A new upper bound for 3-SAT

TL;DR

This paper establishes a new upper bound of approximately 4.49 for the clause-to-variable ratio in random 3-CNF formulas beyond which they are almost surely unsatisfiable, improving previous bounds.

Contribution

The authors present a novel probabilistic approach that tightens the known upper bound for unsatisfiability in random 3-SAT formulas.

Findings

New upper bound of 4.4898 for clause-to-variable ratio

Almost sure unsatisfiability above this ratio

Probabilistic techniques of independent interest

Abstract

We show that a randomly chosen 3-CNF formula over n variables with clauses-to-variables ratio at least 4.4898 is, as n grows large, asymptotically almost surely unsatisfiable. The previous best such bound, due to Dubois in 1999, was 4.506. The first such bound, independently discovered by many groups of researchers since 1983, was 5.19. Several decreasing values between 5.19 and 4.506 were published in the years between. The probabilistic techniques we use for the proof are, we believe, of independent interest.