A Note on Deterministic FPTAS for Partition
Lin Chen, Jiayi Lian, Yuchen Mao, Guochuan Zhang
TL;DR
The paper addresses the Partition problem and delivers a deterministic FPTAS running in $\widetilde{O}(n + 1/\varepsilon)$ time, matching the strongest conditional lower bounds under SETH. Its core idea is to reduce Partition to a Reduced Problem $\mathrm{RP}(\mu,m)$ and solve it deterministically by combining Sparse FFT techniques with Szemerédi–Vu additive-combinatorics results, while introducing $\mu$-canonical sets to manage approximation at different magnitudes without color-coding. The work provides a concrete, near-linear-time deterministic scheme for Partition and implies a weak-approximation result for Subset Sum as a corollary, advancing the theory of deterministic near-linear-time approximation for classic subset-sum problems. Overall, it yields a practical and theoretically optimal approach within the current conditional lower-bound landscape, with implications for related problems and potential extensions to deterministic weak-approximation schemes.
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].