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Tree Decompositions Meet Induced Matchings: Beyond Max Weight Independent Set

Authors Paloma T. Lima , Martin Milanič , Peter Muršič , Karolina Okrasa , Paweł Rzążewski , Kenny Štorgel

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Author Details

Paloma T. Lima
  • IT University of Copenhagen, Denmark
Martin Milanič
  • FAMNIT and IAM, University of Primorska, Koper, Slovenia
Peter Muršič
  • FAMNIT, University of Primorska, Koper, Slovenia
Karolina Okrasa
  • Warsaw University of Technology, Poland
Paweł Rzążewski
  • Warsaw University of Technology, Poland
  • University of Warsaw, Poland
Kenny Štorgel
  • Faculty of Information Studies, Novo mesto, Slovenia
  • FAMNIT, University of Primorska, Koper, Slovenia

Funding

  • Lima, Paloma T.: Supported by the Independent Research Fund Denmark grant agreement number 2098-00012B.
  • Milanič, Martin: Supported in part by the Slovenian Research and Innovation Agency (I0-0035, research program P1-0285 and research projects J1-3001, J1-3002, J1-3003, J1-4008, J1-4084, and N1-0370) and by the research program CogniCom (0013103) at the University of Primorska.
  • Muršič, Peter: Supported in part by the Slovenian Research and Innovation Agency (J1-4008).
  • Okrasa, Karolina: Supported by the Polish National Science Centre grant no. 2021/41/N/ST6/01507.
  • Rzążewski, Paweł: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme grant agreement number 948057.
  • Štorgel, Kenny: Supported in part by the Slovenian Research and Innovation Agency (research program P1-0383 and research projects J1-3002 and J1-4008).

Cite As Get BibTex

Paloma T. Lima, Martin Milanič, Peter Muršič, Karolina Okrasa, Paweł Rzążewski, and Kenny Štorgel. Tree Decompositions Meet Induced Matchings: Beyond Max Weight Independent Set. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 85:1-85:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.85

Abstract

For a tree decomposition 𝒯 of a graph G, by μ(𝒯) we denote the size of a largest induced matching in G all of whose edges intersect one bag of 𝒯. The induced matching treewidth of a graph G is the minimum value of μ(𝒯) over all tree decompositions 𝒯 of G. Yolov [SODA 2018] proved that for graphs of bounded induced matching treewidth, tree decompositions with bounded μ(𝒯) can be computed in polynomial time and Max Weight Independent Set can be solved in polynomial time. 
In this paper we explore what other problems are tractable in such classes of graphs. As our main result, we give a polynomial-time algorithm for Min Weight Feedback Vertex Set. We also provide some positive results concerning packing induced subgraphs, which in particular imply a PTAS for the problem of finding a largest induced subgraph of bounded treewidth.
These results suggest that in graphs of bounded induced matching treewidth, one could find in polynomial time a maximum-weight induced subgraph of bounded treewidth satisfying a given CMSO₂ formula. We conjecture that such a result indeed holds and prove it for graphs of bounded tree-independence number, which form a rich and important family of subclasses of graphs of bounded induced matching treewidth.
We complement these algorithmic results with a number of complexity and structural results concerning induced matching treewidth, including a linear relation to treewidth for graphs with bounded degree.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • induced matching treewidth
  • tree-independence number
  • feedback vertex set
  • induced packing
  • algorithmic meta-theorem

