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Max Weight Independent Set in Sparse Graphs with No Long Claws

Authors Tara Abrishami, Maria Chudnovsky , Marcin Pilipczuk , Paweł Rzążewski

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph algorithms
Keywords
  • Max Weight Independent Set
  • subdivided claw
  • hereditary classes

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Abstract

We revisit the recent polynomial-time algorithm for the Max Weight Independent Set (MWIS) problem in bounded-degree graphs that do not contain a fixed graph whose every component is a subdivided claw as an induced subgraph [Abrishami, Chudnovsky, Dibek, Rzążewski, SODA 2022]. 
First, we show that with an arguably simpler approach we can obtain a faster algorithm with running time n^{𝒪(Δ²)}, where n is the number of vertices of the instance and Δ is the maximum degree. Then we combine our technique with known results concerning tree decompositions and provide a polynomial-time algorithm for MWIS in graphs excluding a fixed graph whose every component is a subdivided claw as an induced subgraph, and a fixed biclique as a subgraph.

Cite As Get BibTex

Tara Abrishami, Maria Chudnovsky, Marcin Pilipczuk, and Paweł Rzążewski. Max Weight Independent Set in Sparse Graphs with No Long Claws. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.STACS.2024.4

Author Details

Tara Abrishami
  • Department of Mathematics, University of Hamburg, Germany
Maria Chudnovsky
  • Princeton University, NJ, USA
Marcin Pilipczuk
  • Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Paweł Rzążewski
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
  • Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland

Funding

  • Abrishami, Tara: Supported by the National Science Foundation Award Number DMS-2303251 and the Alexander von Humboldt Foundation.
  • Chudnovsky, Maria: Supported by NSF-EPSRC Grant DMS-2120644 and by AFOSR grant FA9550-22-1-008.
  • Pilipczuk, Marcin: Supported by Polish National Science Centre SONATA BIS-12 grant number 2022/46/E/ST6/00143.
  • Rzążewski, Paweł: This work is a part of the project BOBR that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 948057).

Acknowledgements

We acknowledge the welcoming and productive atmosphere at Dagstuhl Seminar 22481 “Vertex Partitioning in Graphs: From Structure to Algorithms,” where a crucial part of the work leading to the results in this paper was done.

