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Stretch-Width

Authors Édouard Bonnet , Julien Duron

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LIPIcs.IPEC.2023.8.pdf
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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Contraction sequences
  • twin-width
  • clique-width
  • algorithms
  • algorithmic metatheorems

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Abstract

We introduce a new parameter, called stretch-width, that we show sits strictly between clique-width and twin-width. Unlike the reduced parameters [BKW '22], planar graphs and polynomial subdivisions do not have bounded stretch-width. This leaves open the possibility of efficient algorithms for a broad fragment of problems within Monadic Second-Order (MSO) logic on graphs of bounded stretch-width. In this direction, we prove that graphs of bounded maximum degree and bounded stretch-width have at most logarithmic treewidth. As a consequence, in classes of bounded stretch-width, Maximum Independent Set can be solved in subexponential time 2^{Õ(n^{8/9})} on n-vertex graphs, and, if further the maximum degree is bounded, Existential Counting Modal Logic [Pilipczuk '11] can be model-checked in polynomial time. We also give a polynomial-time O(OPT²)-approximation for the stretch-width of symmetric 0,1-matrices or ordered graphs.
Somewhat unexpectedly, we prove that exponential subdivisions of bounded-degree graphs have bounded stretch-width. This allows to complement the logarithmic upper bound of treewidth with a matching lower bound. We leave as open the existence of an efficient approximation algorithm for the stretch-width of unordered graphs, if the exponential subdivisions of all graphs have bounded stretch-width, and if graphs of bounded stretch-width have logarithmic clique-width (or rank-width).

Cite As Get BibTex

Édouard Bonnet and Julien Duron. Stretch-Width. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.IPEC.2023.8

Author Details

Édouard Bonnet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Julien Duron
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France

References

  1. Tara Abrishami, Maria Chudnovsky, Sepehr Hajebi, and Sophie Spirkl. Induced subgraphs and tree-decompositions III. Three-path-configurations and logarithmic tree-width. Advances in Combinatorics, 2022. Google Scholar
  2. Ambroise Baril, Miguel Couceiro, and Victor Lagerkvist. Linear bounds between cliquewidth and component twin-width and applications, 2023. URL: https://ramics20.lis-lab.fr/slides/slidesAmbroise.pdf.
  3. Pierre Bergé, Édouard Bonnet, and Hugues Déprés. Deciding twin-width at most 4 is NP-complete. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 18:1-18:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.18.
  4. Pierre Bergé, Édouard Bonnet, Hugues Déprés, and Rémi Watrigant. Approximating highly inapproximable problems on graphs of bounded twin-width. In Petra Berenbrink, Patricia Bouyer, Anuj Dawar, and Mamadou Moustapha Kanté, editors, 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023, March 7-9, 2023, Hamburg, Germany, volume 254 of LIPIcs, pages 10:1-10:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.STACS.2023.10.
  5. 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, Florence, Italy, January 22-25, 2023, pages 3282-3304. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch125.
  6. Marthe Bonamy, Edouard Bonnet, Hugues Déprés, Louis Esperet, Colin Geniet, Claire Hilaire, Stéphan Thomassé, and Alexandra Wesolek. Sparse graphs with bounded induced cycle packing number have logarithmic treewidth. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 3006-3028. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch116.
  7. Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width III: Max Independent Set, Min Dominating Set, and Coloring. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 35:1-35:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.35.
  8. Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width II: small classes. Combinatorial Theory, 2(2), 2022. URL: https://doi.org/10.5070/C62257876.
  9. Édouard Bonnet, Ugo Giocanti, Patrice Ossona de Mendez, Pierre Simon, Stéphan Thomassé, and Szymon Torunczyk. Twin-width IV: ordered graphs and matrices. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 924-937. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520037.
  10. Édouard Bonnet, Eun Jung Kim, Amadeus Reinald, and Stéphan Thomassé. Twin-width VI: the lens of contraction sequences. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1036-1056. SIAM, 2022. Google Scholar
  11. Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width I: tractable FO model checking. J. ACM, 69(1):3:1-3:46, 2022. URL: https://doi.org/10.1145/3486655.
  12. Édouard Bonnet, O-joung Kwon, and David R. Wood. Reduced bandwidth: a qualitative strengthening of twin-width in minor-closed classes (and beyond). CoRR, abs/2202.11858, 2022. URL: https://arxiv.org/abs/2202.11858.
  13. 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.
  14. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000. URL: https://doi.org/10.1007/s002249910009.
  15. Zdenek Dvorák and Sergey Norin. Treewidth of graphs with balanced separations. J. Comb. Theory, Ser. B, 137:137-144, 2019. URL: https://doi.org/10.1016/j.jctb.2018.12.007.
  16. Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  17. Petr A. Golovach, Giannos Stamoulis, and Dimitrios M. Thilikos. Model-checking for first-order logic with disjoint paths predicates in proper minor-closed graph classes. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 3684-3699. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch141.
  18. Frank Gurski and Egon Wanke. The tree-width of clique-width bounded graphs without K_n, n. In Ulrik Brandes and Dorothea Wagner, editors, Graph-Theoretic Concepts in Computer Science, 26th International Workshop, WG 2000, Konstanz, Germany, June 15-17, 2000, Proceedings, volume 1928 of Lecture Notes in Computer Science, pages 196-205. Springer, 2000. URL: https://doi.org/10.1007/3-540-40064-8_19.
  19. Russell Impagliazzo and Ramamohan Paturi. On the Complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
  20. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: https://doi.org/10.1006/jcss.2001.1774.
  21. Sang-il Oum. Approximating rank-width and clique-width quickly. ACM Trans. Algorithms, 5(1):10:1-10:20, 2008. URL: https://doi.org/10.1145/1435375.1435385.
  22. Michal Pilipczuk. Problems parameterized by treewidth tractable in single exponential time: A logical approach. In Filip Murlak and Piotr Sankowski, editors, Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Warsaw, Poland, August 22-26, 2011. Proceedings, volume 6907 of Lecture Notes in Computer Science, pages 520-531. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22993-0_47.
  23. Svatopluk Poljak. A note on stable sets and colorings of graphs. Commentationes Mathematicae Universitatis Carolinae, 15(2):307-309, 1974. Google Scholar
  24. Nicole Schirrmacher, Sebastian Siebertz, Giannos Stamoulis, Dimitrios M. Thilikos, and Alexandre Vigny. Model checking disjoint-paths logic on topological-minor-free graph classes. CoRR, abs/2302.07033, 2023. URL: https://doi.org/10.48550/arXiv.2302.07033.
  25. Ni Luh Dewi Sintiari and Nicolas Trotignon. (theta, triangle)-free and (even hole, k_4)-free graphs - part 1: Layered wheels. J. Graph Theory, 97(4):475-509, 2021. URL: https://doi.org/10.1002/jgt.22666.
  26. Szymon Toruńczyk. Flip-width: Cops and robber on dense graphs. CoRR, abs/2302.00352, 2023. URL: https://doi.org/10.48550/arXiv.2302.00352.
  27. Jakub Černý. Coloring circle graphs. Electronic Notes in Discrete Mathematics, 29:457-461, 2007. European Conference on Combinatorics, Graph Theory and Applications. URL: https://doi.org/10.1016/j.endm.2007.07.072.
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