Algorithms and Certificates for Boolean CSP Refutation: "Smoothed is no harder than Random"

TL;DR

This paper introduces an algorithm for strongly refuting smoothed Boolean CSP instances, matching the efficiency of algorithms for random CSPs, and establishes a new connection to hypergraph covers that resolves Feige's conjecture.

Contribution

It presents a novel algorithm for smoothed CSP refutation with optimal trade-offs and links it to hypergraph even covers, resolving a longstanding conjecture.

Findings

Algorithm matches state-of-the-art for random CSPs

Resolves Feige's 2008 hypergraph conjecture

Polynomial-size refutation witnesses below spectral threshold

Abstract

We present an algorithm for strongly refuting smoothed instances of all Boolean CSPs. The smoothed model is a hybrid between worst and average-case input models, where the input is an arbitrary instance of the CSP with only the negation patterns of the literals re-randomized with some small probability. For an $n$-variable smoothed instance of a $k$-arity CSP, our algorithm runs in $n^{O(\ell)}$ time, and succeeds with high probability in bounding the optimum fraction of satisfiable constraints away from $1$, provided that the number of constraints is at least $\tilde{O}(n) (\frac{n}{\ell})^{\frac{k}{2} - 1}$. This matches, up to polylogarithmic factors in $n$, the trade-off between running time and the number of constraints of the state-of-the-art algorithms for refuting fully random instances of CSPs [RRS17]. We also make a surprising new connection between our algorithm and even covers in hypergraphs, which we use to positively resolve Feige's 2008 conjecture, an extremal combinatorics conjecture on the existence of even covers in sufficiently dense hypergraphs that generalizes the well-known Moore bound for the girth of graphs. As a corollary, we show that polynomial-size refutation witnesses exist for arbitrary smoothed CSP instances with number of constraints a polynomial factor below the "spectral threshold" of $n^{k/2}$, extending the celebrated result for random 3-SAT of Feige, Kim and Ofek [FKO06].