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Knapsack and Subset Sum with Small Items

Authors Adam Polak , Lars Rohwedder , Karol Węgrzycki

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Author Details

Adam Polak
  • EPFL, Lausanne, Switzerland
Lars Rohwedder
  • EPFL, Lausanne, Switzerland
Karol Węgrzycki
  • Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany

Funding

  • Polak, Adam: Supported by the Swiss National Science Foundation within the project Lattice Algorithms and Integer Programming (185030). Part of this work was done at Jagiellonian University, supported by Polish National Science Center grant 2017/27/N/ST6/01334.
  • Rohwedder, Lars: Swiss National Science Foundation project 200021-184656.
  • Węgrzycki, Karol: Project TIPEA that has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 850979).

Cite AsGet BibTex

Adam Polak, Lars Rohwedder, and Karol Węgrzycki. Knapsack and Subset Sum with Small Items. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 106:1-106:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ICALP.2021.106

Abstract

Knapsack and Subset Sum are fundamental NP-hard problems in combinatorial optimization. Recently there has been a growing interest in understanding the best possible pseudopolynomial running times for these problems with respect to various parameters. In this paper we focus on the maximum item size s and the maximum item value v. We give algorithms that run in time O(n + s³) and O(n + v³) for the Knapsack problem, and in time Õ(n + s^{5/3}) for the Subset Sum problem. Our algorithms work for the more general problem variants with multiplicities, where each input item comes with a (binary encoded) multiplicity, which succinctly describes how many times the item appears in the instance. In these variants n denotes the (possibly much smaller) number of distinct items. Our results follow from combining and optimizing several diverse lines of research, notably proximity arguments for integer programming due to Eisenbrand and Weismantel (TALG 2019), fast structured (min,+)-convolution by Kellerer and Pferschy (J. Comb. Optim. 2004), and additive combinatorics methods originating from Galil and Margalit (SICOMP 1991).

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic programming
Keywords
  • Knapsack
  • Subset Sum
  • Proximity
  • Additive Combinatorics
  • Multiset

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