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A 1.9999-Approximation Algorithm for Vertex Cover on String Graphs

Authors Daniel Lokshtanov , Fahad Panolan , Saket Saurabh , Jie Xue , Meirav Zehavi

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Author Details

Daniel Lokshtanov
  • University of California, Santa Barbara, CA, USA
Fahad Panolan
  • University of Leeds, UK
Saket Saurabh
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Jie Xue
  • New York University Shanghai, China
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel

Funding

  • Saurabh, Saket: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416), and Swarnajayanti Fellowship (No. DST/SJF/MSA01/2017-18).
  • Zehavi, Meirav: Supported by the ERC grant no. 101039913 (PARAPATH).

Acknowledgements

The authors would like to thank the anonymous reviewers of SoCG'24 for their insightful comments, which helped improve the exposition of the paper.

Cite AsGet BibTex

Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, and Meirav Zehavi. A 1.9999-Approximation Algorithm for Vertex Cover on String Graphs. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 72:1-72:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.72

Abstract

Vertex Cover is a fundamental optimization problem, and is among Karp’s 21 NP-complete problems. The problem aims to compute, for a given graph G, a minimum-size set S of vertices of G such that G - S contains no edge. Vertex Cover admits a simple polynomial-time 2-approximation algorithm, which is the best approximation ratio one can achieve in polynomial time, assuming the Unique Game Conjecture. However, on many restrictive graph classes, it is possible to obtain better-than-2 approximation in polynomial time (or even PTASes) for Vertex Cover. In the club of geometric intersection graphs, examples of such graph classes include unit-disk graphs, disk graphs, pseudo-disk graphs, rectangle graphs, etc. In this paper, we study Vertex Cover on the broadest class of geometric intersection graphs in the plane, known as string graphs, which are intersection graphs of any connected geometric objects in the plane. Our main result is a polynomial-time 1.9999-approximation algorithm for Vertex Cover on string graphs, breaking the natural 2 barrier. Prior to this work, no better-than-2 approximation (in polynomial time) was known even for special cases of string graphs, such as intersection graphs of segments. Our algorithm is simple, robust (in the sense that it does not require the geometric realization of the input string graph to be given), and also works for the weighted version of Vertex Cover. Due to a connection between approximation for Independent Set and approximation for Vertex Cover observed by Har-Peled, our result can be viewed as a first step towards obtaining constant-approximation algorithms for Independent Set on string graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Computational geometry
Keywords
  • vertex cover
  • geometric intersection graphs
  • approximation algorithms

