Document Open Access Logo

Finding Induced Subgraphs from Graphs with Small Mim-Width

Authors Yota Otachi , Akira Suzuki , Yuma Tamura

PDF
Thumbnail PDF

File

LIPIcs.SWAT.2024.38.pdf
  • Filesize: 0.78 MB
  • 16 pages

Document Identifiers

Author Details

Yota Otachi
  • Graduate School of Informatics, Nagoya University, Japan
Akira Suzuki
  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan
Yuma Tamura
  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan

Funding

  • Otachi, Yota: JSPS KAKENHI Grant Numbers JP18H04091, JP20H05793, JP21K11752, JP22H00513.
  • Suzuki, Akira: JSPS KAKENHI Grant Numbers JP20K11666, JP20H05794.
  • Tamura, Yuma: JSPS KAKENHI Grant Number JP21K21278.

Cite AsGet BibTex

Yota Otachi, Akira Suzuki, and Yuma Tamura. Finding Induced Subgraphs from Graphs with Small Mim-Width. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SWAT.2024.38

Abstract

In the last decade, algorithmic frameworks based on a structural graph parameter called mim-width have been developed to solve generally NP-hard problems. However, it is known that the frameworks cannot be applied to the Clique problem, and the complexity status of many problems of finding dense induced subgraphs remains open when parameterized by mim-width. In this paper, we investigate the complexity of the problem of finding a maximum induced subgraph that satisfies prescribed properties from a given graph with small mim-width. We first give a meta-theorem implying that various induced subgraph problems are NP-hard for bounded mim-width graphs. Moreover, we show that some problems, including Clique and Induced Cluster Subgraph, remain NP-hard even for graphs with (linear) mim-width at most 2. In contrast to the intractability, we provide an algorithm that, given a graph and its branch decomposition with mim-width at most 1, solves Induced Cluster Subgraph in polynomial time. We emphasize that our algorithmic technique is applicable to other problems such as Induced Polar Subgraph and Induced Split Subgraph. Since a branch decomposition with mim-width at most 1 can be constructed in polynomial time for block graphs, interval graphs, permutation graphs, cographs, distance-hereditary graphs, convex graphs, and their complement graphs, our positive results reveal the polynomial-time solvability of various problems for these graph classes.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • mim-width
  • graph algorithm
  • NP-hardness
  • induced subgraph problem
  • cluster vertex deletion

