The approximability of three-valued MAX CSP

In the maximum constraint satisfaction problem (Max CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so as to maximize the number (or the total weight, for the weighted case) of satisfied constraints. This problem is NP-hard in general, and, therefore, it is natural to study how restricting the allowed types of constraints affects the approximability of the problem. It is known that every Boolean (that is, two-valued) Max CSP problem with a finite set of allowed constraint types is either solvable exactly in polynomial time or else APX-complete (and hence can have no polynomial time approximation scheme unless P=NP. It has been an open problem for several years whether this result can be extended to non-Boolean Max CSP, which is much more difficult to analyze than the Boolean case. In this paper, we make the first step in this direction by establishing this result for Max CSP over a three-element domain. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description uses the well-known algebraic combinatorial property of supermodularity. We also show that every hard three-valued Max CSP problem contains, in a certain specified sense, one of the two basic hard Max CSP problems which are the Maximum k-colourable subgraph problems for k=2,3

The approximability of MAX CSP with fixed-value constraints

In the maximum constraint satisfaction problem (MAX CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given finite domain to the variables so as to maximize the number (or the total weight, for the weighted case) of satisfied constraints. This problem is NP-hard in general, and, therefore, it is natural to study how restricting the allowed types of constraints affects the approximability of the problem. In this paper, we show that any MAX CSP problem with a finite set of allowed constraint types, which includes all fixed-value constraints (i.e., constraints of the form x=a), is either solvable exactly in polynomial-time or else is APX-complete, even if the number of occurrences of variables in instances are bounded. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description relies on the well-known algebraic combinatorial property of supermodularity

A Combinatorial Algorithm for MAX CSP

We consider the problem max csp over multi-valued domains with variables ranging over sets of size si ^ s and constraints involving kj ^ k variables. We study two algorithms with approximation ratios A and B, respectively, so we obtain a solution with approximation ratio max(A; B). The first algorithm is based on the linear programming algorithm of Serna, Trevisan, and Xhafa [12] and gives ratio A which is bounded below by s 1\Gamma k. For k=2, our bound in terms of the individual set sizes is the minimum over all constraints involving two variables of (