Document Open Access Logo

A Linear Fixed Parameter Tractable Algorithm for Connected Pathwidth

Authors Mamadou Moustapha Kanté , Christophe Paul , Dimitrios M. Thilikos

PDF
Thumbnail PDF

File

LIPIcs.ESA.2020.64.pdf
  • Filesize: 0.7 MB
  • 16 pages

Document Identifiers

Author Details

Mamadou Moustapha Kanté
  • Université Clermont Auvergne, LIMOS, CNRS, Aubière, France
Christophe Paul
  • LIRMM, Université de Montpellier, CNRS, France
Dimitrios M. Thilikos
  • LIRMM, Université de Montpellier, CNRS, France

Funding

  • Kanté, Mamadou Moustapha: DEMOGRAPH (ANR-16-CE40-0028) and ASSK (ANR-18-CE40-0025-01).
  • Paul, Christophe: DEMOGRAPH (ANR-16-CE40-0028) and ESIGMA (ANR-17-CE23-0010).
  • Thilikos, Dimitrios M.: DEMOGRAPH (ANR-16-CE40-0028) and ESIGMA (ANR-17-CE23-0010).

Cite As Get BibTex

Mamadou Moustapha Kanté, Christophe Paul, and Dimitrios M. Thilikos. A Linear Fixed Parameter Tractable Algorithm for Connected Pathwidth. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 64:1-64:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ESA.2020.64

Abstract

The graph parameter of pathwidth can be seen as a measure of the topological resemblance of a graph to a path. A popular definition of pathwidth is given in terms of node search where we are given a system of tunnels (represented by a graph) that is contaminated by some infectious substance and we are looking for a search strategy that, at each step, either places a searcher on a vertex or removes a searcher from a vertex and where an edge is cleaned when both endpoints are simultaneously occupied by searchers. It was proved that the minimum number of searchers required for a successful cleaning strategy is equal to the pathwidth of the graph plus one. Two desired characteristics for a cleaning strategy is to be monotone (no recontamination occurs) and connected (clean territories always remain connected). Under these two demands, the number of searchers is equivalent to a variant of pathwidth called connected pathwidth. We prove that connected pathwidth is fixed parameter tractable, in particular we design a 2^O(k²)⋅n time algorithm that checks whether the connected pathwidth of G is at most k. This resolves an open question by [Dereniowski, Osula, and Rzążewski, Finding small-width connected path-decompositions in polynomial time. Theor. Comput. Sci., 794:85–100, 2019]. For our algorithm, we enrich the typical sequence technique that is able to deal with the connectivity demand. Typical sequences have been introduced in [Bodlaender and Kloks. Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms, 21(2):358–402, 1996] for the design of linear parameterized algorithms for treewidth and pathwidth. While this technique has been later applied to other parameters, none of its advancements was able to deal with the connectivity demand, as it is a "global" demand that concerns an unbounded number of parts of the graph of unbounded size. The proposed extension is based on an encoding of the connectivity property that is quite versatile and may be adapted so to deliver linear parameterized algorithms for the connected variants of other width parameters as well. An immediate consequence of our result is a 2^O(k²)⋅n time algorithm for the monotone and connected version of the edge search number.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph theory
  • Theory of computation → Fixed parameter tractability
Keywords
  • Graph decompositions
  • Parameterized algorithms
  • Typical sequences
  • Pathwidth
  • Graph searching

