Statistical mechanics analysis of the continuous number partitioning problem

The number partitioning problem consists of partitioning a sequence of positive numbers {a1,a2,…,aN} into two disjoint sets, A and B, such that the absolute value of the difference of the sums of aj over the two sets is minimized. We use statistical mechanics tools to study analytically the linear programming relaxation of this NP-complete integer programming. In particular, we calculate the probability distribution of the difference between the cardinalities of A and B and show that this difference is not self-averaging.