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Max Weight Independent Set in Graphs with No Long Claws: An Analog of the Gyárfás' Path Argument

Authors Konrad Majewski , Tomáš Masařík , Jana Novotná , Karolina Okrasa , Marcin Pilipczuk , Paweł Rzążewski , Marek Sokołowski

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Author Details

Konrad Majewski
  • Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Tomáš Masařík
  • Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Jana Novotná
  • Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Karolina Okrasa
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
  • Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
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
Marek Sokołowski
  • Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland

Funding

This research is part of projects that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme Grant Agreement 714704 (JN, KO, MP, PRz) and 948057 (KM, TM, MS).
  • Rzążewski, Paweł: Partially supported by Polish National Science Centre grant no. 2018/31/D/ST6/00062.

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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 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 93:1-93:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ICALP.2022.93

Abstract

We revisit recent developments for the Maximum Weight Independent Set problem in graphs excluding a subdivided claw S_{t,t,t} as an induced subgraph [Chudnovsky, Pilipczuk, Pilipczuk, Thomassé, SODA 2020] and provide a subexponential-time algorithm with improved running time 2^𝒪(√nlog n) and a quasipolynomial-time approximation scheme with improved running time 2^𝒪(ε^{-1} log⁵ n). The Gyárfás' path argument, a powerful tool that is the main building block for many algorithms in P_t-free graphs, ensures that given an n-vertex P_t-free graph, in polynomial time we can find a set P of at most t-1 vertices, such that every connected component of G-N[P] has at most n/2 vertices. Our main technical contribution is an analog of this result for S_{t,t,t}-free graphs: given an n-vertex S_{t,t,t}-free graph, in polynomial time we can find a set P of 𝒪(t log n) vertices and an extended strip decomposition (an appropriate analog of the decomposition into connected components) of G-N[P] such that every particle (an appropriate analog of a connected component to recurse on) of the said extended strip decomposition has at most n/2 vertices.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Approximation algorithms
Keywords
  • Max Independent Set
  • subdivided claw
  • QPTAS
  • subexponential-time algorithm

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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. CoRR, abs/2107.05434, 2021. URL: http://arxiv.org/abs/2107.05434.
  2. 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.
  3. Tara Abrishami, Maria Chudnovsky, Cemil Dibek, Stéphan Thomassé, Nicolas Trotignon, and Kristina Vušković. Graphs with polynomially many minimal separators. J. Comb. Theory, Ser. B, 152:248-280, 2022. URL: https://doi.org/10.1016/j.jctb.2021.10.003.
  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. 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
  6. 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.
  7. 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.
  8. Brenda S. Baker. Approximation algorithms for NP-complete problems on planar graphs. J. ACM, 41(1):153-180, 1994. URL: https://doi.org/10.1145/174644.174650.
  9. 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.
  10. 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: http://arxiv.org/abs/1907.04585.
  11. 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.
  12. Maria Chudnovsky and Paul D. Seymour. The structure of claw-free graphs. In Bridget S. Webb, editor, Surveys in Combinatorics, 2005 [invited lectures from the Twentieth British Combinatorial Conference, Durham, UK, July 2005], volume 327 of London Mathematical Society Lecture Note Series, pages 153-171. Cambridge University Press, 2005. URL: https://doi.org/10.1017/cbo9780511734885.008.
  13. Maria Chudnovsky and Paul D. Seymour. The three-in-a-tree problem. Comb., 30(4):387-417, 2010. URL: https://doi.org/10.1007/s00493-010-2334-4.
  14. Jack Edmonds. Paths, trees, and flowers. Canadian Journal of Mathematics, 17:449-467, 1965. URL: https://doi.org/10.4153/CJM-1965-045-4.
  15. 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.
  16. 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.
  17. 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.
  18. András Gyárfás. On Ramsey covering-numbers. In Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. II, number 10 in Colloq. Math. Soc. Janos Bolyai, pages 801-816. North-Holland, Amsterdam, 1975. Google Scholar
  19. András Gyárfás. Problems from the world surrounding perfect graphs. In Proceedings of the International Conference on Combinatorial Analysis and its Applications, (Pokrzywna, 1985), number 19 in Zastos. Mat., pages 413-441, 1987. URL: https://doi.org/10.4064/am-19-3-4-413-441.
  20. Johan Håstad. Clique is hard to approximate within n^1-ε. Acta Math., 182(1):105-142, 1999. URL: https://doi.org/10.1007/BF02392825.
  21. Daniel Lokshtanov, 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.
  22. 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, 2022. URL: https://doi.org/10.48550/ARXIV.2203.04836.
  23. 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.
  24. 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.
  25. 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.
  26. David Zuckerman. Linear degree extractors and the inapproximability of Max Clique and Chromatic Number. Theory of Computing, 3(1):103-128, 2007. URL: https://doi.org/10.4086/toc.2007.v003a006.
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