Metrics

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

Related Versions

References

  1. Isolde Adler, Christophe Paul, and Dimitrios M. Thilikos. Connected search for a lazy robber. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS, volume 150 of Leibniz International Proceedings in Informatics, pages 7:1-7:14, 2019. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2019.7.
  2. Stefan Arnborg, Derek G. Corneil, and Andrzej Proskurowski. Complexity of finding embeddings in a k-tree. SIAM Journal on Algebraic Discrete Methods, 8(2):277-284, 1987. Google Scholar
  3. Lali Barrière, Paola Flocchini, Fedor V. Fomin, Pierre Fraigniaud, Nicolas Nisse, Nicola Santoro, and Dimitrios M. Thilikos. Connected graph searching. Information and Computation, 219:1-16, 2012. URL: https://doi.org/10.1016/j.ic.2012.08.004.
  4. Lali Barrière, Paola Flocchini, Pierre Fraigniaud, and Nicola Santoro. Capture of an intruder by mobile agents. In 14th Annual ACM Symposium on Parallel Algorithms and Architectures, SPAA, pages 200-209. ACM, 2002. URL: https://doi.org/10.1145/564870.564906.
  5. Lali Barrière, Pierre Fraigniaud, Nicola Santoro, and Dimitrios M. Thilikos. Searching is not jumping. In 29th International Workshop on Graph-Theoretic Concepts in Computer Science, WG, volume 2880 of Lecture Notes in Computer Science, pages 34-45, 2003. URL: https://doi.org/10.1007/978-3-540-39890-5_4.
  6. Hans L. Bodlaender, Michael R. Fellows, and Dimitrios M. Thilikos. Derivation of algorithms for cutwidth and related graph layout parameters. Journal of Computer and System Sciences, 75(4):231-244, 2009. URL: https://doi.org/10.1016/j.jcss.2008.10.003.
  7. Hans L. Bodlaender, Lars Jaffke, and Jan Arne Telle. Typical sequences revisited - computing width parameters of graphs. In 37th International Symposium on Theoretical Aspects of Computer Science, STACS, volume 154 of Leibniz International Proceedings in Informatics, pages 57:1-57:16, 2020. URL: https://doi.org/10.4230/LIPIcs.STACS.2020.57.
  8. Hans L. Bodlaender and Ton Kloks. Efficient and constructive algorithms for the pathwidth and treewidth of graphs. Journal of Algorithms, 21(2):358-402, 1996. URL: https://doi.org/10.1006/jagm.1996.0049.
  9. Hans L. Bodlaender and Dimitrios M. Thilikos. Constructive linear time algorithms for branchwidth. In 24th International Colloquium on Automata, Languages and Programming, ICALP, volume 1256 of Lecture Notes in Computer Science, pages 627-637, 1997. URL: https://doi.org/10.1007/3-540-63165-8_217.
  10. Hans L. Bodlaender and Dimitrios M. Thilikos. Computing small search numbers in linear time. In First International Workshop on Parameterized and Exact Computation, IWPEC, volume 3162 of Lecture Notes in Computer Science, pages 37-48, 2004. URL: https://doi.org/10.1007/978-3-540-28639-4_4.
  11. Mikołaj Bojańczyk and Michal Pilipczuk. Optimizing tree decompositions in MSO. In 34th Symposium on Theoretical Aspects of Computer Science, STACS, volume 66 of Leibniz International Proceedings in Informatics, pages 15:1-15:13, 2017. URL: https://doi.org/10.4230/LIPIcs.STACS.2017.15.
  12. R. Breisch. An intuitive approach to speleotopology. Southwestern Cavers (A publication of the Southwestern Region of the National Speleological Society), VI(5):72-78, 1967. Google Scholar
  13. Gary Chartrand, Ping Zhang, Teresa W. Haynes, Michael A. Henning, Fred R. McMorris, and Robert C. Brigham. Graphical measurement. In Handbook of Graph Theory, pages 872-951. Chapman & Hall / Taylor & Francis, 2003. URL: https://doi.org/10.1201/9780203490204.ch9.
  14. 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.
  15. Bruno Courcelle and Jens Lagergren. Equivalent definitions of recognizability for sets of graphs of bounded tree-width. Mathematical Structure for Computer Sciecnce, 6(2):141-165, 1996. URL: https://doi.org/10.1017/S096012950000092X.
  16. Dariusz Dereniowski. From pathwidth to connected pathwidth. SIAM Journal on Discrete Mathematics, 26(4):1709-1732, 2012. URL: https://doi.org/10.1137/110826424.
  17. Dariusz Dereniowski, Dorota Osula, and Paweł Rzążewski. Finding small-width connected path decompositions in polynomial time. Theoretical Computer Science, 794:85-100, 2019. URL: https://doi.org/10.1016/j.tcs.2019.03.039.
  18. Fedor V. Fomin. Complexity of connected search when the number of searchers is small. https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=2ahUKEwiWrP77jfXoAhXFiVwKHW3gAQ4QFjABegQIBRAB&url=http%3A%2F%2Ffiles.thilikos.info%2Fdata%2Fconferences%2FGRASTA2017%2Fopen_problems%2FGRASTA2017-open_problems.pdf&usg=AOvVaw1lbev892ugbeQtQhMWUwdO, 2017.
  19. Fedor V. Fomin and Dimitrios M. Thilikos. An annotated bibliography on guaranteed graph searching. Theoretical Computer Science, 399(3):236-245, 2008. URL: https://doi.org/10.1016/j.tcs.2008.02.040.
