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


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)


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
  • approximation algorithms
  • knapsack
  • subset sum


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  1. Alok Aggarwal, Maria M. Klawe, Shlomo Moran, Peter Shor, and Robert Wilber. Geometric applications of a matrix-searching algorithm. Algorithmica, 2(1):195-208, November 1987. URL:
  2. David Bremner, Timothy M. Chan, Erik D. Demaine, Jeff Erickson, Ferran Hurtado, John Iacono, Stefan Langerman, Mihai Pǎtraşcu, and Perouz Taslakian. Necklaces, Convolutions, and X+Y. Algorithmica, 69(2):294-314, June 2014. URL:
  3. Timothy M. Chan. Approximation Schemes for 0-1 Knapsack. In Proceedings of the 1st Symposium on Simplicity in Algorithms (SOSA), pages 5:1-5:12, 2018. URL:
  4. Timothy M. Chan and Ryan Williams. Deterministic APSP, Orthogonal Vectors, and More: Quickly Derandomizing Razborov-Smolensky. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1246-1255, 2016. URL:
  5. Marek Cygan, Marcin Mucha, Karol Węgrzycki, and Michał Włodarczyk. On Problems Equivalent to (Min,+)-Convolution. ACM Trans. Algorithms, 15(1):14:1-14:25, January 2019. URL:
  6. Oscar H. Ibarra and Chul E. Kim. Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems. Journal of the ACM (JACM), 22(4):463-468, October 1975. URL:
  7. Klaus Jansen and Stefan E.J. Kraft. A faster FPTAS for the Unbounded Knapsack Problem. European Journal of Combinatorics, 68:148-174, 2018. URL:
  8. Richard M. Karp. Reducibility among combinatorial problems. In Complexity of Computer Computations, pages 85-103. Springer US, 1972. URL:
  9. Hans Kellerer, Renata Mansini, Ulrich Pferschy, and Maria Grazia Speranza. An efficient fully polynomial approximation scheme for the Subset-Sum Problem. Journal of Computer and System Sciences, 66(2):349-370, 2003. URL:
  10. Hans Kellerer and Ulrich Pferschy. A New Fully Polynomial Time Approximation Scheme for the Knapsack Problem. Journal of Combinatorial Optimization, 3(1):59-71, July 1999. URL:
  11. Hans Kellerer and Ulrich Pferschy. Improved Dynamic Programming in Connection with an FPTAS for the Knapsack Problem. Journal of Combinatorial Optimization, 8(1):5-11, March 2004. URL:
  12. Marvin Künnemann, Ramamohan Paturi, and Stefan Schneider. On the Fine-Grained Complexity of One-Dimensional Dynamic Programming. In Proceedings of the 44th International Colloquium on Automata, Languages, and Programming (ICALP), pages 21:1-21:15, 2017. URL:
  13. Eugene L. Lawler. Fast Approximation Algorithms for Knapsack Problems. Mathematics of Operations Research, 4(4):339-356, 1979. URL:
  14. Marcin Mucha, Karol Węgrzycki, and Michał Włodarczyk. Subquadratic Approximation Scheme for Partition. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 70-88, 2019. URL:
  15. Donguk Rhee. Faster fully polynomial approximation schemes for knapsack problems. Master’s thesis, Massachusetts Institute of Technology, 2015. URL:
  16. Ryan Williams. Faster All-pairs Shortest Paths via Circuit Complexity. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC), pages 664-673, 2014. URL: