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The Precise Complexity of Finding Rainbow Even Matchings

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Algebraic Informatics (CAI 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11545))

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Abstract

A progress in complexity lower bounds might be achieved by studying problems where a very precise complexity is conjectured. In this note we propose one such problem: Given a planar graph on n vertices and disjoint pairs of its edges \(p_1, \ldots , p_g\), perfect matching M is Rainbow Even Matching (REM) if \(|M\cap p_i|\) is even for each \(i= 1, \ldots , g\). A straightforward algorithm finds a REM or asserts that no REM exists in \(2^g\times \mathrm {poly}(n)\) steps and we conjecture that no deterministic or randomised algorithm has complexity asymptotically smaller than \(2^g\). Our motivation is also to pinpoint the curse of dimensionality of the Max-Cut problem for graphs embedded into orientable surfaces: a basic problem of statistical physics.

The author was partially supported by the H2020-MSCA-RISE project CoSP- GA No. 823748.

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Acknowledgement

This project initially started as a joint work with Marcos Kiwi. I would like to thank Marcos for many helpful discussions.

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Authors and Affiliations

  1. Department of Applied Mathematics, Charles University, Prague, Czech Republic

    Martin Loebl

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  1. Martin Loebl
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Correspondence to Martin Loebl .

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Editors and Affiliations

  1. University of Niš, Niš, Serbia

    Miroslav Ćirić

  2. University of Leipzig , Leipzig, Germany

    Manfred Droste

  3. Université Paris Denis Diderot and CNRS, Paris, France

    Jean-Éric Pin

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Loebl, M. (2019). The Precise Complexity of Finding Rainbow Even Matchings. In: Ćirić, M., Droste, M., Pin, JÉ. (eds) Algebraic Informatics. CAI 2019. Lecture Notes in Computer Science(), vol 11545. Springer, Cham. https://doi.org/10.1007/978-3-030-21363-3_16

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  • DOI: https://doi.org/10.1007/978-3-030-21363-3_16

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-21362-6

  • Online ISBN: 978-3-030-21363-3

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