Dense Subset Sum may be the hardest
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by
Per Austrin, Mikko Koivisto, Petteri Kaski, Jesper Nederlof
2015
Abstract
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
ϵ > 0 we give a truly faster algorithm for instances with β≤
2^(0.5-ϵ)n, as well as instances with β≥ 2^0.661n.
Consequently, we also obtain a characterization in terms of the popular density
parameter n/_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.
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