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 pro blem is given by m constraints applied to n Boolean variables. Different co mputational problems arise from constraint satisfaction problems depending on the nature of the underlying constraints as well as on the goal of the o ptimization task. Here we consider four possible goals: Max CSP (Min CSP) i s the class of problems where the goal is to nd an assignment maximizing th e number of satis ed constraints (minimizing the number of unsatisfied cons traints). Max Ones (Min Ones) is the class of optimization problems where t he goal is to nd an assignment satisfying all constraints with maximum (min imum) 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 o f finite Boolean functions that describe the possible constraints that may be used. Tight bounds on the approximability of every problem in Max CSP were obtain ed by Creignou [J. Comput. System Sci., 51 (1995), pp. 511-522]. In this wo rk we determine tight bounds on the approximability (i.e., the ratio to wit hin which each problem may be approximated in polynomial time) of every pro blem in Max Ones, Min CSP, and Min Ones. Combined with the result of Creign ou, this completely classifies all optimization problems derived from Boole an constraint satisfaction. Our results capture a diverse collection of opt imization problems such as MAX 3-SAT, Max Cut, Max Clique, Min Cut, Nearest Codeword, etc. Our results unify recent results on the (in-) approximabili ty of these optimization problems and yield a compact presentation of most known results. Moreover, these results provide a formal basis to many state ments on the behavior of natural optimization problems that have so far bee n observed only empirically.