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Polynomial-Time Algorithms for the Longest Induced Path and Induced Disjoint Paths Problems on Graphs of Bounded Mim-Width

Authors Lars Jaffke, O-joung Kwon, Jan Arne Telle

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Lars Jaffke
O-joung Kwon
Jan Arne Telle

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Lars Jaffke, O-joung Kwon, and Jan Arne Telle. Polynomial-Time Algorithms for the Longest Induced Path and Induced Disjoint Paths Problems on Graphs of Bounded Mim-Width. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 21:1-21:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.IPEC.2017.21

Abstract

We give the first polynomial-time algorithms on graphs of bounded maximum induced matching width (mim-width) for problems that are not locally checkable. In particular, we give n^O(w)-time algorithms on graphs of mim-width at most w, when given a decomposition, for the following problems: Longest Induced Path, Induced Disjoint Paths and H-Induced Topological Minor for fixed H. Our results imply that the following graph classes have polynomial-time algorithms for these three problems: Interval and Bi-Interval graphs, Circular Arc, Per- mutation and Circular Permutation graphs, Convex graphs, k-Trapezoid, Circular k-Trapezoid, k-Polygon, Dilworth-k and Co-k-Degenerate graphs for fixed k.
Keywords
  • graph width parameters
  • dynamic programming
  • graph classes
  • induced paths
  • induced topological minors

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References

  1. Rémy Belmonte, Petr A. Golovach, Pinar Heggernes, Pim van 't Hof, Marcin Kaminski, and Daniël Paulusma. Detecting fixed patterns in chordal graphs in polynomial time. Algorithmica, 69(3):501-521, 2014. URL: http://dx.doi.org/10.1007/s00453-013-9748-5.
  2. Rémy Belmonte and Martin Vatshelle. Graph classes with structured neighborhoods and algorithmic applications. Theor. Comput. Sci., 511:54-65, 2013. URL: http://dx.doi.org/10.1016/j.tcs.2013.01.011.
  3. Binh-Minh Bui-Xuan, Jan Arne Telle, and Martin Vatshelle. Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems. Theoret. Comput. Sci., 511:66-76, 2013. Google Scholar
  4. Jirí Fiala, Marcin Kaminski, Bernard Lidický, and Daniël Paulusma. The k-in-a-path problem for claw-free graphs. Algorithmica, 62(1-2):499-519, 2012. URL: http://dx.doi.org/10.1007/s00453-010-9468-z.
  5. Fanica Gavril. Algorithms for maximum weight induced paths. Inf. Process. Lett., 81(4):203-208, 2002. URL: http://dx.doi.org/10.1016/S0020-0190(01)00222-8.
  6. Petr A. Golovach, Daniël Paulusma, and Erik Jan van Leeuwen. Induced disjoint paths in at-free graphs. In Fedor V. Fomin and Petteri Kaski, editors, Algorithm Theory - SWAT 2012 - 13th Scandinavian Symposium and Workshops, Helsinki, Finland, July 4-6, 2012. Proceedings, volume 7357 of Lecture Notes in Computer Science, pages 153-164. Springer, 2012. URL: http://dx.doi.org/10.1007/978-3-642-31155-0_14.
  7. Petr A. Golovach, Daniël Paulusma, and Erik Jan van Leeuwen. Induced disjoint paths in circular-arc graphs in linear time. Theor. Comput. Sci., 640:70-83, 2016. URL: http://dx.doi.org/10.1016/j.tcs.2016.06.003.
  8. Petr Hlinený, Sang-il Oum, Detlef Seese, and Georg Gottlob. Width parameters beyond tree-width and their applications. Comput. J., 51(3):326-362, 2008. URL: http://dx.doi.org/10.1093/comjnl/bxm052.
  9. Tetsuya Ishizeki, Yota Otachi, and Koichi Yamazaki. An improved algorithm for the longest induced path problem on k-chordal graphs. Discrete Applied Mathematics, 156(15):3057-3059, 2008. URL: http://dx.doi.org/10.1016/j.dam.2008.01.019.
  10. Lars Jaffke, O-joung Kwon, and Jan Arne Telle. Polynomial-time algorithms for the longest induced path and induced disjoint paths problems on graphs of bounded mim-width. ArXiv e-prints, 2017. arXiv:1708.04536. Google Scholar
  11. Dong Yeap Kang, O-joung Kwon, Torstein J. F. Strømme, and Jan Arne Telle. A width parameter useful for chordal and co-comparability graphs. In Sheung-Hung Poon, Md. Saidur Rahman, and Hsu-Chun Yen, editors, WALCOM: Algorithms and Computation, 11th International Conference and Workshops, WALCOM 2017, Hsinchu, Taiwan, March 29-31, 2017, Proceedings., volume 10167 of Lecture Notes in Computer Science, pages 93-105. Springer, 2017. URL: http://dx.doi.org/10.1007/978-3-319-53925-6_8.
  12. Ken-ichi Kawarabayashi and Yusuke Kobayashi. The induced disjoint paths problem. In Andrea Lodi, Alessandro Panconesi, and Giovanni Rinaldi, editors, Integer Programming and Combinatorial Optimization, 13th International Conference, IPCO 2008, Bertinoro, Italy, May 26-28, 2008, Proceedings, volume 5035 of Lecture Notes in Computer Science, pages 47-61. Springer, 2008. URL: http://dx.doi.org/10.1007/978-3-540-68891-4_4.
  13. Dieter Kratsch, Haiko Müller, and Ioan Todinca. Feedback vertex set and longest induced path on at-free graphs. In Hans L. Bodlaender, editor, Graph-Theoretic Concepts in Computer Science, 29th International Workshop, WG 2003, Elspeet, The Netherlands, June 19-21, 2003, Revised Papers, volume 2880 of Lecture Notes in Computer Science, pages 309-321. Springer, 2003. URL: http://dx.doi.org/10.1007/978-3-540-39890-5_27.
  14. Sridhar Natarajan and Alan P. Sprague. Disjoint paths in circular arc graphs. Nordic J. of Computing, 3(3):256-270, 1996. URL: http://dl.acm.org/citation.cfm?id=642150.642154.
  15. Neil Robertson and Paul D. Seymour. Graph minors .xiii. the disjoint paths problem. J. Comb. Theory, Ser. B, 63(1):65-110, 1995. URL: http://dx.doi.org/10.1006/jctb.1995.1006.
  16. Sigve Hortemo Sæther and Martin Vatshelle. Hardness of computing width parameters based on branch decompositions over the vertex set. Theor. Comput. Sci., 615:120-125, 2016. Google Scholar
  17. Jan Arne Telle and Andrzej Proskurowski. Algorithms for vertex partitioning problems on partial k-trees. SIAM J. Discrete Math., 10(4):529-550, 1997. URL: http://dx.doi.org/10.1137/S0895480194275825.
  18. Martin Vatshelle. New Width Parameters of Graphs. PhD thesis, University of Bergen, 2012. Google Scholar
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