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Enumeration of Minimal Tropical Connected Sets

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Algorithms and Complexity (CIAC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13898))

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Abstract

A subset of vertices in a vertex-colored graph is called tropical if vertices of each color present in the subset. This paper is dedicated to the enumeration of all minimal tropical connected sets in various classes of graphs. We show that all minimal tropical connected sets can be enumerated in \(\mathcal {O}(1.7142^n)\) time on n-vertex interval graph which improves previous \(\mathcal {O}(1.8613^n)\) upper bound obtained by Kratsch et al. Moreover, for chordal and general class of graphs we present algorithms with running times in \(\mathcal {O}(1.937^n)\) and \(\mathcal {O}(1.999958^n)\), respectively. The last two algorithms answer question implicitly asked in the paper [Kratsch et al. SOFSEM 2017]: «Is the number of tropical sets significantly smaller than the trivial upper bound \(2^n\)?».

Work of Ivan Bliznets is supported by the project CRACKNP that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 853234).

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Acknowledgements

We would like to thank Lucas Meijer and anonymous reviewers for comments that helped to improve the paper.

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Authors and Affiliations

  1. Utrecht University, Utrecht, Netherlands

    Ivan Bliznets

  2. St. Petersburg Department of Steklov Institute of Mathematics of the RAS, Saint Petersburg, Russia

    Danil Sagunov

  3. St. Petersburg State University, Saint Petersburg, Russia

    Eugene Tagin

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  1. Ivan Bliznets
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  2. Danil Sagunov
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  3. Eugene Tagin
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Correspondence to Ivan Bliznets .

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  1. University of Cyprus, Nicosia, Cyprus

    Marios Mavronicolas

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Bliznets, I., Sagunov, D., Tagin, E. (2023). Enumeration of Minimal Tropical Connected Sets. In: Mavronicolas, M. (eds) Algorithms and Complexity. CIAC 2023. Lecture Notes in Computer Science, vol 13898. Springer, Cham. https://doi.org/10.1007/978-3-031-30448-4_10

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  • DOI: https://doi.org/10.1007/978-3-031-30448-4_10

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