Maximum Constraint Satisfaction: Relaxations and Upper Bounds

Abstract

Constraint satisfaction problems ( CSPs) may be viewed as a generalization of satisfiability (SAT) to include cases where, instead of taking binary values only (0– 1 or true-false) the variables may take on a finite number (> 2) of given possible values. For an infeasible CSP, a relevant question, both theoretically and practically, is to determine an assignment of values to variables such that the number of satisfied constraints is the largest possible. This is the so-called maximum constraint satisfaction problem (MAX-CSP), which generalizes in a natural way maximum satisfiability (MAX-SAT). Since MAX-2SAT is NP-complete (see e.g. [12, pp. 259–260]) even the subclass of MAX-CSP corresponding to binary CSPs (those problems with constraints involving pairs of variables only) is NP-complete. Therefore, for very large instances such as those arising from practical applications (e.g. the RLFAP discussed below) one can only hope for approximate solutions using some of the currently available heuristic approaches such as: simulated annealing, tabu search, genetic algorithms, or local search of various kinds. However, for many applications, getting an approximate solution without any information about the quality of this solution (e.g. measured by the difference between the cost of this solution and the optimal cost) may be of little value. We address in this paper the problem of computing upper bounds to the optimum cost of MAX-CSP problems from which estimates on the quality of heuristic solutions can be derived. The article is organized as follows. Basic definitions about CSPs and MAX-CSPs are recalled in the second section. Modeling the so-called radio link frequency assignment problem (RLFAP) in terms of CSP and MAX-CSP is addressed in the third section. Then we present a general class of relaxations for MAX-CSP problems and its specialization to the computation of MAX-CSP bounds for RLFAP. Finally results of extensive computational experiments carried out on series of both real test problems and realistic randomly generated test problems are presented. To our knowledge, this is the first time extensive computational results of this kind are reported for such large scale MAX-CSP problems.