An Improved FPTAS for 0-1 Knapsack

Abstract

The 0-1 knapsack problem is an important NP-hard problem that admits fully polynomial-time approximation schemes (FPTASs). Previously the fastest FPTAS by Chan (2018) with approximation factor 1 + ε runs in Õ(n + (1/ε) 12/5 ) time, where Õ hides polylogarithmic factors. In this paper we present an improved algorithm in Õ(n + (1/ε) 9/4 ) time, with only a (1/ε) 1/4 gap from the quadratic conditional lower bound based on (min, +)-convolution. Our improvement comes from a multi-level extension of Chan's number-theoretic construction, and a greedy lemma that reduces unnecessary computation spent on cheap items. ACM Subject Classification Theory of computation → Algorithm design techniquesKeywords and phrases approximation algorithms, knapsack, subset sum Acknowledgements Part of this research was done while visiting Harvard University. I would like to thank Professor Jelani Nelson for introducing this problem to me, advising this project, and giving many helpful comments on my writeup.