  20. Martin Fürer. Faster computation of path-width. In 27th International Workshop on Combinatorial Algorithms, IWOCA, Lecture Notes in Computer Science, pages 385-396, 2016. URL: https://doi.org/10.1007/978-3-319-44543-4_30.
  21. P. A. Golovach. Equivalence of two formalizations of a search problem on a graph (Russian). Vestnik Leningrad. Univ. Mat. Mekh. Astronom., vyp. 1:10-14, 122, 1989. translation in Vestnik Leningrad Univ. Math. 22 (1989), no. 1, 13-19. Google Scholar
  22. Petr A. Golovach (П. А. Головач). A topological invariant in pursuit problems, Об одном топологическом инварианте в задачах преследования. Differentsialânye Uravneniya (Differential Equations), Дифференц. уравнения, 25(6):923-929, 1989. URL: http://mi.mathnet.ru/de6861.
  23. Jisu Jeong, Eun Jung Kim, and Sang-il Oum. Constructive algorithm for path-width of matroids. In 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 1695-1704, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch116.
  24. Jisu Jeong, Eun Jung Kim, and Sang-il Oum. The "art of trellis decoding" is fixed-parameter tractable. IEEE Transactions on Information Theory, 63(11):7178-7205, 2017. URL: https://doi.org/10.1109/TIT.2017.2740283.
  25. Jisu Jeong, Eun Jung Kim, and Sang-il Oum. Finding branch-decompositions of matroids, hypergraphs, and more. In 45th International Colloquium on Automata, Languages, and Programming, ICALP, volume 107 of Leibniz International Proceedings in Informatics, pages 80:1-80:14, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.80.
  26. Mamadou Moustapha Kanté, Christophe Paul, and Dimitrios M. Thilikos. A linear fixed parameter tractable algorithm for connected pathwidth. CoRR, abs/2004.11937, 2020. URL: http://arxiv.org/abs/2004.11937.
  27. Nancy G. Kinnersley. The vertex separation number of a graph equals its path-width. Information Processing Letters, 42(6):345-350, 1992. URL: https://doi.org/10.1016/0020-0190(92)90234-M.
  28. Lefteris M. Kirousis and Christos H. Papadimitriou. Interval graphs and seatching. Discrete Mathematics, 55(2):181-184, 1985. URL: https://doi.org/10.1016/0012-365X(85)90046-9.
  29. Lefteris M. Kirousis and Christos H. Papadimitriou. Searching and pebbling. Theoretical Computer Science, 47(3):205-218, 1986. URL: https://doi.org/10.1016/0304-3975(86)90146-5.
  30. Jens Lagergren and Stefan Arnborg. Finding minimal forbidden minors using a finite congruence. In 18th International Colloquium on Automata, Languages and Programming, ICALP, volume 510 of Lecture Notes in Computer Science, pages 532-543, 1991. URL: https://doi.org/10.1007/3-540-54233-7_161.
  31. Rolf H. Möhring. Graph problems related to gate matrix layout and PLA folding. In Computational graph theory, volume 7 of Comput. Suppl., pages 17-51. Springer, 1990. Google Scholar
  32. Ronan Pardo Soares. Pursuit-Evasion, Decompositions and Convexity on Graphs. Theses, Université Nice Sophia Antipolis, November 2013. URL: https://tel.archives-ouvertes.fr/tel-00908227.
  33. Torrence D. Parsons. Pursuit-evasion in a graph. In International Conference on Theory and applications of graphs, volume 642 of Lecture Notes in Mathematics, pages 426-441. Springer, 1978. Google Scholar
  34. Torrence D. Parsons. The search number of a connected graph. In 9th Southeastern Conference on Combinatorics, Graph Theory and Computing, volume XXI of Congress. Numer., XXI, pages 549-554. Utilitas Math., 1978. Google Scholar
  35. Nikolai N. Petrov (Н. Н. Петров). A problem of pursuit in the absence of information on the pursued, (Задачи преследования при отсутствии информации об убегающем). Differentsial'nye Uravneniya (Differential Equations), (Дифференц. уравнения), 18(8):1345-1352, 1468, 1982. URL: http://mi.mathnet.ru/de4628.
  36. Neil Robertson and Paul D. Seymour. Graph minors. I. excluding a forest. Journal of Combinatorial Theory, Series B, 35(1):39-61, 1983. URL: https://doi.org/10.1016/0095-8956(83)90079-5.
  37. Neil Robertson and Paul D. Seymour. Graph minors. XX. wagner’s conjecture. Journal of Combinatorial Theory, Series B, 92(2):325-357, 2004. URL: https://doi.org/10.1016/j.jctb.2004.08.001.
  38. Dimitrios M. Thilikos, Maria J. Serna, and Hans L. Bodlaender. Constructive linear time algorithms for small cutwidth and carving-width. In 11th International Conference on Algorithms and Computation, ISAAC, volume 1969 of Lecture Notes in Computer Science, pages 192-203, 2000. URL: https://doi.org/10.1007/3-540-40996-3_17.
  39. Dimitrios M. Thilikos, Maria J. Serna, and Hans L. Bodlaender. Cutwidth I: A linear time fixed parameter algorithm. Journal of Algorithms, 56(1):1-24, 2005. URL: https://doi.org/10.1016/j.jalgor.2004.12.001.
  40. Dimitrios M. Thilikos, Maria J. Serna, and Hans L. Bodlaender. Cutwidth II: algorithms for partial w-trees of bounded degree. Journal of Algorithms, 56(1):25-49, 2005. URL: https://doi.org/10.1016/j.jalgor.2004.12.003.
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 About DROPS Imprint Privacy Contact