On the Complexity of Variations of Equal Sum Subsets

Abstract

The EQUAL SUM SUBSETS problem, where we are given a set of positive integers and we ask for two nonempty disjoint subsets such that their elements add up to the same total, is known to be NP-hard. In this paper we give (pseudo-)polynomial algorithms and/or (strong) NP-hardness proofs for several natural variations of EQUAL SUM SUBSETS. Among others we present (i) a framework for obtaining NP-hardness proofs and pseudopolynomial time algorithms for EQUAL SUM SUBSETS variations, which we apply to variants of the problem with additional selection restrictions, (ii) a proof of NP-hardness and a pseudo-polynomial time algorithm for the case where we ask for two subsets such that the ratio of their sums is some fixed rational r > 0, (iii) a pseudo-polynomial time algorithm for finding k subsets of equal sum, with k = O(1), and a proof of strong NP-hardness for the same problem with k = Ω(n), (iv) algorithms and hardness results for finding k equal sum subsets under the additional requirement that the subsets should be of equal cardinality. Our results are a step towards determining the dividing lines between polynomial time solvability, pseudo-polynomial time solvability, and strong NP-completeness of subset-sum related problems.