A Faster Pseudopolynomial Time Algorithm for Subset Sum

TL;DR

This paper introduces a new divide-and-conquer algorithm that significantly speeds up the computation of subset sums, outperforming traditional dynamic programming methods for various input sizes.

Contribution

The paper presents the fastest known algorithm for subset sum, improving the time complexity to $ ilde{O}( ext{min}\{ ext{sqrt}(n)u, u^{4/3}, ext{sum of elements} ight"), using a novel divide-and-conquer approach.

Findings

Achieves faster computation of subset sums than dynamic programming.

Provides a new algorithm with improved theoretical time bounds.

Extends the approach to cyclic groups for broader applications.

Abstract

Given a multiset $S$ of $n$ positive integers and a target integer $t$, the subset sum problem is to decide if there is a subset of $S$ that sums up to $t$. We present a new divide-and-conquer algorithm that computes all the realizable subset sums up to an integer $u$ in $\widetilde{O}\!\left(\min\{\sqrt{n}u,u^{4/3},\sigma\}\right)$, where $\sigma$ is the sum of all elements in $S$ and $\widetilde{O}$ hides polylogarithmic factors. This result improves upon the standard dynamic programming algorithm that runs in $O(nu)$ time. To the best of our knowledge, the new algorithm is the fastest general algorithm for this problem. We also present a modified algorithm for cyclic groups, which computes all the realizable subset sums within the group in $\widetilde{O}\!\left(\min\{\sqrt{n}m,m^{5/4}\}\right)$ time, where $m$ is the order of the group.