Probabilistic Estimates for the Generalized Maximum Satisfiability Problem

The generalized maximum satisfiability problem (GMAXSAT) deals with variables taking their values in a finite set. A set of logical clauses is given and the goal is to find an assignment of values to the variables, minimizing the number j3 of unsatisfied clauses. For randomly generated instances of a uniform type we study the distribution of p, as well as the distribution of the maximal size a of a satisfiable subproblem, by means of the first and second moment method (Spencer, 1987). Numerical estimates for the distribution of tl and p are given for some instances. In relation with the asymptotic behavior, we show that CI has almost surely three possible values only. Furthermore, in the spirit of Burkard and Fincke (1985), we show that for some sequences of random instances, the size of which tends to co, the relative error of any algorithm for GMAXSAT tends almost surely towards zero.