Approximating Partition in Near-Linear Time

TL;DR

The paper presents a randomized near-linear-time ($\widetilde{O}(n+1/\varepsilon)$) weak approximation scheme for Subset Sum, which directly yields a near-linear-time FPTAS for Partition. Central to the approach is a novel sumset approximation framework that blends deterministic sparse convolution with additive combinatorics (Szemerédi–Vu) and a two-layer color-coding scheme to keep the number of computational levels small. When levels are sparse, the algorithm uses output-sensitive sumset computations; when levels are dense, arithmetic progressions provide accurate approximations, enabling effective reconstruction of near-optimal solutions. This combination achieves the best-known (up to polylog factors) time bound under SETH for Partition and advances the understanding of near-linear-time approximation schemes for NP-hard problems with strong structure and tight lower bounds.

Abstract

We propose an $\widetilde{O}(n + 1/\eps)$-time FPTAS (Fully Polynomial-Time Approximation Scheme) for the classical Partition problem. This is the best possible (up to a polylogarithmic factor) assuming SETH (Strong Exponential Time Hypothesis) [Abboud, Bringmann, Hermelin, and Shabtay'22]. Prior to our work, the best known FPTAS for Partition runs in $\widetilde{O}(n + 1/\eps^{5/4})$ time [Deng, Jin and Mao'23, Wu and Chen'22]. Our result is obtained by solving a more general problem of weakly approximating Subset Sum.

Theorems & Definitions (45)

• Theorem 1
• Theorem 2
• Definition 3
• Lemma 4
• Lemma 5: BFN22
• Lemma 6
• proof
• Definition 7: The Reduced Problem $\mathrm{RP}(\beta)$
• Lemma 7
• Lemma 8: Bri17