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References

  1. Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl, and Kristina Vušković. Tree independence number I. (Even hole, diamond, pyramid)-free graphs. Journal of Graph Theory, 106(4):923-943, 2024. URL: https://doi.org/10.1002/jgt.23104.
  2. Tara Abrishami, Marcin Briański, Jadwiga Czyżewska, Rose McCarty, Martin Milanič, Paweł Rzążewski, and Bartosz Walczak. Excluding a clique or a biclique in graphs of bounded induced matching treewidth. CoRR, abs/2405.04617, 2024. URL: https://arxiv.org/abs/2405.04617.
  3. Tara Abrishami, Maria Chudnovsky, Sepehr Hajebi, and Sophie Spirkl. Induced subgraphs and tree decompositions III. Three-path-configurations and logarithmic treewidth. Advances in Combinatorics, 2022:6:1-29, 2022. URL: https://doi.org/10.19086/aic.2022.6.
  4. Tara Abrishami, Maria Chudnovsky, Marcin Pilipczuk, Paweł Rzążewski, and Paul D. Seymour. Induced subgraphs of bounded treewidth and the container method. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 1948-1964. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.116.
  5. Isolde Adler. Width functions for hypertree decompositions. PhD thesis, Albert-Ludwigs-Universität Freiburg im Breisgau, 2006. Google Scholar
  6. Betül Ahat, Tınaz Ekim, and Z. Caner Taşkın. Integer programming formulations and benders decomposition for the maximum induced matching problem. INFORMS J. Comput., 30(1):43-56, 2018. URL: https://doi.org/10.1287/ijoc.2017.0764.
  7. 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, April 1987. URL: https://doi.org/10.1137/0608024.
  8. Rangaswami Balakrishnan and P. Paulraja. Powers of chordal graphs. J. Austral. Math. Soc. Ser. A, 35(2):211-217, 1983. Google Scholar
  9. Walid Ben-Ameur, Mohamed-Ahmed Mohamed-Sidi, and José Neto. The k-separator problem: polyhedra, complexity and approximation results. J. Comb. Optim., 29(1):276-307, 2015. URL: https://doi.org/10.1007/s10878-014-9753-x.
  10. Benjamin Bergougnoux, Édouard Bonnet, Nick Brettell, and O-joung Kwon. Close relatives of feedback vertex set without single-exponential algorithms parameterized by treewidth. In Yixin Cao and Marcin Pilipczuk, editors, 15th International Symposium on Parameterized and Exact Computation, IPEC 2020, December 14-18, 2020, Hong Kong, China (Virtual Conference), volume 180 of LIPIcs, pages 3:1-3:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPICS.IPEC.2020.3.
  11. 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.
  12. Benjamin Bergougnoux, Tuukka Korhonen, and Igor Razgon. New width parameters for independent set: One-sided-mim-width and neighbor-depth. In Daniël Paulusma and Bernard Ries, editors, Graph-Theoretic Concepts in Computer Science - 49th International Workshop, WG 2023, Fribourg, Switzerland, June 28-30, 2023, Revised Selected Papers, volume 14093 of Lecture Notes in Computer Science, pages 72-85. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-43380-1_6.
  13. Umberto Bertele and Francesco Brioschi. Nonserial dynamic programming. Academic Press, 1972. Google Scholar
  14. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, 1996. URL: https://doi.org/10.1137/S0097539793251219.
  15. Hans L. Bodlaender, Édouard Bonnet, Lars Jaffke, Dušan Knop, Paloma T. Lima, Martin Milanič, Sebastian Ordyniak, Sukanya Pandey, and Ondrej Suchý. Treewidth is NP-complete on cubic graphs. In Neeldhara Misra and Magnus Wahlström, editors, 18th International Symposium on Parameterized and Exact Computation, IPEC 2023, September 6-8, 2023, Amsterdam, The Netherlands, volume 285 of LIPIcs, pages 7:1-7:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.IPEC.2023.7.
  16. Hans L. Bodlaender, Pål Grønås Drange, Markus S. Dregi, Fedor V. Fomin, Daniel Lokshtanov, and Michal Pilipczuk. A c^k n 5-approximation algorithm for treewidth. SIAM J. Comput., 45(2):317-378, 2016. URL: https://doi.org/10.1137/130947374.
  17. Hans L. Bodlaender and Ton Kloks. Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms, 21(2):358-402, 1996. URL: https://doi.org/10.1006/jagm.1996.0049.
  18. 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.
  19. Vincent Bouchitté and Ioan Todinca. Treewidth and minimum fill-in: Grouping the minimal separators. SIAM J. Comput., 31(1):212-232, 2001. URL: https://doi.org/10.1137/S0097539799359683.
  20. Nick Brettell, Andrea Munaro, Daniël Paulusma, and Shizhou Yang. Comparing width parameters on graph classes. CoRR, abs/2308.05817, 2023. URL: https://arxiv.org/abs/2308.05817.
