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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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Subject Classification

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

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Abstract

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.

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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) https://doi.org/10.4230/LIPIcs.IPEC.2019.23

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

Funding

  • Novotná, Jana: Supported by student grants GAUK 1277018, SVV-2017-260452.
  • Pilipczuk, Michał: This work is a part of project TOTAL that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 677651).
  • Rzążewski, Paweł: Partially supported by Polish National Science Centre grant no. 2018/31/D/ST6/00062.
  • Walczak, Bartosz: Partially supported by Polish National Science Centre grant no. 2015/17/B/ST6/01873.

Acknowledgements

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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