Sharp Threshold and Scaling Window for the Integer Partitioning Problem

We consider the problem of partitioning n integers chosen randomly between 1 and 2m into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized. A partition is called perfect if the optimum discrepancy is 0 when the sum of all n integers in the original set is even, or 1 when the sum is odd. Parameterizing the random problem in terms of  = m=n, we prove that the problem has a sharp threshold at  = 1, in the sense that for  < 1, there are many perfect partitions with probability tending to 1 as n ! 1, while for  > 1, there are no perfect partitions with probability tending to 1. Moreover, we show that the derivative of the so-called entropy is discontinuous at  = 1. We also determine the scaling window about the transition point: n = 1 ???? (2n)????1 log2 n + n=n, by showing that the probability of a perfect partition tends to 0, 1, or some explicitly computable p() 2 (0; 1), depending on whether n tends to ????1, 1, or  2 (????1;1), respectively. For n ! ????1 fast enough, we show that the number of perfect partitions is Gaussian in the limit. For n ! 1, we prove that with high probability the optimum partition is unique, and that the optimum discrepancy is (n). Within the window, i.e., if jnj is bounded, we prove that the optimum discrepancy is bounded. Both for n ! 1 and within the window, the limiting distribution of the (scaled) discrepancy is found.