A Note on Deterministic FPTAS for Partition

2025-01-22

AI Summary

This paper addresses the integer Partition problem and presents the first deterministic fully polynomial-time approximation scheme (FPTAS) with runtime $widetilde{O}(n + 1/varepsilon)$. It breaks the prior deterministic lower bound of $widetilde{O}(n + 1/varepsilon^{5/4})$, achieving—for the first time—the same asymptotic time complexity as randomized FPTASes. Technically, the algorithm integrates dynamic programming with aggressive pruning, bucket-based approximate encoding, and rigorous error propagation analysis to guarantee both determinism and controllable approximation accuracy. Crucially, this runtime is proven conditionally optimal under the Strong Exponential Time Hypothesis (SETH): any improvement beyond polylogarithmic factors would falsify SETH. This result establishes a fundamental theoretical barrier for deterministic approximation algorithms for Partition, resolving a long-standing open question in fine-grained complexity and approximation algorithms.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial 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
• Security and Privacy: Privacy-enhancing technologies

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].

Problem

• Deterministic FPTAS
• Partition Problem
• Strong Exponential Time Hypothesis

Innovation

• Deterministic FPTAS
• Sublinear Time Complexity
• Strong Exponential Time Hypothesis