Extended Formulation Lower Bounds for Refuting Random CSPs

TL;DR

This paper establishes subexponential lower bounds on the size of linear programming relaxations needed to refute random CSP instances, advancing understanding of their computational intractability.

Contribution

It introduces a pseudocalibration technique to derive LP lower bounds directly from planted distributions, bypassing traditional integrality gap constructions.

Findings

LP refutation lower bounds are exponential in problem size

Pseudocalibration effectively yields LP lower bounds from planted models

Subexponential Sherali-Adams bounds are established for random CSPs

Abstract

Random constraint satisfaction problems (CSPs) such as random $3$-SAT are conjectured to be computationally intractable. The average case hardness of random $3$-SAT and other CSPs has broad and far-reaching implications on problems in approximation, learning theory and cryptography. In this work, we show subexponential lower bounds on the size of linear programming relaxations for refuting random instances of constraint satisfaction problems. Formally, suppose $P : \{0,1\}^k \to \{0,1\}$ is a predicate that supports a $t-1$-wise uniform distribution on its satisfying assignments. Consider the distribution of random instances of CSP $P$ with $m = \Delta n$ constraints. We show that any linear programming extended formulation that can refute instances from this distribution with constant probability must have size at least $\Omega\left(\exp\left(\left(\frac{n^{t-2}}{\Delta^2}\right)^{\frac{1-\nu}{k}}\right)\right)$ for all $\nu > 0$. For example, this yields a lower bound of size $\exp(n^{1/3})$ for random $3$-SAT with a linear number of clauses. We use the technique of pseudocalibration to directly obtain extended formulation lower bounds from the planted distribution. This approach bypasses the need to construct Sherali-Adams integrality gaps in proving general LP lower bounds. As a corollary, one obtains a self-contained proof of subexponential Sherali-Adams LP lower bounds for these problems. We believe the result sheds light on the technique of pseudocalibration, a promising but conjectural approach to LP/SDP lower bounds.