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Enumerating Minimal Solution Sets for Metric Graph Problems

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Graph-Theoretic Concepts in Computer Science (WG 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14760))

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Abstract

Problems from metric graph theory like Metric Dimension, Geodetic Set, and Strong Metric Dimension have recently had a strong impact in parameterized complexity by being the first known problems in NP to admit double-exponential lower bounds in the treewidth, and even in the vertex cover number for the latter, assuming the Exponential Time Hypothesis. We initiate the study of enumerating minimal solution sets for these problems and show that they are also of great interest in enumeration. Specifically, we show that enumerating minimal resolving sets in graphs and minimal geodetic sets in split graphs are equivalent to enumerating minimal transversals in hypergraphs (denoted Trans-Enum), whose solvability in total-polynomial time is one of the most important open problems in algorithmic enumeration. This provides two new natural examples to a question that emerged in recent works: for which vertex (or edge) set graph property \(\varPi \) is the enumeration of minimal (or maximal) subsets satisfying \(\varPi \) equivalent to Trans-Enum? As very few properties are known to fit within this context—namely, those related to minimal domination—our results make significant progress in characterizing such properties, and provide new angles to approach Trans-Enum. In contrast, we observe that minimal strong resolving sets can be enumerated with polynomial delay. Additionally, we consider cases where our reductions do not apply, namely graphs with no long induced paths, and show both positive and negative results related to the enumeration and extension of partial solutions.

This work is an extended abstract of [10].

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Notes

  1. 1.

    The different notions from enumeration complexity are defined in Sect. 2.

  2. 2.

    An enumeration problem is at least as hard as another enumeration problem if a total-polynomial-time algorithm for the first implies a total-polynomial-time algorithm for the second; the problems are (polynomially) equivalent if the reverse direction also holds.

  3. 3.

    As Trans-Enum is in \(\textsf{QP}\) (quasi-polynomial time) and it is believed that \(\textsf {NP}\not \subseteq \textsf{QP}\).

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Acknowledgements

This work was supported by the ANR project DISTANCIA (ANR-17-CE40-0015) and the Austrian Science Fund (FWF, project Y1329). We would like to thank the anonymous reviewers for their careful reading.

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Authors and Affiliations

  1. Institute of Informatics, University of Warsaw, Warsaw, Poland

    Benjamin Bergougnoux

  2. Aix Marseille Université, CNRS, LIS, 13009, Marseille, France

    Oscar Defrain

  3. Algorithms and Complexity Group, Technische Universität Wien, Vienna, Austria

    Fionn Mc Inerney

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  1. Benjamin Bergougnoux
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  2. Oscar Defrain
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Correspondence to Benjamin Bergougnoux .

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  1. Masaryk University, Brno, Czech Republic

    Daniel Kráľ

  2. University of Primorska, Koper, Slovenia

    Martin Milanič

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Bergougnoux, B., Defrain, O., Mc Inerney, F. (2025). Enumerating Minimal Solution Sets for Metric Graph Problems. In: Kráľ, D., Milanič, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2024. Lecture Notes in Computer Science, vol 14760. Springer, Cham. https://doi.org/10.1007/978-3-031-75409-8_4

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