Document Open Access Logo

Complexity of Finding Maximum Locally Irregular Induced Subgraphs

Authors Foivos Fioravantes, Nikolaos Melissinos, Theofilos Triommatis

PDF
Thumbnail PDF

File

LIPIcs.SWAT.2022.24.pdf
  • Filesize: 0.81 MB
  • 20 pages

Document Identifiers

Author Details

Foivos Fioravantes
  • Université Côte d'Azur, Inria, CNRS, I3S, Valbonne, France
Nikolaos Melissinos
  • Université Paris-Dauphine, Université PSL, CNRS, LAMSADE, 75016, Paris, France
Theofilos Triommatis
  • School of Electrical Engineering, Electronics and Computer Science, University of Liverpool, UK

Funding

  • Fioravantes, Foivos: Supported by the French government through the UCA JEDI (ANR-15-IDEX-01) and the EUR DS4H (ANR-17-EURE-004) Investments in the Future projects.
  • Triommatis, Theofilos: Supported by EP/S023445/1 EPSRC CDT in Distributed Algorithms, University of Liverpool.

Cite AsGet BibTex

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). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SWAT.2022.24

Abstract

If a graph G is such that no two adjacent vertices of G have the same degree, we say that G is locally irregular. In this work we introduce and study the problem of identifying a largest induced subgraph of a given graph G that is locally irregular. Equivalently, given a graph G, find a subset S of V(G) with minimum order, such that by deleting the vertices of S from G results in a locally irregular graph; we denote with I(G) the order of such a set S. We first examine some easy graph families, namely paths, cycles, trees, complete bipartite and complete graphs. However, we show that the decision version of the introduced problem is NP-Complete, even for restricted families of graphs, such as subcubic planar bipartite, or cubic bipartite graphs. We then show that we can not even approximate an optimal solution within a ratio of 𝒪(n^{1-1/k}), where k ≥ 1 and n is the order the graph, unless 𝒫=NP, even when the input graph is bipartite. Then, looking for more positive results, we turn our attention towards computing I(G) through the lens of parameterised complexity. In particular, we provide two algorithms that compute I(G), each one considering different parameters. The first one considers the size of the solution k and the maximum degree Δ of G with running time (2Δ)^kn^{𝒪(1)}, while the second one considers the treewidth tw and Δ of G, and has running time Δ^{2tw}n^{𝒪(1)}. Therefore, we show that the problem is FPT by both k and tw if the graph has bounded maximum degree Δ. Since these algorithms are not FPT for graphs with unbounded maximum degree (unless we consider Δ + k or Δ + tw as the parameter), it is natural to wonder if there exists an algorithm that does not include additional parameters (other than k or tw) in its dependency. We answer negatively, to this question, by showing that our algorithms are essentially optimal. In particular, we prove that there is no algorithm that computes I(G) with dependence f(k)n^{o(k)} or f(tw)n^{o(tw)}, unless the ETH fails.

Subject Classification

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

Metrics

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

Related Versions

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. Yousef Alavi, Alfred Boals, Gary Chartrand, Ortrud Oellermann, and Paul Erdős. K-path irregular graphs. Congressus Numerantium, 65, January 1988. Google Scholar
  3. Yousef Alavi, Gary Chartrand, Fan R. K. Chung, Paul Erdös, Ronald L. Graham, and Ortrud R. Oellermann. Highly irregular graphs. Journal of Graph Theory, 11(2):235-249, 1987. URL: https://doi.org/10.1002/jgt.3190110214.
  4. Akhbar Ali, Gary Chartrand, and Ping Zhang. Irregularity in Graphs. Springer briefs in mathematics. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-67993-4.
  5. 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.
  6. 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.
  7. 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.
  8. Julien Bensmail, Martin Merker, and Carsten Thomassen. Decomposing graphs into a constant number of locally irregular subgraphs. European Journal of Combinatorics, 60:124-134, 2017. URL: https://doi.org/10.1016/j.ejc.2016.09.011.
  9. Piotr Berman, Marek Karpinski, and Alex D. Scott. Approximation hardness of short symmetric instances of MAX-3SAT. Electronic Colloquium on Computational Complexity, 2003. URL: https://eccc.weizmann.ac.il/eccc-reports/2003/TR03-049/index.html.
  10. Hans L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science, 209(1):1-45, 1998. URL: https://doi.org/10.1016/S0304-3975(97)00228-4.
  11. 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.
  12. Gary Chartrand, Paul Erdös, and Ortrud Oellermann. How to define an irregular graph. The College Mathematics Journal, 19, January 1988. URL: https://doi.org/10.2307/2686701.
  13. Gary Chartrand, Michael Jacobon, Jenö Lehel, Ortrud Oellermann, Sergio Ruiz, and Farrokh Saba. Irregular networks. Congressus Numerantium, 64, January 1986. Google Scholar
  14. Jianer Chen, Benny Chor, Mike Fellows, Xiuzhen Huang, David W. Juedes, Iyad A. Kanj, and Ge Xia. Tight lower bounds for certain parameterized np-hard problems. Information and Computation, 201(2):216-231, 2005. URL: https://doi.org/10.1109/CCC.2004.1313826.
  15. Jianer Chen, Xiuzhen Huang, Iyad A. Kanj, and Ge Xia. On the computational hardness based on linear FPT-reductions. Journal of Combinatorial Optimization, 11(2):231-247, 2006. URL: https://doi.org/10.1007/s10878-006-7137-6.
  16. Jianer Chen, Xiuzhen Huang, Iyad A. Kanj, and Ge Xia. Strong computational lower bounds via parameterized complexity. Journal of Computer and System Sciences, 72(8):1346-1367, 2006. URL: https://doi.org/10.1016/j.jcss.2006.04.007.
  17. Miroslav Chlebík and Janka Chlebíková. Complexity of approximating bounded variants of optimization problems. Theor. Comput. Sci., 354(3):320-338, 2006. URL: https://doi.org/10.1016/j.tcs.2005.11.029.
  18. 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.
  19. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: https://doi.org/10.1007/978-1-4471-5559-1.
  20. Alan M. Frieze, Ronald J. Gould, Michal Karonski, and Florian Pfender. On graph irregularity strength. Journal of Graph Theory, 41(2):120-137, 2002. URL: https://doi.org/10.1002/jgt.10056.
  21. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  22. 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.
  23. 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.
  24. 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.
  25. Carla Negri Lintzmayer, Guilherme Oliveira Mota, and Maycon Sambinelli. Decomposing split graphs into locally irregular graphs. Discrete Applied Mathematics, 292:33-44, 2021. URL: https://doi.org/10.1016/j.entcs.2019.08.053.
  26. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Lower bounds based on the exponential time hypothesis. Bulletin of the European Association for Theoretical Computer Science, 105:41-72, 2011. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/92.
  27. Bojan Mohar. Face covers and the genus problem for apex graphs. J. Comb. Theory, Ser. B, 82(1):102-117, 2001. URL: https://doi.org/10.1006/jctb.2000.2026.
  28. 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.
  29. Jakub Przybyło. Irregularity strength of regular graphs. Electronic Journal of Combinatorics, 15, June 2008. URL: https://doi.org/10.37236/806.
  30. Jakub Przybyło. On decomposing graphs of large minimum degree into locally irregular subgraphs. Electronic Journal of Combinatorics, 23(2):2-31, 2016. URL: https://doi.org/10.37236/5173.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact