Statistical mechanics of an NP-complete problem: subset sum

We study the statistical properties of an NP-complete problem, the subset sum, using the methods and concepts of statistical mechanics. The problem is a generalization of the number partitioning problem, which is also an NP-complete problem and has been studied in the physics literature. The asymptotic expressions for the number of solutions are obtained. These results, applied to the number partitioning problem as a special case, are compared with those which were previously obtained by a different method. We discuss the limit of applicability of the techniques of statistical mechanics to the present problem.