References

  1. Tara Abrishami, Maria Chudnovsky, Cemil Dibek, and Paweł Rzążewski. Polynomial-time algorithm for maximum independent set in bounded-degree graphs with no long induced claws. In Niv Buchbinder Joseph (Seffi) Naor, editor, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference, January 9-12, 2022, pages 1448-1470. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.61.
  2. Tara Abrishami, Maria Chudnovsky, Marcin Pilipczuk, and Paweł Rzążewski. Max Weight Independent Set in sparse graphs with no long claws. CoRR, abs/2309.16995, 2023. URL: https://doi.org/10.48550/ARXIV.2309.16995.
  3. Vladimir E. Alekseev. The effect of local constraints on the complexity of determination of the graph independence number. Combinatorial-algebraic methods in applied mathematics, pages 3-13, 1982. Google Scholar
  4. Vladimir E. Alekseev. On easy and hard hereditary classes of graphs with respect to the independent set problem. Discret. Appl. Math., 132(1-3):17-26, 2003. URL: https://doi.org/10.1016/S0166-218X(03)00387-1.
  5. Vladimir E. Alekseev. Polynomial algorithm for finding the largest independent sets in graphs without forks. Discrete Applied Mathematics, 135(1):3-16, 2004. Russian Translations II. URL: https://doi.org/10.1016/S0166-218X(02)00290-1.
  6. Gábor Bacsó, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Zsolt Tuza, and Erik Jan van Leeuwen. Subexponential-time algorithms for Maximum Independent Set in P_t-free and broom-free graphs. Algorithmica, 81(2):421-438, 2019. URL: https://doi.org/10.1007/s00453-018-0479-5.
  7. Andreas Brandstädt and Raffaele Mosca. Maximum weight independent set for 𝓁-claw-free graphs in polynomial time. Discret. Appl. Math., 237:57-64, 2018. URL: https://doi.org/10.1016/j.dam.2017.11.029.
  8. Andreas Brandstädt and Raffaele Mosca. Maximum weight independent sets for (P₇, triangle)-free graphs in polynomial time. Discret. Appl. Math., 236:57-65, 2018. URL: https://doi.org/10.1016/j.dam.2017.10.003.
  9. Andreas Brandstädt and Raffaele Mosca. Maximum weight independent sets for (S_1,2,4,triangle)-free graphs in polynomial time. Theor. Comput. Sci., 878-879:11-25, 2021. URL: https://doi.org/10.1016/j.tcs.2021.05.027.
  10. Christoph Brause. A subexponential-time algorithm for the Maximum Independent Set Problem in P_t-free graphs. Discret. Appl. Math., 231:113-118, 2017. URL: https://doi.org/10.1016/j.dam.2016.06.016.
  11. Rajesh Chitnis, Marek Cygan, MohammadTaghi Hajiaghayi, Marcin Pilipczuk, and Michał Pilipczuk. Designing FPT algorithms for cut problems using randomized contractions. SIAM J. Comput., 45(4):1171-1229, 2016. URL: https://doi.org/10.1137/15M1032077.
  12. Maria Chudnovsky, Marcin Pilipczuk, Michał Pilipczuk, and Stéphan Thomassé. Quasi-polynomial time approximation schemes for the Maximum Weight Independent Set Problem in H-free graphs. CoRR, abs/1907.04585, 2019. URL: https://arxiv.org/abs/1907.04585.
  13. Maria Chudnovsky, Marcin Pilipczuk, Michał Pilipczuk, and Stéphan Thomassé. Quasi-polynomial time approximation schemes for the Maximum Weight Independent Set Problem in H-free graphs. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 2260-2278. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.139.
  14. Maria Chudnovsky, Stéphan Thomassé, Nicolas Trotignon, and Kristina Vuskovic. Maximum independent sets in (pyramid, even hole)-free graphs. CoRR, abs/1912.11246, 2019. URL: https://arxiv.org/abs/1912.11246.
  15. M. Cygan, F. Fomin, Ł. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized algorithms. Springer, 2015. Google Scholar
  16. Peter Gartland and Daniel Lokshtanov. Independent set on P_k-free graphs in quasi-polynomial time. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 613-624. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00063.
  17. Peter Gartland, Daniel Lokshtanov, Tomáš Masařík, Marcin Pilipczuk, Michał Pilipczuk, and Paweł Rzążewski. Maximum weight independent set in graphs with no long claws in quasi-polynomial time. CoRR, abs/2305.15738, 2023. URL: https://doi.org/10.48550/arXiv.2305.15738.
  18. Michael U. Gerber, Alain Hertz, and Vadim V. Lozin. Stable sets in two subclasses of banner-free graphs. Discrete Applied Mathematics, 132(1-3):121-136, 2003. URL: https://doi.org/10.1016/S0166-218X(03)00395-0.
  19. Andrzej Grzesik, Tereza Klimošová, Marcin Pilipczuk, and Michał Pilipczuk. Polynomial-time algorithm for Maximum Weight Independent Set on P₆-free graphs. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1257-1271. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.77.
  20. M. Grötschel, L. Lovász, and A. Schrijver. Polynomial algorithms for perfect graphs. In C. Berge and V. Chvátal, editors, Topics on Perfect Graphs, volume 88 of North-Holland Mathematics Studies, pages 325-356. North-Holland, 1984. URL: https://doi.org/10.1016/S0304-0208(08)72943-8.
