The modified differencing method for the set partitioning problem with cardinality constraints Original Research Article

The set partitioning problem requires partitioning a set of n numbers into K subsets so that the difference between the maximum and minimum subset sums is minimized. This paper investigates the problem with the additional constraint that the subset cardinalities should be as balanced as possible. It modifies the differencing method of Karmarkar and Karp to ensure that the cardinality of each subset is either ⌈nK⌉ or ⌊nK⌋. The modified algorithm preserves the asymptotic quality of the differencing method, that is, the difference between the maximum and minimum subset sums does not exceed n−c log n, almost surely.