Dense Subset Sum may be the hardest

TL;DR

This paper investigates the computational complexity of the Subset Sum problem, providing new algorithms that are faster under certain conditions related to the maximum bin size and density, challenging previous assumptions about problem hardness.

Contribution

It introduces a truly faster algorithm for Subset Sum based on maximum bin size and density parameters, advancing understanding of problem hardness and algorithmic limits.

Findings

Faster algorithms for instances with specific bin size bounds.

Characterization of problem hardness in terms of density parameter.

Challenging the notion that density 1 instances are the hardest.

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", $O^*(2^{(0.5-\delta)n})$-time algorithm, with some constant $\delta > 0$. Continuing an earlier work [STACS 2015], we study Subset Sum parameterized by the maximum bin size $\beta$, defined as the largest number of subsets of the $n$ input integers that yield the same sum. For every $\epsilon > 0$ we give a truly faster algorithm for instances with $\beta \leq 2^{(0.5-\epsilon)n}$, as well as instances with $\beta \geq 2^{0.661n}$. 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.