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Equal-Subset-Sum Faster Than the Meet-in-the-Middle

Authors Marcin Mucha , Jesper Nederlof , Jakub Pawlewicz , Karol Węgrzycki

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  • Mathematics of computing → Combinatorial algorithms
Keywords
  • Equal-Subset-Sum
  • Subset-Sum
  • meet-in-the-middle
  • enumeration technique
  • randomized algorithm

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Abstract

In the Equal-Subset-Sum problem, we are given a set S of n integers and the problem is to decide if there exist two disjoint nonempty subsets A,B subseteq S, whose elements sum up to the same value. The problem is NP-complete. The state-of-the-art algorithm runs in O^*(3^(n/2)) <= O^*(1.7321^n) time and is based on the meet-in-the-middle technique. In this paper, we improve upon this algorithm and give O^*(1.7088^n) worst case Monte Carlo algorithm. This answers a question suggested by Woeginger in his inspirational survey. 
Additionally, we analyse the polynomial space algorithm for Equal-Subset-Sum. A naive polynomial space algorithm for Equal-Subset-Sum runs in O^*(3^n) time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in O^*(2.6817^n) time and polynomial space.

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Marcin Mucha, Jesper Nederlof, Jakub Pawlewicz, and Karol Węgrzycki. Equal-Subset-Sum Faster Than the Meet-in-the-Middle. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 73:1-73:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ESA.2019.73

Author Details

Marcin Mucha
  • Institute of Informatics, University of Warsaw, Poland
Jesper Nederlof
  • Eindhoven University of Technology, The Netherlands
Jakub Pawlewicz
  • Institute of Informatics, University of Warsaw, Poland
Karol Węgrzycki
  • Institute of Informatics, University of Warsaw, Poland

Funding

  • Mucha, Marcin: Supported by project TOTAL that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 677651).
  • Nederlof, Jesper: Supported by the Netherlands Organization for Scientific Research under project no. 024.002.003 and the European Research Council under project no. 617951.
  • Węgrzycki, Karol: Supported by the grants 2016/21/N/ST6/01468 and 2018/28/T/ST6/00084 of the Polish National Science Center and project TOTAL that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 677651).

Acknowledgements

The authors would like to thank anonymous reviewers for their remarks and suggestions. This research has been initiated during Parameterized Algorithms Retreat of University of Warsaw 2019, Karpacz, 25.02-01.03.2019.

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