  21. Kathie Cameron and Pavol Hell. Independent packings in structured graphs. Math. Program., 105(2-3, Ser. B):201-213, 2006. URL: https://doi.org/10.1007/s10107-005-0649-5.
  22. Maria Chudnovsky, Rose McCarty, Marcin Pilipczuk, Michał Pilipczuk, and Paweł Rzążewski. Sparse induced subgraphs in P₆-free graphs. In David P. Woodruff, editor, Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024, pages 5291-5299. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.190.
  23. Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
  24. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, MichałPilipczuk, and Saket Saurabh. Parameterized algorithms. Springer, Cham, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  25. Clément Dallard, Martin Milanič, and Kenny Štorgel. Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure. Journal of Combinatorial Theory, Series B, 167:338-391, 2024. URL: https://doi.org/10.1016/j.jctb.2024.03.005.
  26. Clément Dallard, Martin Milanič, and Kenny Štorgel. Treewidth versus clique number. I. Graph classes with a forbidden structure. SIAM J. Discrete Math., 35(4):2618-2646, 2021. URL: https://doi.org/10.1137/20M1352119.
  27. Clément Dallard, Martin Milanič, and Kenny Štorgel. Treewidth versus clique number. II. Tree-independence number. J. Combin. Theory Ser. B, 164:404-442, 2024. URL: https://doi.org/10.1016/j.jctb.2023.10.006.
  28. Clément Dallard, Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, and Martin Milanič. Computing tree decompositions with small independence number. In Karl Bringmann, Martin Grohe, Gabriele Puppis, and Ola Svensson, editors, 51th International Colloquium on Automata, Languages, and Programming, ICALP 2024, July 8-12, 2024, Tallinn, Estonia, volume 297 of LIPIcs, pages 51:1-51:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.ICALP.2024.51.
  29. Irit Dinur. Mildly exponential reduction from gap 3SAT to polynomial-gap label-cover. Electron. Colloquium Comput. Complex., TR16-128, 2016. URL: https://eccc.weizmann.ac.il/report/2016/128, URL: https://arxiv.org/abs/TR16-128.
  30. Pierre Duchet. Classical perfect graphs: an introduction with emphasis on triangulated and interval graphs. In Topics on perfect graphs, volume 88 of North-Holland Math. Stud., pages 67-96. North-Holland, Amsterdam, 1984. URL: https://doi.org/10.1016/S0304-0208(08)72924-4.
  31. Hiroshi Eto, Fengrui Guo, and Eiji Miyano. Distance-d independent set problems for bipartite and chordal graphs. J. Comb. Optim., 27(1):88-99, 2014. URL: https://doi.org/10.1007/s10878-012-9594-4.
  32. Fedor V. Fomin, Ioan Todinca, and Yngve Villanger. Large induced subgraphs via triangulations and CMSO. SIAM J. Comput., 44(1):54-87, 2015. URL: https://doi.org/10.1137/140964801.
  33. Peter Gartland, Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, and Paweł Rzążewski. Finding large induced sparse subgraphs in C_> t-free graphs in quasipolynomial time. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 330-341. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451034.
  34. Peter Gartland, Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, and Paweł Rzążewski. Finding large induced sparse subgraphs in C_> t-free graphs in quasipolynomial time. CoRR, abs/2007.11402, 2021. URL: https://arxiv.org/abs/2007.11402.
  35. Andrzej Grzesik, Tereza Klimošová, Marcin Pilipczuk, and Michał Pilipczuk. Polynomial-time algorithm for Maximum Weight Independent Set on P₆-free graphs. ACM Trans. Algorithms, 18(1):4:1-4:57, 2022. URL: https://doi.org/10.1145/3414473.
  36. Rudolf Halin. S-functions for graphs. Journal of Geometry, 8(1):171-186, 1976. URL: https://doi.org/10.1007/BF01917434.
  37. Frédéric Havet, Ross J. Kang, and Jean-Sébastien Sereni. Improper coloring of unit disk graphs. Networks, 54(3):150-164, 2009. URL: https://doi.org/10.1002/net.20318.
  38. Bart M. P. Jansen, Daniel Lokshtanov, and Saket Saurabh. A near-optimal planarization algorithm. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1802-1811. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.130.
  39. Dong Yeap Kang, O-joung Kwon, Torstein J. F. Strømme, and Jan Arne Telle. A width parameter useful for chordal and co-comparability graphs. Theoret. Comput. Sci., 704:1-17, 2017. URL: https://doi.org/10.1016/j.tcs.2017.09.006.
  40. Tuukka Korhonen. A single-exponential time 2-approximation algorithm for treewidth. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 184-192. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00026.
