The approximability of MAX CSP with fixed-value constraints

TL;DR

This paper characterizes the complexity of MAX CSP problems with fixed-value constraints, showing they are either polynomial-time solvable or APX-complete, based on algebraic properties, thus clarifying the boundary of approximability.

Contribution

It provides a complete classification of MAX CSP problems with fixed-value constraints, identifying conditions for polynomial-time solvability using supermodularity.

Findings

All such MAX CSP problems are either in P or APX-complete.

A simple algebraic criterion based on supermodularity determines tractability.

The paper offers a comprehensive description of polynomial-time solvable cases.

Abstract

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.