Derandomizing Pseudopolynomial Algorithms for Subset Sum

2026-01-04

arXiv.org

AI Summary

This work addresses the challenge of derandomizing the best-known randomized algorithms for the Subset Sum problem by presenting the first deterministic algorithm with a running time of Õ(t). Leveraging a novel deterministic pseudopolynomial-time technique, combined with output-sensitive analysis and fine-grained complexity reductions, the proposed method not only achieves an efficient deterministic solution to Subset Sum but also successfully derandomizes Bringmann’s output-sensitive algorithm and the reduction framework of Cygan et al. for the 0–1 Knapsack problem. This advancement significantly broadens the applicability of deterministic approaches in related domains of fine-grained complexity and combinatorial optimization.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content

Abstract

We reexamine the classical subset sum problem: given a set $X$ of $n$ positive integers and a number $t$, decide whether there exists a subset of $X$ that sums to $t$; or more generally, compute the set $\mbox{out}$ of all numbers $y\in\{0,\ldots,t\}$ for which there exists a subset of $X$ that sums to $y$. Standard dynamic programming solves the problem in $O(tn)$ time. In SODA'17, two papers appeared giving the current best deterministic and randomized algorithms, ignoring polylogarithmic factors: Koiliaris and Xu's deterministic algorithm runs in $\widetilde{O}(t\sqrt{n})$ time, while Bringmann's randomized algorithm runs in $\widetilde{O}(t)$ time. We present the first deterministic algorithm running in $\widetilde{O}(t)$ time. Our technique has a number of other applications: for example, we can also derandomize the more recent output-sensitive algorithms by Bringmann and Nakos [STOC'20] and Bringmann, Fischer, and Nakos [SODA'25] running in $\widetilde{O}(|\mbox{out}|^{4/3})$ and $\widetilde{O}(|\mbox{out}|\sqrt{n})$ time, and we can derandomize a previous fine-grained reduction from 0-1 knapsack to min-plus convolution by Cygan et al. [ICALP'17].

Problem

• Subset Sum
• Derandomization
• Deterministic Algorithm
• Pseudopolynomial Time
• Dynamic Programming

Innovation

• Subset Sum
• Derandomization
• Deterministic Algorithm
• Fine-grained Complexity
• Output-sensitive Algorithm