A high-performance analog Max-SAT solver and its application to Ramsey numbers

TL;DR

This paper presents a novel continuous-time analog MaxSAT solver that predicts optimal solutions via escape rate dynamics and applies it to estimate the Ramsey number R(5,5), achieving new bounds.

Contribution

It introduces a high-performance analog MaxSAT solver utilizing escape rate dynamics and applies it to the Ramsey number problem, providing new insights and bounds.

Findings

Predicts global optima using escape rate scaling.

Successfully applied to hard MaxSAT problems.

Provides new bounds for the Ramsey number R(5,5).

Abstract

We introduce a continuous-time analog solver for MaxSAT, a quintessential class of NP-hard discrete optimization problems, where the task is to find a truth assignment for a set of Boolean variables satisfying the maximum number of given logical constraints. We show that the scaling of an invariant of the solver's dynamics, the escape rate, as function of the number of unsatisfied clauses can predict the global optimum value, often well before reaching the corresponding state. We demonstrate the performance of the solver on hard MaxSAT competition problems. We then consider the two-color Ramsey number $R(m,m)$ problem, translate it to SAT, and apply our algorithm to the still unknown $R(5,5)$. We find edge colorings without monochromatic 5-cliques for complete graphs up to 42 vertices, while on 43 vertices we find colorings with only two monochromatic 5-cliques, the best coloring found so far, supporting the conjecture that $R(5,5) = 43$.