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Bipartizing (Pseudo-)Disk Graphs: Approximation with a Ratio Better than 3

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

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Author Details

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

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Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, and Meirav Zehavi. Bipartizing (Pseudo-)Disk Graphs: Approximation with a Ratio Better than 3. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.6

Abstract

In a disk graph, every vertex corresponds to a disk in ℝ² and two vertices are connected by an edge whenever the two corresponding disks intersect. Disk graphs form an important class of geometric intersection graphs, which generalizes both planar graphs and unit-disk graphs. We study a fundamental optimization problem in algorithmic graph theory, Bipartization (also known as Odd Cycle Transversal), on the class of disk graphs. The goal of Bipartization is to delete a minimum number of vertices from the input graph such that the resulting graph is bipartite. A folklore (polynomial-time) 3-approximation algorithm for Bipartization on disk graphs follows from the classical framework of Goemans and Williamson [Combinatorica'98] for cycle-hitting problems. For over two decades, this result has remained the best known approximation for the problem (in fact, even for Bipartization on unit-disk graphs). In this paper, we achieve the first improvement upon this result, by giving a (3-α)-approximation algorithm for Bipartization on disk graphs, for some constant α > 0. Our algorithm directly generalizes to the broader class of pseudo-disk graphs. Furthermore, our algorithm is robust in the sense that it does not require a geometric realization of the input graph to be given.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • bipartization
  • geometric intersection graphs
  • approximation algorithms

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