  41. Tuukka Korhonen and Daniel Lokshtanov. An improved parameterized algorithm for treewidth. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 528-541. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585245.
  42. Euiwoong Lee. Partitioning a graph into small pieces with applications to path transversal. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 1546-1558. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.101.
  43. Euiwoong Lee. Partitioning a graph into small pieces with applications to path transversal. Math. Program., 177(1-2, Ser. A):1-19, 2019. URL: https://doi.org/10.1007/s10107-018-1255-7.
  44. Paloma T. Lima, Martin Milanič, Peter Muršič, Karolina Okrasa, Paweł Rzążewski, and Kenny Štorgel. Tree decompositions meet induced matchings: beyond max weight independent set. CoRR, abs/2402.15834, 2024. URL: https://arxiv.org/abs/2402.15834.
  45. Daniel Lokshantov, Martin Vatshelle, and Yngve Villanger. Independent set in P₅-free graphs in polynomial time. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 570-581. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.43.
  46. Pasin Manurangsi and Prasad Raghavendra. A birthday repetition theorem and complexity of approximating dense CSPs. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 78:1-78:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPICS.ICALP.2017.78.
  47. Dániel Marx. Tractable hypergraph properties for constraint satisfaction and conjunctive queries. J. ACM, 60(6):42:1-42:51, 2013. URL: https://doi.org/10.1145/2535926.
  48. Pranabendu Misra, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. Parameterized algorithms for even cycle transversal. In Martin Charles Golumbic, Michal Stern, Avivit Levy, and Gila Morgenstern, editors, Graph-Theoretic Concepts in Computer Science - 38th International Workshop, WG 2012, Jerusalem, Israel, June 26-28, 2012, Revised Selcted Papers, volume 7551 of Lecture Notes in Computer Science, pages 172-183. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-34611-8_19.
  49. Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-27875-4.
  50. Stavros D. Nikolopoulos and Leonidas Palios. Minimal separators in P₄-sparse graphs. Discrete Math., 306(3):381-392, 2006. URL: https://doi.org/10.1016/j.disc.2005.12.008.
  51. Yury Orlovich, Alexandre Dolgui, Gerd Finke, Valery Gordon, and Frank Werner. The complexity of dissociation set problems in graphs. Discrete Appl. Math., 159(13):1352-1366, 2011. URL: https://doi.org/10.1016/j.dam.2011.04.023.
  52. Giacomo Paesani, Daniël Paulusma, and Paweł Rzążewski. Feedback Vertex Set and Even Cycle Transversal for H-free graphs: Finding large block graphs. SIAM J. Discret. Math., 36(4):2453-2472, 2022. URL: https://doi.org/10.1137/22m1468864.
  53. B. S. Panda, Arti Pandey, Juhi Chaudhary, Piyush Dane, and Manav Kashyap. Maximum weight induced matching in some subclasses of bipartite graphs. J. Comb. Optim., 40(3):713-732, 2020. URL: https://doi.org/10.1007/s10878-020-00611-2.
  54. Marcin Pilipczuk. A tight lower bound for Vertex Planarization on graphs of bounded treewidth. Discret. Appl. Math., 231:211-216, 2017. URL: https://doi.org/10.1016/j.dam.2016.05.019.
  55. Neil Robertson and Paul D. Seymour. Graph minors. III. Planar tree-width. J. Comb. Theory, Ser. B, 36(1):49-64, 1984. URL: https://doi.org/10.1016/0095-8956(84)90013-3.
  56. Neil Robertson and Paul D. Seymour. Graph Minors. XIII. The Disjoint Paths Problem. J. Comb. Theory, Ser. B, 63(1):65-110, 1995. URL: https://doi.org/10.1006/jctb.1995.1006.
  57. Robert Scheidweiler and Sebastian Wiederrecht. On chordal graph and line graph squares. Discrete Appl. Math., 243:239-247, 2018. URL: https://doi.org/10.1016/j.dam.2018.02.013.
  58. Alex Scott and Paul D. Seymour. A survey of χ-boundedness. J. Graph Theory, 95(3):473-504, 2020. URL: https://doi.org/10.1002/jgt.22601.
  59. Michalis Yannakakis. Node-deletion problems on bipartite graphs. SIAM J. Comput., 10(2):310-327, 1981. URL: https://doi.org/10.1137/0210022.
  60. Nikola Yolov. Minor-matching hypertree width. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 219-233. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.16.
  61. Jie You, Jianxin Wang, and Yixin Cao. Approximate association via dissociation. Discret. Appl. Math., 219:202-209, 2017. URL: https://doi.org/10.1016/J.DAM.2016.11.007.
  62. David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput., 3:103-128, 2007. URL: https://doi.org/10.4086/toc.2007.v003a006.
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