Approximating Rectangles by Juntas and Weakly-Exponential Lower Bounds for LP Relaxations of CSPs

TL;DR

This paper establishes that sub-exponential LP relaxations are as powerful as high-round Sherali-Adams hierarchies for CSPs, providing nearly exponential lower bounds for problems like MAX-CUT and MAX-3SAT.

Contribution

It introduces a new structural result on high-entropy rectangles and extends communication complexity techniques to prove strong lower bounds for LP relaxations of CSPs.

Findings

Sub-exponential LP relaxations are as powerful as high-round Sherali-Adams hierarchies.

Sub-exponential size lower bounds are proved for LP relaxations beating random guessing.

A new structural result on high-entropy rectangles is developed.

Abstract

We show that for constraint satisfaction problems (CSPs), sub-exponential size linear programming relaxations are as powerful as $n^{\Omega(1)}$-rounds of the Sherali-Adams linear programming hierarchy. As a corollary, we obtain sub-exponential size lower bounds for linear programming relaxations that beat random guessing for many CSPs such as MAX-CUT and MAX-3SAT. This is a nearly-exponential improvement over previous results, previously, it was only known that linear programs of size $n^{o(\log n)}$ cannot beat random guessing for any CSP (Chan-Lee-Raghavendra-Steurer 2013). Our bounds are obtained by exploiting and extending the recent progress in communication complexity for "lifting" query lower bounds to communication problems. The main ingredient in our results is a new structural result on "high-entropy rectangles" that may of independent interest in communication complexity.