Performance Ratios for the Differencing Method Applied to the Balanced Number Partitioning Problem

We consider the problem of partitioning a set of n numbers into m subsets of cardinality k n m or n m , such that the maximum subset sum is minimal. We prove that the performance ratios of the Differencing Method of Karmarkar and Karp for k 3 4 5 and 6 are precisely 4/3, 19/12, 103/60, and 643/360, respectively, by means of a novel approach in which the ratios are explicitly calculated using mixed integer linear programming. Moreover, we show that for k 7 the performance ratio lies between 2 2 k and 2 1 k 1 . For the case that m is given instead of k, we prove a performance ratio of precisely 2 1 m. The results settle the problem of determining the worst-case performance of the Differencing Method.