A Note on Deterministic FPTAS for Partition

TL;DR

This paper introduces a deterministic FPTAS for the Partition problem with near-optimal running time, matching the best possible bounds under the Strong Exponential Time Hypothesis, improving over previous randomized and deterministic algorithms.

Contribution

It presents the first deterministic FPTAS for Partition with a running time nearly optimal under the Strong Exponential Time Hypothesis.

Findings

Achieves $ ilde{O}(n + 1/\varepsilon)$ running time

Matches the lower bound assuming the Strong Exponential Time Hypothesis

Improves deterministic algorithms over previous work

Abstract

We consider the Partition problem and propose a deterministic FPTAS (Fully Polynomial-Time Approximation Scheme) that runs in $\widetilde{O}(n + 1/\varepsilon)$-time. This is the best possible (up to a polylogarithmic factor) assuming the Strong Exponential Time Hypothesis~[Abboud, Bringmann, Hermelin, and Shabtay'22]. Prior to our work, only a randomized algorithm can achieve a running time of $\widetilde{O}(n + 1/\varepsilon)$~[Chen, Lian, Mao and Zhang '24], while the best deterministic algorithm runs in $\widetilde{O}(n+1/\varepsilon^{5/4})$ time~[Deng, Jin and Mao '23] and [Wu and Chen '22].