Sharp threshold and scaling window for the integer partitioning problem

• Authors: Christian Borgs;Jennifer Chayes;Boris Pittel
• Venue: STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
• Year: 2001

We consider the problem of partitioning n integers chosen randomly between 1 and 2^m 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 &kgr; = m/n, we prove that the problem has a sharp threshold at &kgr; = 1, in the sense that for &kgr; , there are many perfect partitions with probability tending to 1 as n \to \infty, 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 &kgr;=1.We also determine the scaling window about the transition point: &kgr;_n = 1 - (2n)^{-1}\log_2 n + &lgr;_n / n, by showing that the probability of a perfect partition tends to 0, 1, or some explicitly computable p(&lgr;) \in (0,1), depending on whether &lgr;_n tends to -\infty$, $\infty$, or $&lgr; \in (-\infty, \infty), respectively. For &lgr;_n \to -\infty fast enough, we show that the number of perfect partitions is Gaussian in the limit. For &lgr;_n \to \infty, we prove that with high probability the optimum partition is unique, and that the optimum discrepancy is &thgr;(&lgr;_n). Within the window, i.e., if |&lgr;_n| is bounded, we prove that the optimum discrepancy is bounded. Both for &lgr;_n \to \infty and within the window, the limiting distribution of the (scaled) discrepancy