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References

  1. Anna Adamaszek, Sariel Har-Peled, and Andreas Wiese. Approximation schemes for independent set and sparse subsets of polygons. J. ACM, 66(4):29:1-29:40, 2019. Google Scholar
  2. Noga Alon, Raphy Yuster, and Uri Zwick. Color-coding: a new method for finding simple paths, cycles and other small subgraphs within large graphs. In Proceedings of the twenty-sixth annual ACM symposium on Theory of computing, pages 326-335, 1994. Google Scholar
  3. Brenda S. Baker. Approximation algorithms for np-complete problems on planar graphs. J. ACM, 41(1):153-180, 1994. Google Scholar
  4. Reuven Bar-Yehuda, Danny Hermelin, and Dror Rawitz. Minimum vertex cover in rectangle graphs. Computational Geometry, 44(6-7):356-364, 2011. Google Scholar
  5. Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discret. Comput. Geom., 48(2):373-392, 2012. Google Scholar
  6. Miroslav Chlebík and Janka Chlebíková. Crown reductions for the minimum weighted vertex cover problem. Discrete Applied Mathematics, 156(3):292-312, 2008. Google Scholar
  7. Anuj Dawar, Martin Grohe, Stephan Kreutzer, and Nicole Schweikardt. Approximation schemes for first-order definable optimisation problems. In 21th IEEE Symposium on Logic in Computer Science (LICS 2006), 12-15 August 2006, Seattle, WA, USA, Proceedings, pages 411-420. IEEE Computer Society, 2006. Google Scholar
  8. Erik D. Demaine and Mohammad Taghi Hajiaghayi. Bidimensionality: new connections between FPT algorithms and ptass. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, Vancouver, British Columbia, Canada, January 23-25, 2005, pages 590-601. SIAM, 2005. Google Scholar
  9. Thomas Erlebach, Klaus Jansen, and Eike Seidel. Polynomial-time approximation schemes for geometric intersection graphs. SIAM J. Comput., 34(6):1302-1323, 2005. Google Scholar
  10. Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Excluded grid minors and efficient polynomial-time approximation schemes. J. ACM, 65(2):10:1-10:44, 2018. Google Scholar
  11. Jacob Fox and János Pach. Computing the independence number of intersection graphs. In Proceedings of the twenty-second annual ACM-SIAM Symposium on Discrete algorithms, pages 1161-1165. SIAM, 2011. Google Scholar
  12. Waldo Gálvez, Arindam Khan, Mathieu Mari, Tobias Mömke, Madhusudhan Reddy Pittu, and Andreas Wiese. A 3-approximation algorithm for maximum independent set of rectangles. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9-12, 2022, pages 894-905. SIAM, 2022. Google Scholar
  13. Martin Grohe. Local tree-width, excluded minors, and approximation algorithms. Comb., 23(4):613-632, 2003. Google Scholar
  14. Magnús M. Halldórsson and Jaikumar Radhakrishnan. Greed is good: Approximating independent sets in sparse and bounded-degree graphs. Algorithmica, 18(1):145-163, 1997. Google Scholar
  15. Sariel Har-Peled. Approximately: Independence implies vertex cover. arXiv preprint, 2023. URL: https://arxiv.org/abs/2308.00840.
  16. Sariel Har-Peled and Kent Quanrud. Approximation algorithms for polynomial-expansion and low-density graphs. SIAM J. Comput., 46(6):1712-1744, 2017. Google Scholar
  17. Johan Håstad. Clique is hard to approximate within n^1-ε. Acta Mathematica, 182:105-142, 1999. Google Scholar
  18. John E Hopcroft and Richard M Karp. An n^5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on computing, 2(4):225-231, 1973. Google Scholar
  19. Richard M. Karp. Reducibility among combinatorial problems. In Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA, pages 85-103, 1972. Google Scholar
  20. Marek Karpinski and Alexander Zelikovsky. Approximating dense cases of covering problems. Citeseer, 1996. Google Scholar
  21. Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2-ε. Journal of Computer and System Sciences, 74(3):335-349, 2008. Google Scholar
  22. Philip Klein, Serge A Plotkin, and Satish Rao. Excluded minors, network decomposition, and multicommodity flow. In Proceedings of the twenty-fifth annual ACM symposium on Theory of computing, pages 682-690, 1993. Google Scholar
  23. Jan Kratochvíl and Jirí Matousek. String graphs requiring exponential representations. J. Comb. Theory, Ser. B, 53(1):1-4, 1991. Google Scholar
  24. James R Lee. Separators in region intersection graphs. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://arxiv.org/abs/1608.01612.
  25. Wenjun Li and Binhai Zhu. A 2k-kernelization algorithm for vertex cover based on crown decomposition. Theoretical computer science, 739:80-85, 2018. Google Scholar
  26. Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, and Meirav Zehavi. A framework for approximation schemes on disk graphs. In 34rd Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, 2023. Google Scholar
  27. Joseph S. B. Mitchell. Approximating maximum independent set for rectangles in the plane. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 339-350. IEEE, 2021. Google Scholar
  28. George L Nemhauser and Leslie E Trotter Jr. Vertex packings: structural properties and algorithms. Mathematical Programming, 8(1):232-248, 1975. Google Scholar
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