Random MAX SAT, random MAX CUT, and their phase transitions

Abstract

With random inputs, certain decision problems undergo a "phase transition." We prove similar behavior in an optimization context. Given a conjunctive normal form (CNF) formula F on n variables and with m k-variable clauses, denote by max F the maximum number of clauses satisfiable by a single assignment of the variables. (Thus the decision problem k-SAT is to determine if max F is equal to m.) With the formula F chosen at random, the expectation of max F is trivially bounded by (3/4)m ≤ double-struck E sign max F ≤ m. We prove that for random formulas with m = ⌊cn⌋ clauses: for constants c < 1, double-struck E sign max F is ⌊cn⌋ - θ(1/n); for large c, it approaches ((3/4)c + θ(√c))n; and in the "window" c = 1 + θ(n -1/3), it is cn θ(1). Our full results are more detailed, but this already shows that the optimization problem MAX 2-SAT undergoes a phase transition just as the 2-SAT decision problem does, and at the same critical value c = 1. Most of our results are established without reference to the analogous propositions for decision 2-SAT, and can be used to reproduce them. We consider "online" versions of MAX 2-SAT, and show that for one version the obvious greedy algorithm is optimal; all other natural questions remain open. We can extend only our simplest MAX 2-SAT results to MAX k-SAT, but we conjecture a "MAX k-SAT limiting function conjecture" analogous to the folklore "satisfiability threshold conjecture," but open even for k = 2. Neither conjecture immediately implies the other, but it is natural to further conjecture a connection between them. We also prove analogous results for random MAX CUT. © 2004 Wiley Periodicals, Inc.