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Equal Sum Subsets: Complexity of Variations

By Mark Cieliebak, Stephan Eidenbenz, Aris Pagourtzis and Konrad Schlude


We start an investigation into the complexity of variations of the Equal Sum Subsets problem, a basic problem in which we are given a set of numbers and are asked to find two disjoint subsets of the numbers that add up to the same sum. While Equal Sum Subsets is known to be NP -complete, only very few studies have investigated the complexity of its variations. In this paper, we show NP -completeness for two very natural variations, namely Factor-r Sum Subsets, where we need to find two subsets such that the ratio of their sums is exactly r, and k Equal Sum Subsets, where we need to find k subsets of equal sum. In an effort to gain an intuitive understanding of what makes a variation of Equal Sum Subsets NP -hard, we study several variations of Equal Sum Subsets in which we introduce additional requirements that a solution must fulfill (e.g., the cardinalities of the two sets must differ by exactly one), and prove NP -hardness for these variations. Finally, we investigate and show NP -hardness for the Equal Sum Subsets from Two Sets problem and its variations, where we are given two sets and we need to find two subsets of equal sum

Year: 2007
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