Phase Transition in Matched Formulas and a Heuristic for Biclique Satisfiability

A matched formula is a CNF formula whose incidence graph admits a matching which matches a distinct variable to every clause. We study phase transition in a context of matched formulas and their generalization of biclique satisfiable formulas. We have performed experiments to find a phase transition of property “being matched” with respect to the ratio \(m/n\) where \(m\) is the number of clauses and \(n\) is the number of variables of the input formula \(\varphi \). We compare the results of experiments to a theoretical lower bound which was shown by Franco and Van Gelder [11]. Any matched formula is satisfiable, and it remains satisfiable even if we change polarities of any literal occurrences. Szeider [17] generalized matched formulas into two classes having the same property—var-satisfiable and biclique satisfiable formulas. A formula is biclique satisfiable if its incidence graph admits covering by pairwise disjoint bounded bicliques. Recognizing if a formula is biclique satisfiable is NP-complete. In this paper we describe a heuristic algorithm for recognizing whether a formula is biclique satisfiable and we evaluate it by experiments on random formulas. We also describe an encoding of the problem of checking whether a formula is biclique satisfiable into SAT and we use it to evaluate the performance of our heuristic.