  21. Ararat Harutyunyan, Michael Lampis, Vadim V. Lozin, and Jérôme Monnot. Maximum independent sets in subcubic graphs: New results. Theor. Comput. Sci., 846:14-26, 2020. URL: https://doi.org/10.1016/j.tcs.2020.09.010.
  22. Johan Håstad. Clique is hard to approximate within n^(1-ε). In Acta Mathematica, pages 627-636, 1996. Google Scholar
  23. Richard M. Karp. Reducibility among combinatorial problems. In Complexity of Computer Computations, pages 85-103. Springer US, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  24. Ken-ichi Kawarabayashi and Mikkel Thorup. The minimum k-way cut of bounded size is fixed-parameter tractable. In Rafail Ostrovsky, editor, IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 160-169. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/FOCS.2011.53.
  25. T. Kövari, V. Sós, and P. Turán. On a problem of K. Zarankiewicz. Colloquium Mathematicae, 3(1):50-57, 1954. URL: http://eudml.org/doc/210011.
  26. Ngoc Chi Lê, Christoph Brause, and Ingo Schiermeyer. The Maximum Independent Set Problem in subclasses of S_i,j,k-free graphs. Electron. Notes Discret. Math., 49:43-49, 2015. URL: https://doi.org/10.1016/j.endm.2015.06.008.
  27. 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.
  28. Vadim V. Lozin and Martin Milanič. A polynomial algorithm to find an independent set of maximum weight in a fork-free graph. J. Discrete Algorithms, 6(4):595-604, 2008. URL: https://doi.org/10.1016/j.jda.2008.04.001.
  29. Vadim V. Lozin, Martin Milanič, and Christopher Purcell. Graphs without large apples and the Maximum Weight Independent Set problem. Graphs Comb., 30(2):395-410, 2014. URL: https://doi.org/10.1007/s00373-012-1263-y.
  30. Vadim V. Lozin, Jérôme Monnot, and Bernard Ries. On the maximum independent set problem in subclasses of subcubic graphs. J. Discrete Algorithms, 31:104-112, 2015. URL: https://doi.org/10.1016/j.jda.2014.08.005.
  31. Vadim V. Lozin and Raffaele Mosca. Independent sets in extensions of 2K₂-free graphs. Discret. Appl. Math., 146(1):74-80, 2005. URL: https://doi.org/10.1016/j.dam.2004.07.006.
  32. Vadim V. Lozin and Dieter Rautenbach. Some results on graphs without long induced paths. Inf. Process. Lett., 88(4):167-171, 2003. URL: https://doi.org/10.1016/j.ipl.2003.07.004.
  33. Vadim V. Lozin and Igor Razgon. Tree-width dichotomy. Eur. J. Comb., 103:103517, 2022. URL: https://doi.org/10.1016/j.ejc.2022.103517.
  34. Frédéric Maffray and Lucas Pastor. Maximum weight stable set in (P₇, bull)-free graphs. CoRR, abs/1611.09663, 2016. URL: http://arxiv.org/abs/1611.09663, URL: https://arxiv.org/abs/1611.09663.
  35. Konrad Majewski, Tomáš Masařík, Jana Novotná, Karolina Okrasa, Marcin Pilipczuk, Paweł Rzążewski, and Marek Sokołowski. Max Weight Independent Set in Graphs with No Long Claws: An Analog of the Gyárfás' Path Argument. In Mikołaj Bojańczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), volume 229 of Leibniz International Proceedings in Informatics (LIPIcs), pages 93:1-93:19, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.93.
  36. George J. Minty. On maximal independent sets of vertices in claw-free graphs. Journal of Combinatorial Theory, Series B, 28(3):284-304, 1980. URL: https://doi.org/10.1016/0095-8956(80)90074-X.
  37. Raffaele Mosca. Stable sets in certain P₆-free graphs. Discret. Appl. Math., 92(2-3):177-191, 1999. URL: https://doi.org/10.1016/S0166-218X(99)00046-3.
  38. Raffaele Mosca. Stable sets of maximum weight in (P₇, banner)-free graphs. Discrete Mathematics, 308(1):20-33, 2008. URL: https://doi.org/10.1016/j.disc.2007.03.044.
  39. Raffaele Mosca. Independent sets in (P₆, diamond)-free graphs. Discret. Math. Theor. Comput. Sci., 11(1):125-140, 2009. URL: https://doi.org/10.46298/dmtcs.473.
  40. Raffaele Mosca. Maximum weight independent sets in (P₆, co-banner)-free graphs. Inf. Process. Lett., 113(3):89-93, 2013. URL: https://doi.org/10.1016/j.ipl.2012.10.004.
  41. Raffaele Mosca. Independent sets in (P_4+4, triangle)-free graphs. Graphs Comb., 37(6):2173-2189, 2021. URL: https://doi.org/10.1007/s00373-021-02340-7.
  42. Marcin Pilipczuk, Michał Pilipczuk, and Paweł Rzążewski. Quasi-polynomial-time algorithm for independent set in P_t-free graphs via shrinking the space of induced paths. In Hung Viet Le and Valerie King, editors, 4th Symposium on Simplicity in Algorithms, SOSA 2021, Virtual Conference, January 11-12, 2021, pages 204-209. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976496.23.
  43. Najiba Sbihi. Algorithme de recherche d'un stable de cardinalite maximum dans un graphe sans etoile. Discrete Mathematics, 29(1):53-76, 1980. URL: https://doi.org/10.1016/0012-365X(90)90287-R.
  44. Daniel Weißauer. On the block number of graphs. SIAM J. Discret. Math., 33(1):346-357, 2019. URL: https://doi.org/10.1137/17M1145306.
  45. 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.
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