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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

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