Lokshtanov, D, Panolan, F orcid.org/0000-0001-6213-8687, Saurabh, S et al. (2 more authors) (Accepted: 2024) A 1.9999-Approximation Algorithm for Vertex Cover on String Graphs. In: 40th International Symposium on Computational Geometry (SoCG 2024). 40th International Symposium on Computational Geometry (SoCG 2024), 11-14 Jun 2024, Athens, Greece. LIPIcs . (In Press)
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 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
Metadata
Item Type: | Proceedings Paper |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | This is an author produced version of a conference paper accepted for publication in the 40th International Symposium on Computational Geometry (SoCG 2024), made available under the terms of the Creative Commons Attribution License (CC-BY), which permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited. |
Keywords: | vertex cover, geometric intersection graphs, approximation algorithms |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 14 Mar 2024 11:53 |
Last Modified: | 14 Mar 2024 15:55 |
Status: | In Press |
Publisher: | LIPIcs |