The Subset Sum problem asks whether a given set of n positive integers contains a subset of elements that sum up to a given target t. It is an outstanding open question whether the O * (2 n/2 )-time algorithm for Subset Sum by Horowitz and Sahni [J. ACM 1974] can be beaten in the worst-case setting by a "truly faster", O * (2 (0.5−δ)n )-time algorithm, with some constant δ > 0. Continuing an earlier work [STACS 2015], we study Subset Sum parameterized by the maximum bin size β, defined as the largest number of subsets of the n input integers that yield the same sum. For every ǫ > 0 we give a truly faster algorithm for instances with β ≤ 2 (0.5−ǫ)n , as well as instances with β ≥ 2 0.661n . Consequently, we also obtain a characterization in terms of the popular density parameter n/ log 2 t: if all instances of density at least 1.003 admit a truly faster algorithm, then so does every instance. This goes against the current intuition that instances of density 1 are the hardest, and therefore is a step toward answering the open question in the affirmative. Our results stem from novel combinations of earlier algorithms for Subset Sum and a study of an extremal question in additive combinatorics connected to the problem of Uniquely Decodable Code Pairs in information theory.
Subsequently, Austrin et al. REF showed that one can get a faster algorithm for Subset-Sum than meet-in-the-middle if β ≤ 2 (0.5−ε)n or β ≥ 2 0.661n .
Dense Subset Sum may be the hardest
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