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Half-Space Separation in Monophonic Convexity

Authors Mohammed Elaroussi , Lhouari Nourine , Simon Vilmin

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Author Details

Mohammed Elaroussi
  • Université de Bejaia, Faculté des Sciences Exactes, Unité de Recherche LaMOS, 06000 Bejaia, Algeria
Lhouari Nourine
  • Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-Ferrand, France
Simon Vilmin
  • Aix-Marseille Université, CNRS, LIS, Marseille, France

Funding

The first two authors have been funded by the CMEP Tassili project: "Argumentation et jeux: Structures et Algorithmes", Codes: 46085QH-21MDU320 PHC, 2021-2023. This research is also supported by the French government IDEXISITE initiative 16-IDEX-0001 (CAP 20-25).

Acknowledgements

We thank the reviewers for their comments. We are also grateful to Victor Chepoi for suggesting the term "shadow" instead of "extension" as well as pointing us to further references, especially the recent works on monophonic convexity [Bressan et al., 2024; Chepoi, 2024].

Cite As Get BibTex

Mohammed Elaroussi, Lhouari Nourine, and Simon Vilmin. Half-Space Separation in Monophonic Convexity. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 51:1-51:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.51

Abstract

We study half-space separation in the convexity of chordless paths of a graph, i.e., monophonic convexity. In this problem, one is given a graph and two (disjoint) subsets of vertices and asks whether these two sets can be separated by complementary convex sets, called half-spaces. While it is known this problem is NP-complete for geodesic convexity - the convexity of shortest paths - we show that it can be solved in polynomial time for monophonic convexity.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Paths and connectivity problems
  • Mathematics of computing → Graph algorithms
Keywords
  • chordless paths
  • monophonic convexity
  • separation
  • half-space

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References

  1. Liliana Alcón, Boštjan Brešar, Tanja Gologranc, Marisa Gutierrez, Tadeja Kraner Šumenjak, Iztok Peterin, and Aleksandra Tepeh. Toll convexity. European Journal of Combinatorics, 46:161-175, 2015. URL: https://doi.org/10.1016/j.ejc.2015.01.002.
  2. Hans-Jürgen Bandelt. Graphs with intrinsic S₃ convexities. Journal of graph theory, 13(2):215-228, 1989. URL: https://doi.org/10.1002/jgt.3190130208.
  3. Bernhard E Boser, Isabelle M Guyon, and Vladimir N Vapnik. A training algorithm for optimal margin classifiers. In Proceedings of the fifth annual workshop on Computational learning theory, pages 144-152, 1992. URL: https://doi.org/10.1145/130385.130401.
  4. Marco Bressan, Emmanuel Esposito, and Maximilian Thiessen. Efficient algorithms for learning monophonic halfspaces in graphs. arXiv preprint arXiv:2405.00853, 2024. URL: https://doi.org/10.48550/arXiv.2405.00853.
  5. Manoj Changat and Joseph Mathew. On triangle path convexity in graphs. Discrete Mathematics, 206(1-3):91-95, 1999. URL: https://doi.org/10.1016/S0012-365X(98)00394-X.
  6. Victor Chepoi. Some properties of domain finite convexity structures (in russian), res. Algebra, Geometry and Appl.(Moldova State University), pages 142-148, 1986. Google Scholar
  7. Victor Chepoi. Separation of two convex sets in convexity structures. Journal of Geometry, 50:30-51, 1994. URL: https://doi.org/10.1007/BF01222661.
  8. Victor Chepoi. Separation axiom S₃ for geodesic convexity in graphs. arXiv preprint arXiv:2405.07512, 2024. URL: https://doi.org/10.48550/arXiv.2405.07512.
  9. Mitre C. Dourado, Marisa Gutierrez, Fábio Protti, and Silvia Tondato. Weakly toll convexity and proper interval graphs. Discrete Mathematics & Theoretical Computer Science, vol. 26:2, April 2024. URL: https://doi.org/10.46298/dmtcs.9837.
  10. Mitre C Dourado, Fábio Protti, and Jayme L Szwarcfiter. Complexity results related to monophonic convexity. Discrete Applied Mathematics, 158(12):1268-1274, 2010. URL: https://doi.org/10.1016/j.dam.2009.11.016.
  11. Feodor F Dragan, Falk Nicolai, and Andreas Brandstädt. Convexity and hhd-free graphs. SIAM Journal on Discrete Mathematics, 12(1):119-135, 1999. URL: https://doi.org/10.1137/S0895480195321718.
  12. Pierre Duchet. Convex sets in graphs, ii. minimal path convexity. Journal of Combinatorial Theory, Series B, 44(3):307-316, 1988. URL: https://doi.org/10.1016/0095-8956(88)90039-1.
  13. Mohammed Elaroussi, Lhouari Nourine, and Simon Vilmin. Half-space separation in monophonic convexity. arXiv preprint arXiv:2404.17564, 2024. URL: https://doi.org/10.48550/arXiv.2404.17564.
  14. JW Ellis. A general set-separation theorem. Duke Math. J., 19(1):417-421, 1952. URL: https://doi.org/10.1215/S0012-7094-52-01941-8.
  15. Martin Farber and Robert E Jamison. Convexity in graphs and hypergraphs. SIAM Journal on Algebraic Discrete Methods, 7(3):433-444, 1986. URL: https://doi.org/10.1137/0607049.
  16. Yoav Freund and Robert E Schapire. Large margin classification using the perceptron algorithm. In Proceedings of the eleventh annual conference on Computational learning theory, pages 209-217, 1998. URL: https://doi.org/10.1145/279943.279985.
  17. Lucía M González, Luciano N Grippo, Martín D Safe, and Vinicius F dos Santos. Covering graphs with convex sets and partitioning graphs into convex sets. Information Processing Letters, 158:105944, 2020. URL: https://doi.org/10.1016/j.ipl.2020.105944.
  18. David Kay and Eugene W Womble. Axiomatic convexity theory and relationships between the carathéodory, helly, and radon numbers. Pacific Journal of Mathematics, 38(2):471-485, 1971. URL: https://doi.org/10.2140/pjm.1971.38.471.
  19. László Lovász. Coverings and colorings of hypergraphs. In Proc. 4th Southeastern Conference of Combinatorics, Graph Theory, and Computing, pages 3-12. Utilitas Mathematica Publishing, 1973. Google Scholar
  20. Ignacio M Pelayo. Geodesic convexity in graphs. Springer, 2013. URL: https://doi.org/10.1007/978-1-4614-8699-2.
  21. Florian Seiffarth, Tamás Horváth, and Stefan Wrobel. Maximal closed set and half-space separations in finite closure systems. Theoretical Computer Science, 973:114105, 2023. URL: https://doi.org/10.1016/j.tcs.2023.114105.
  22. Maximilian Thiessen and Thomas Gärtner. Active learning of convex halfspaces on graphs. Advances in Neural Information Processing Systems, 34:23413-23425, 2021. URL: https://proceedings.neurips.cc/paper_files/paper/2021/file/c4bf1e24f3e6f92ca9dfd9a7a1a1049c-Paper.pdf.
  23. Marcel van de Vel. Binary convexities and distributive lattices. Proceedings of the London Mathematical Society, 3(1):1-33, 1984. URL: https://doi.org/10.1112/plms/s3-48.1.1.
  24. Marcel van De Vel. Theory of convex structures. Elsevier, 1993. Google Scholar
  25. Vladimir Vapnik. Statistical learning theory. Wiley New York, 1998. Google Scholar
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