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

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