A universally fastest algorithm for Max 2-Sat, Max 2-CSP, and everything in between

Abstract

In this paper we introduce "hybrid" Max 2-CSP formulas consisting of "simple clauses", namely conjunctions and disjunctions of pairs of variables, and general 2-variable clauses, which can be any integer-valued functions of pairs of boolean variables. This allows an algorithm to use both efficient reductions specific to AND and OR clauses, and other powerful reductions that require the general CSP setting. We use new reductions introduced here, and recent reductions such as "clause-learning" and "2-reductions" generalized to our setting's mixture of simple and general clauses. Parametrizing an instance by the fraction p of non-simple clauses, we give an exact (exponential-time) algorithm that is the fastest known polynomial-space algorithm for p=0 (which includes the well-studied Max 2-Sat problem but also instances with arbitrary mixtures of AND and OR clauses); the only efficient algorithm for mixtures of AND, OR, and general integer-valued clauses; and tied for fastest for general Max 2-CSP (p=1). Since a pure 2-Sat input instance may be transformed to a general CSP instance in the course of being solved, the algorithm's efficiency and generality go hand in hand. Our algorithm analysis and optimization are a variation on the familiar measure-and-conquer approach, resulting in an optimizing mathematical program that is convex not merely quasi-convex, and thus can be solved efficiently and with a certificate of optimality. We produce a family of running-time upper-bound formulas, each optimized for instances with a particular value of p but valid for all instances.