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An Improved FPTAS for 0-1 Knapsack

Author Ce Jin

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Ce Jin
  • Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China

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.

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Ce Jin. An Improved FPTAS for 0-1 Knapsack. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 76:1-76:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.76

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+epsilon runs in O~(n + (1/epsilon)^{12/5}) time, where O~ hides polylogarithmic factors. In this paper we present an improved algorithm in O~(n+(1/epsilon)^{9/4}) time, with only a (1/epsilon)^{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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithm design techniques
Keywords
  • approximation algorithms
  • knapsack
  • subset sum

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