An Improved Pseudopolynomial Time Algorithm for Subset Sum

TL;DR

This work tackles Subset Sum in the pseudo-polynomial regime by delivering a randomized $\tilde{O}(n + \sqrt{wt})$-time algorithm. It combines a dense-vs-sparse decomposition with a key lemma that partitions the input into parts with controlled sum and a divisor structure, enabling distinct handling of dense and sparse instances via additive combinatorics and sparse convolution. In the dense case, the sumset contains all relevant multiples of a common divisor within a short interval, while in the sparse case a three-phase, random-partitioned, color-coded convolution yields a compact witness set containing the target with high probability. The result improves prior $\tilde{O}(n+t)$-time bounds in the regime $t \ge w$ and points toward the long-standing goal of $\tilde{O}(n+w)$-time solutions; however, reconstructing an actual subset remains an open challenge due to non-constructive ingredients.

Abstract

We investigate pseudo-polynomial time algorithms for Subset Sum. Given a multi-set $X$ of $n$ positive integers and a target $t$, Subset Sum asks whether some subset of $X$ sums to $t$. Bringmann proposes an $\tilde{O}(n + t)$-time algorithm [Bringmann SODA'17], and an open question has naturally arisen: can Subset Sum be solved in $O(n + w)$ time? Here $w$ is the maximum integer in $X$. We make a progress towards resolving the open question by proposing an $\tilde{O}(n + \sqrt{wt})$-time algorithm.

Theorems & Definitions (67)

• Theorem 1.1
• Remark
• Corollary 1.2
• Remark
• Lemma 2.1
• Lemma 2.2: BFN22
• Lemma 3.0
• Lemma 3.0
• Lemma 3.0
• Theorem 3.1