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 , while for > 1, there are no perfect

• Type: Conference
• Year: 2001
• Where: STOC
• Authors: Christian Borgs, Jennifer T. Chayes, Boris Pittel
• Keywords: Algorithms, Optimum Discrepancy, Perfect Partitions, Random Problem, STOC 2001