Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs

Authors Jana Novotná, Karolina Okrasa, Michał Pilipczuk, Paweł Rzążewski , Erik Jan van Leeuwen, Bartosz Walczak

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Author Details

Jana Novotná
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Karolina Okrasa
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
Michał 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
Erik Jan van Leeuwen
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Bartosz Walczak
  • Department of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland


The results presented in this paper were obtained during the Parameterized Algorithms Retreat of the algorithms group of the University of Warsaw (PARUW), held in Karpacz in February 2019. This Retreat was financed by the project CUTACOMBS, which has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 714704).

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Jana Novotná, Karolina Okrasa, Michał Pilipczuk, Paweł Rzążewski, Erik Jan van Leeuwen, and Bartosz Walczak. Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 23:1-23:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Let C and D be hereditary graph classes. Consider the following problem: given a graph G in D, find a largest, in terms of the number of vertices, induced subgraph of G that belongs to C. We prove that it can be solved in 2^{o(n)} time, where n is the number of vertices of G, if the following conditions are satisfied: - the graphs in C are sparse, i.e., they have linearly many edges in terms of the number of vertices; - the graphs in D admit balanced separators of size governed by their density, e.g., O(Delta) or O(sqrt{m}), where Delta and m denote the maximum degree and the number of edges, respectively; and - the considered problem admits a single-exponential fixed-parameter algorithm when parameterized by the treewidth of the input graph. This leads, for example, to the following corollaries for specific classes C and D: - a largest induced forest in a P_t-free graph can be found in 2^{O~(n^{2/3})} time, for every fixed t; and - a largest induced planar graph in a string graph can be found in 2^{O~(n^{3/4})} time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Problems, reductions and completeness
  • subexponential algorithm
  • feedback vertex set
  • P_t-free graphs
  • string graphs


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  1. 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. (in Russian). Google Scholar
  2. 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. Google Scholar
  3. Ivan Bliznets, Fedor V. Fomin, Michał Pilipczuk, and Yngve Villanger. Largest Chordal and Interval Subgraphs Faster than 2ⁿ. Algorithmica, 76(2):569-594, 2016. URL:
  4. Hans L. Bodlaender, Pål Grønås Drange, Markus S. Dregi, Fedor V. Fomin, Daniel Lokshtanov, and Michał Pilipczuk. A c^k n 5-Approximation Algorithm for Treewidth. SIAM J. Comput., 45(2):317-378, 2016. URL:
  5. John Adrian Bondy and Miklós Simonovits. Cycles of even length in graphs. J. Combin. Theory Ser. B, 16(2):97-105, 1974. Google Scholar
  6. Édouard Bonnet and Paweł Rzążewski. Optimality Program in Segment and String Graphs. Algorithmica, 81(7):3047-3073, 2019. URL:
  7. Nina Chiarelli, Tatiana Romina Hartinger, Matthew Johnson, Martin Milanič, and Daniël Paulusma. Minimum connected transversals in graphs: New hardness results and tractable cases using the price of connectivity. Theor. Comput. Sci., 705:75-83, 2018. URL:
  8. Maria Chudnovsky, Marcin Pilipczuk, Michał Pilipczuk, and Stéphan Thomassé. On the Maximum Weight Independent Set Problem in graphs without induced cycles of length at least five. CoRR, abs/1903.04761, 2019. URL:
  9. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL:
  10. Marek Cygan, Dániel Marx, Marcin Pilipczuk, and Michal Pilipczuk. Hitting forbidden subgraphs in graphs of bounded treewidth. Inf. Comput., 256:62-82, 2017. URL:
  11. Zdeněk Dvořák and Sergey Norin. Treewidth of graphs with balanced separations. J. Combin. Theory Ser. B, 137:137-144, 2019. URL:
  12. Fedor V. Fomin, Serge Gaspers, Daniel Lokshtanov, and Saket Saurabh. Exact algorithms via monotone local search. In STOC 2016, pages 764-775. ACM, 2016. Google Scholar
  13. Fedor V. Fomin, Ioan Todinca, and Yngve Villanger. Exact Algorithm for the Maximum Induced Planar Subgraph Problem. In ESA 2011, volume 6942 of LNCS, pages 287-298. Springer, 2011. Google Scholar
  14. Fedor V. Fomin, Ioan Todinca, and Yngve Villanger. Large Induced Subgraphs via Triangulations and CMSO. SIAM J. Comput., 44(1):54-87, 2015. Google Scholar
  15. Andrzej Grzesik, Tereza Klimošová, Marcin Pilipczuk, and Michał Pilipczuk. Polynomial-time algorithm for Maximum Weight Independent Set on P₆-free graphs. In SODA 2019, pages 1257-1271. SIAM, 2019. Google Scholar
  16. Tomasz Kociumaka and Marcin Pilipczuk. Deleting vertices to graphs of bounded genus. CoRR, abs/1706.04065, 2017. URL:
  17. Christian Komusiewicz. Tight Running Time Lower Bounds for Vertex Deletion Problems. ACM Trans. on Comput. Theory (TOCT), 10(2):6:1-6:18, 2018. Google Scholar
  18. James R. Lee. Separators in Region Intersection Graphs. In ITCS 2017, volume 67 of LIPIcs, pages 1:1-1:8. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. Google Scholar
  19. John M. Lewis and Mihalis Yannakakis. The Node-Deletion Problem for Hereditary Properties is NP-Complete. J. Comput. Syst. Sci., 20(2):219-230, 1980. URL:
  20. 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. Google Scholar
  21. Marcin Pilipczuk and Michał Pilipczuk. Finding a Maximum Induced Degenerate Subgraph Faster Than 2ⁿ. In IPEC 2012, volume 7535 of LNCS, pages 3-12. Springer, 2012. Google Scholar
  22. Michał Pilipczuk. Problems Parameterized by Treewidth Tractable in Single Exponential Time: A Logical Approach. In MFCS 2011, volume 6907, pages 520-531. Springer, 2011. Google Scholar
  23. Ewald Speckenmeyer. Untersuchungen zum Feedback Vertex Set Problem in ungerichteten Graphen. PhD thesis, Universität Paderborn, 1983. In German. Google Scholar
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