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An 11/6-Approximation Algorithm for Vertex Cover on String Graphs

Authors Édouard Bonnet , Paweł Rzążewski

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Approximation algorithms
  • string graphs
  • Vertex Cover
  • Coloring
  • odd girth

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Abstract

We present a 1.8334-approximation algorithm for Vertex Cover on string graphs given with a representation, which takes polynomial time in the size of the representation; the exact approximation factor is 11/6. Recently, the barrier of 2 was broken by Lokshtanov, Panolan, Saurabh, Xue, and Zehavi [SoGC '24] with a 1.9999-approximation algorithm. Thus we increase by three orders of magnitude the distance of the approximation ratio to the trivial bound of 2. Our algorithm is very simple. The intricacies reside in its analysis, where we mainly establish that string graphs without odd cycles of length at most 11 are 8-colorable. Previously, Chudnovsky, Scott, and Seymour [JCTB '21] showed that string graphs without odd cycles of length at most 7 are 80-colorable, and string graphs without odd cycles of length at most 5 have bounded chromatic number.

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Édouard Bonnet and Paweł Rzążewski. An 11/6-Approximation Algorithm for Vertex Cover on String Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.24

Author Details

Édouard Bonnet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Paweł Rzążewski
  • Warsaw University of Technology, Poland
  • University of Warsaw, Poland

Funding

For the purpose of Open Access, the author has applied a CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.
  • Bonnet, Édouard: supported by the French National Research Agency through the project TWIN-WIDTH with reference number ANR-21-CE48-0014.
  • Rzążewski, Paweł: funded in part by National Science Centre SONATA BIS grant number 2024/54/E/ST6/00094.

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