An Empirical Study of MAX-2-SAT Phase Transitions

The decision version of the maximum satisfiability problem (MAX-SAT) is stated as follows: Given a set S of propositional clauses and an integer g, decide if there exists a truth assignment that falsifies at most g clauses in S, where g is called the allowance for false clauses. We conduct an extensive experiment on over a million of random instances of 2-SAT and identify statistically the relationship between g, n (number of variables) and m (number of clauses). In our experiment, we apply an efficient decision procedure based on the branch-and-bound method. The statistical data of the experiment confirm not only the “scaling window” of MAX-2-SAT discovered by Chayes, Kim and Borgs, but also the recent results of Coppersmith et al. While there is no easy-hard-easy pattern for the complexity of 2-SAT at the phase transition, we show that there is such a pattern for the decision problem of MAX-2-SAT associated with the phase transition. We also identify that the hardest problems are among those with high allowance for false clauses but low number of clauses.