Metrics

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

Related Versions

References

  1. Rémy Belmonte and Martin Vatshelle. Graph classes with structured neighborhoods and algorithmic applications. Theoretical Computer Science, 511:54-65, 2013. URL: https://doi.org/10.1016/j.tcs.2013.01.011.
  2. Benjamin Bergougnoux, Jan Dreier, and Lars Jaffke. A logic-based algorithmic meta-theorem for mim-width. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms (SODA 2023), pages 3282-3304, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch125.
  3. Benjamin Bergougnoux and M. Moustapha Kanté. More applications of the d-neighbor equivalence: Acyclicity and connectivity constraints. SIAM Journal on Discrete Mathematics, 35(3):1881-1926, 2021. URL: https://doi.org/10.1137/20m1350571.
  4. Benjamin Bergougnoux, Charis Papadopoulos, and Jan Arne Telle. Node multiway cut and subset feedback vertex set on graphs of bounded mim-width. Algorithmica, 84(5):1385-1417, 2022. URL: https://doi.org/10.1007/s00453-022-00936-w.
  5. Flavia Bonomo-Braberman, Nick Brettell, Andrea Munaro, and Daniël Paulusma. Solving problems on generalized convex graphs via mim-width. J. Comput. Syst. Sci., 140:103493, 2024. URL: https://doi.org/10.1016/J.JCSS.2023.103493.
  6. Johann Brault-Baron, Florent Capelli, and Stefan Mengel. Understanding model counting for beta-acyclic CNF-formulas. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, volume 30 of LIPIcs, pages 143-156. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.STACS.2015.143.
  7. Nick Brettell, Jake Horsfield, Andrea Munaro, Giacomo Paesani, and Daniël Paulusma. Bounding the mim-width of hereditary graph classes. Journal of Graph Theory, 99(1):117-151, 2022. URL: https://doi.org/10.1002/jgt.22730.
  8. Nick Brettell, Jake Horsfield, Andrea Munaro, and Daniël Paulusma. List k-colouring P_t-free graphs: A mim-width perspective. Information Processing Letters, 173:106168, 2022. URL: https://doi.org/10.1016/j.ipl.2021.106168.
  9. Binh-Minh Bui-Xuan, Jan Arne Telle, and Martin Vatshelle. Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems. Theoretical Computer Science, 511:66-76, 2013. URL: https://doi.org/10.1016/j.tcs.2013.01.009.
  10. Yixin Cao, Yuping Ke, Yota Otachi, and Jie You. Vertex deletion problems on chordal graphs. Theoretical Computer Science, 745:75-86, 2018. URL: https://doi.org/10.1016/j.tcs.2018.05.039.
  11. Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas. The strong perfect graph theorem. Annals of Mathematics, 164(1):51-229, 2006. URL: https://doi.org/10.4007/annals.2006.164.51.
  12. Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Information and Computation, 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
  13. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, 33(2):125-150, 2000. URL: https://doi.org/10.1007/s002249910009.
  14. Fedor V. Fomin, Petr A. Golovach, and Jean-Florent Raymond. On the tractability of optimization problems on H-graphs. Algorithmica, 82(9):2432-2473, 2020. URL: https://doi.org/10.1007/s00453-020-00692-9.
  15. Esther Galby, Paloma T. Lima, and Bernard Ries. Reducing the domination number of graphs via edge contractions and vertex deletions. Discrete Mathematics, 344(1):112169, 2021. URL: https://doi.org/10.1016/j.disc.2020.112169.
  16. Esther Galby, Andrea Munaro, and Bernard Ries. Semitotal domination: New hardness results and a polynomial-time algorithm for graphs of bounded mim-width. Theoretical Computer Science, 814:28-48, 2020. URL: https://doi.org/10.1016/j.tcs.2020.01.007.
  17. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, 1988. URL: https://doi.org/10.1007/978-3-642-97881-4.
  18. Peter L. Hammer, Uri N. Peled, and Xiaorong Sun. Difference graphs. Discrete Applied Mathematics, 28(1):35-44, 1990. URL: https://doi.org/10.1016/0166-218X(90)90092-Q.
  19. Sun-Yuan Hsieh, Hoàng-Oanh Le, Van Bang Le, and Sheng-Lung Peng. On the d-claw vertex deletion problem. Algorithmica, 86(2):505-525, 2024. URL: https://doi.org/10.1007/S00453-023-01144-W.
  20. Sang il Oum. Rank-width and vertex-minors. Journal of Combinatorial Theory, Series B, 95(1):79-100, 2005. URL: https://doi.org/10.1016/j.jctb.2005.03.003.
  21. Lars Jaffke, O-joung Kwon, Torstein J. F. Strømme, and Jan Arne Telle. Mim-width III. Graph powers and generalized distance domination problems. Theoretical Computer Science, 796:216-236, 2019. URL: https://doi.org/10.1016/j.tcs.2019.09.012.
  22. Lars Jaffke, O-joung Kwon, and Jan Arne Telle. A unified polynomial-time algorithm for feedback vertex set on graphs of bounded mim-width. In Rolf Niedermeier and Brigitte Vallée, editors, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018), volume 96 of Leibniz International Proceedings in Informatics (LIPIcs), pages 42:1-42:14, Dagstuhl, Germany, 2018. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.STACS.2018.42.
  23. Lars Jaffke, O-joung Kwon, and Jan Arne Telle. Mim-width I. Induced path problems. Discrete Applied Mathematics, 278:153-168, 2020. URL: https://doi.org/10.1016/j.dam.2019.06.026.
  24. Lars Jaffke, O-joung Kwon, and Jan Arne Telle. Mim-width II. The feedback vertex set problem. Algorithmica, 82(1):118-145, 2020. URL: https://doi.org/10.1007/s00453-019-00607-3.
  25. Lars Jaffke, Paloma T. Lima, and Roohani Sharma. Structural parameterizations of b-coloring. In Satoru Iwata and Naonori Kakimura, editors, 34th International Symposium on Algorithms and Computation (ISAAC 2023), volume 283 of Leibniz International Proceedings in Informatics (LIPIcs), pages 40:1-40:14, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2023.40.
  26. Dong Yeap Kang, O-joung Kwon, Torstein J.F. Strømme, and Jan Arne Telle. A width parameter useful for chordal and co-comparability graphs. Theoretical Computer Science, 704:1-17, 2017. URL: https://doi.org/10.1016/j.tcs.2017.09.006.
  27. Hoang-Oanh Le and Van Bang Le. Complexity of the cluster vertex deletion problem on H-free graphs. In Stefan Szeider, Robert Ganian, and Alexandra Silva, editors, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022), volume 241 of Leibniz International Proceedings in Informatics (LIPIcs), pages 68:1-68:10. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.MFCS.2022.68.
  28. 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.
  29. Stefan Mengel. Lower bounds on the mim-width of some graph classes. Discrete Applied Mathematics, 248:28-32, 2018. URL: https://doi.org/10.1016/j.dam.2017.04.043.
  30. Andrea Munaro and Shizhou Yang. On algorithmic applications of sim-width and mim-width of (H₁,H₂)-free graphs. Theoretical Computer Science, 955:113825, 2023. URL: https://doi.org/10.1016/j.tcs.2023.113825.
  31. Svatopluk Poljak. A note on stable sets and colorings of graphs. Commentationes Mathematicae Universitatis Carolinae, 15(2):307-309, 1974. Google Scholar
  32. Sigve Hortemo Sæther and Martin Vatshelle. Hardness of computing width parameters based on branch decompositions over the vertex set. Theoretical Computer Science, 615:120-125, 2016. URL: https://doi.org/10.1016/j.tcs.2015.11.039.
  33. Martin Vatshelle. New Width Parameters of Graphs. PhD thesis, University of Bergen, 2012. Google Scholar
  34. Mihalis Yannakakis. Node-deletion problems on bipartite graphs. SIAM Journal on Computing, 10(2):310-327, 1981. URL: https://doi.org/10.1137/0210022.
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact