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Heuristic, Branch-and-Bound Solver and Improved Space Reduction for the Median of Permutations Problem

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Combinatorial Algorithms (IWOCA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10765))

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Abstract

Given a set \(\mathcal {A}\subseteq \mathbb {S}_n\) of m permutations of \(\{1,2,\ldots ,n\}\) and a distance function d, the median problem consists of finding a permutation \(\pi ^*\) that is the “closest” of the m given permutations. Here, we study the problem under the Kendall-\(\tau \) distance which counts the number of order disagreements between pairs of elements of permutations. In this article, we explore this NP-hard problem using three different approaches: a well parameterized heuristic, an improved space search reduction technique and a refined branch-and-bound solver.

This work is supported by a grant from the National Sciences and Engineering Research Council of Canada (NSERC) through an Individual Discovery Grant RGPIN-2016-04576 (Hamel) and by Fonds Nature et Technologies (FRQNT) through a Doctoral scholarship (Milosz).

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Notes

  1. 1.

    Note that all appendices and the source code (Java) for testing can be found online at http://www-etud.iro.umontreal.ca/~miloszro/iwoca/iwoca.html.

  2. 2.

    More detailed explanations and some examples for these Major Order Theorems can be found in Sect. 4 of [18].

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Acknowledgements

We would like to thanks our anonymous reviewers for their careful and inspiring comments. Be sure that the suggestions that were not included here, due to time and space constraints, will be integrate in the journal version of this article.

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Authors and Affiliations

  1. DIRO, Université de Montréal, C.P. 6128 Succursale Centre-Ville, Montréal, QC, H3C 3J7, Canada

    Robin Milosz & Sylvie Hamel

Authors
  1. Robin Milosz
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  2. Sylvie Hamel
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Correspondence to Sylvie Hamel .

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Editors and Affiliations

  1. University of Newcastle Australia, Callaghan, New South Wales, Australia

    Ljiljana Brankovic

  2. University of Newcastle Australia, Callaghan, New South Wales, Australia

    Joe Ryan

  3. McMaster University, Hamilton, Ontario, Canada

    William F. Smyth

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Milosz, R., Hamel, S. (2018). Heuristic, Branch-and-Bound Solver and Improved Space Reduction for the Median of Permutations Problem. In: Brankovic, L., Ryan, J., Smyth, W. (eds) Combinatorial Algorithms. IWOCA 2017. Lecture Notes in Computer Science(), vol 10765. Springer, Cham. https://doi.org/10.1007/978-3-319-78825-8_25

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  • DOI: https://doi.org/10.1007/978-3-319-78825-8_25

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