Document Open Access Logo

Parameterised Distance to Local Irregularity

Authors Foivos Fioravantes , Nikolaos Melissinos , Theofilos Triommatis

PDF

File

Thumbnail PDF
PDF
LIPIcs.IPEC.2024.18.pdf
  • Filesize: 0.81 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Locally irregular
  • largest induced subgraph
  • FPT
  • W-hardness

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

A graph G is locally irregular if no two of its adjacent vertices have the same degree. The authors of [Fioravantes et al. Complexity of finding maximum locally irregular induced subgraph. SWAT, 2022] introduced and provided some initial algorithmic results on the problem of finding a locally irregular induced subgraph of a given graph G of maximum order, or, equivalently, computing a subset S of V(G) of minimum order, whose deletion from G results in a locally irregular graph; S is called an optimal vertex-irregulator of G. In this work we provide an in-depth analysis of the parameterised complexity of computing an optimal vertex-irregulator of a given graph G. Moreover, we introduce and study a variation of this problem, where S is a subset of the edges of G; in this case, S is denoted as an optimal edge-irregulator of G. We prove that computing an optimal vertex-irregulator of a graph G is in FPT when parameterised by various structural parameters of G, while it is W[1]-hard when parameterised by the feedback vertex set number or the treedepth of G. Moreover, computing an optimal edge-irregulator of a graph G is in FPT when parameterised by the vertex integrity of G, while it is NP-hard even if G is a planar bipartite graph of maximum degree 6, and W[1]-hard when parameterised by the size of the solution, the feedback vertex set or the treedepth of G. Our results paint a comprehensive picture of the tractability of both problems studied here.

Cite As Get BibTex

Foivos Fioravantes, Nikolaos Melissinos, and Theofilos Triommatis. Parameterised Distance to Local Irregularity. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.IPEC.2024.18

Author Details

Foivos Fioravantes
  • Department of Theoretical Computer Science, FIT, Czech Technical University in Prague, Czech Republic
Nikolaos Melissinos
  • Department of Theoretical Computer Science, FIT, Czech Technical University in Prague, Czech Republic
Theofilos Triommatis
  • School of Electrical Engineering, Electronics and Computer Science University of Liverpool, UK

Funding

This work was co-funded by the European Union under the project Robotics and advanced industrial production (reg. no. CZ.02.01.01/00/22_008/0004590). Foivos Fioravantes and Nikolaos Melissinos are also supported by the CTU Global postdoc fellowship program.

Acknowledgements

The authors would like to thank Dániel Marx for his important contribution towards rendering the proof of Theorem 7 more elegant.

