Efficient Approximation Algorithms for the Subset-Sums Equality Problem

摘要

We investigate the problem of finding two nonempty disjoint subsets of a set of n positive integers, with the objective that the sums of the numbers in the two subsets be as close as possible. In two versions of this problem, the quality of a solution is measured by the ratio and the difference of the two partial sums, respectively. Answering a problem of G. J. Woeginger and Z. Yu (1992, Inform. Process. Lett.42, 299–302) in the affirmative, we give a fully polynomial-time approximation scheme for the case where the value to be optimized is the ratio between the sums of the numbers in the two sets. On the other hand, we show that in the case where the value of a solution is the positive difference between the two partial sums, the problem is not 2nk-approximable in polynomial time unless P=NP, for any constant k. In the positive direction, we give a polynomial time algorithm that finds two subsets for which the difference of the two sums does not exceed K/nΩ(log n), where K is the greatest number in the instance.