log \log n)$ levels of the Sherali-Adams hierarchy cannot achieve an approximation better than a random assignment. We simplify their proof and strengthen the bound to $\Omega\left(\frac{\log n}{\log \log n}\right)$ levels.