References

  1. Agostinho Agra, Geir Dahl, Torkel Andreas Haufmann, and Sofia J. Pinheiro. The k-regular induced subgraph problem. Discretete Applied Mathematics, 222:14-30, 2017. URL: https://doi.org/10.1016/j.dam.2017.01.029.
  2. Noga Alon, Michael Krivelevich, and Wojciech Samotij. Largest subgraph from a hereditary property in a random graph. Discrete Mathematics, 346(9):113480, 2023. URL: https://doi.org/10.1016/J.DISC.2023.113480.
  3. Yuichi Asahiro, Hiroshi Eto, Takehiro Ito, and Eiji Miyano. Complexity of finding maximum regular induced subgraphs with prescribed degree. Theoretical Computer Science, 550:21-35, 2014. URL: https://doi.org/10.1016/j.tcs.2014.07.008.
  4. Olivier Baudon, Julien Bensmail, Jakub Przybyło, and Mariusz Wozniak. On decomposing regular graphs into locally irregular subgraphs. European Journal of Combinatorics, 49:90-104, 2015. URL: https://doi.org/10.1016/j.ejc.2015.02.031.
  5. Rémy Belmonte and Ignasi Sau. On the complexity of finding large odd induced subgraphs and odd colorings. Algorithmica, 83(8):2351-2373, 2021. URL: https://doi.org/10.1007/s00453-021-00830-x.
  6. Julien Bensmail. Maximising 1’s through Proper Labellings. Research report, Côte d’Azur University, 2023. URL: https://hal.science/hal-04086726.
  7. Robert Bredereck, Sepp Hartung, André Nichterlein, and Gerhard J. Woeginger. The complexity of finding a large subgraph under anonymity constraints. In Algorithms and Computation - 24th International Symposium, ISAAC 2013, volume 8283 of Lecture Notes in Computer Science, pages 152-162. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-45030-3_15.
  8. Leizhen Cai. Fixed-parameter tractability of graph modification problems for hereditary properties. Information Processing Letters, 58(4):171-176, 1996. URL: https://doi.org/10.1016/0020-0190(96)00050-6.
  9. Gruia Călinescu, Cristina G. Fernandes, Hemanshu Kaul, and Alexander Zelikovsky. Maximum series-parallel subgraph. Algorithmica, 63(1-2):137-157, 2012. URL: https://doi.org/10.1007/S00453-011-9523-4.
  10. Markus Chimani, Ivo Hedtke, and Tilo Wiedera. Exact algorithms for the maximum planar subgraph problem: New models and experiments. ACM Journal of Experimental Algorithmics, 24(1):2.1:1-2.1:21, 2019. URL: https://doi.org/10.1145/3320344.
  11. Elias Dahlhaus, David S. Johnson, Christos H. Papadimitriou, Paul D. Seymour, and Mihalis Yannakakis. The complexity of multiterminal cuts. SIAM Journal on Computing, 23(4):864-894, 1994. URL: https://doi.org/10.1137/S0097539792225297.
  12. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. URL: https://doi.org/10.1007/978-3-662-53622-3.
  13. Pål Grønås Drange, Markus S. Dregi, and Pim van 't Hof. On the computational complexity of vertex integrity and component order connectivity. Algorithmica, 76(4):1181-1202, 2016. URL: https://doi.org/10.1007/S00453-016-0127-X.
  14. Baris Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma. Faster exponential-time approximation algorithms using approximate monotone local search. In 30th Annual European Symposium on Algorithms, ESA 2022, volume 244 of LIPIcs, pages 50:1-50:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.ESA.2022.50.
  15. Michael R. Fellows, Fedor V. Fomin, Daniel Lokshtanov, Frances A. Rosamond, Saket Saurabh, Stefan Szeider, and Carsten Thomassen. On the complexity of some colorful problems parameterized by treewidth. Information and Computation, 209(2):143-153, 2011. URL: https://doi.org/10.1016/J.IC.2010.11.026.
  16. Foivos Fioravantes, Nikolaos Melissinos, and Theofilos Triommatis. Complexity of finding maximum locally irregular induced subgraphs. In 18th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2022, volume 227 of LIPIcs, pages 24:1-24:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.SWAT.2022.24.
  17. Fedor V. Fomin, Serge Gaspers, Daniel Lokshtanov, and Saket Saurabh. Exact algorithms via monotone local search. Journal of the ACM, 66(2):8:1-8:23, 2019. URL: https://doi.org/10.1145/3284176.
  18. Robert Ganian, Fabian Klute, and Sebastian Ordyniak. On structural parameterizations of the bounded-degree vertex deletion problem. Algorithmica, 83(1):297-336, 2021. URL: https://doi.org/10.1007/S00453-020-00758-8.
  19. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  20. Gregory Z. Gutin, Eun Jung Kim, Arezou Soleimanfallah, Stefan Szeider, and Anders Yeo. Parameterized complexity results for general factors in bipartite graphs with an application to constraint programming. Algorithmica, 64(1):112-125, 2012. URL: https://doi.org/10.1007/S00453-011-9548-8.
  21. Falk Hüffner, Christian Komusiewicz, Hannes Moser, and Rolf Niedermeier. Fixed-parameter algorithms for cluster vertex deletion. Theory of Computing Systems, 47(1):196-217, 2010. URL: https://doi.org/10.1007/S00224-008-9150-X.
  22. Mark Jones, Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh, and Ondrej Suchý. Parameterized complexity of directed steiner tree on sparse graphs. SIAM J. Discret. Math., 31(2):1294-1327, 2017. URL: https://doi.org/10.1137/15M103618X.
  23. Michał Karoński, Tomasz Łuczak, and Andrew Thomason. Edge weights and vertex colors. Journal of Combinatorial Theory, 91:151-157, May 2004. URL: https://doi.org/10.1016/j.jctb.2003.12.001.
  24. Ralph Keusch. A solution to the 1-2-3 conjecture. Journal of Combinatorial Theory, Series B, 166:183-202, 2024. URL: https://doi.org/10.1016/J.JCTB.2024.01.002.
  25. Subhash Khot and Venkatesh Raman. Parameterized complexity of finding subgraphs with hereditary properties. Theoretical Computer Science, 289(2):997-1008, 2002. URL: https://doi.org/10.1016/S0304-3975(01)00414-5.
  26. Michael Lampis. Algorithmic meta-theorems for restrictions of treewidth. Algorithmica, 64(1):19-37, 2012. URL: https://doi.org/10.1007/S00453-011-9554-X.
  27. Michael Lampis and Manolis Vasilakis. Structural parameterizations for two bounded degree problems revisited. In 31st Annual European Symposium on Algorithms, ESA 2023, September 4-6, 2023, Amsterdam, The Netherlands, volume 274 of LIPIcs, pages 77:1-77:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ESA.2023.77.
  28. H. W. Lenstra Jr. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8(4):538-548, 1983. URL: https://doi.org/10.1287/MOOR.8.4.538.
  29. John M. Lewis and Mihalis Yannakakis. The node-deletion problem for hereditary properties is np-complete. Journal of Computer and System Sciences, 20(2):219-230, 1980. URL: https://doi.org/10.1016/0022-0000(80)90060-4.
  30. David Lichtenstein. Planar formulae and their uses. SIAM journal on computing, 11(2):329-343, 1982. URL: https://doi.org/10.1137/0211025.
  31. Hannes Moser and Dimitrios M. Thilikos. Parameterized complexity of finding regular induced subgraphs. Journal of Discrete Algorithms, 7(2):181-190, 2009. URL: https://doi.org/10.1016/j.jda.2008.09.005.
  32. J. Barkley Rosser and Lowell Schoenfeld. Approximate formulas for some functions of prime numbers. Illinois Journal of Mathematics, 6(1):64-94, 1962. Google Scholar
  33. A. Schrijver. Combinatorial Optimization: Polyhedra and Efficiency. Springer-Verlag Berlin and Heidelberg GmbH & Co., 2003. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact