The approximability of constraint satisfaction problems

We study optimization problems that may be expressed as Boolean constraint satisfaction problems. An instance of a Boolean constraint satisfaction problem is given by m constraints applied to n Boolean variables. Different computational problems arise from constraint satisfaction problems depending on the nature of the underlying constraints as well as on the goal of the optimization task. Here we consider four possible goals: Max CSP (Min CSP) is the class of problems where the goal is to nd an assignment maximizing the number of satis ed constraints (minimizing the number of unsatisfied constraints). Max Ones (Min Ones) is the class of optimization problems where the goal is to nd an assignment satisfying all constraints with maximum (minimum) number of variables set to 1. Each class consists of infinitely many problems and a problem within a class is specified by a finite collection of finite Boolean functions that describe the possible constraints that may be used. Tight bounds on the approximability of every problem in Max CSP were obtained by Creignou [J. Comput. System Sci., 51 (1995), pp. 511-522]. In this work we determine tight bounds on the approximability (i.e., the ratio to within which each problem may be approximated in polynomial time) of every problem in Max Ones, Min CSP, and Min Ones. Combined with the result of Creignou, this completely classifies all optimization problems derived from Boolean constraint satisfaction. Our results capture a diverse collection of optimization problems such as MAX 3-SAT, Max Cut, Max Clique, Min Cut, Nearest Codeword, etc. Our results unify recent results on the (in-) approximability of these optimization problems and yield a compact presentation of most known results. Moreover, these results provide a formal basis to many statements on the behavior of natural optimization problems that have so far been observed only empirically.