Dense Subset Sum May Be the Hardest

Per Austrin, Petteri Kaski, Mikko Koivisto, Jesper Nederlof
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
more » ... est number of subsets of the n input integers that yield the same sum. For every > 0 we give a truly faster algorithm for instances with β ≤ 2 (0.5−)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 a novel combinatorial analysis of mixings 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.