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

TL;DR

This paper presents a new approximation algorithm for the Bipartization problem on disk and pseudo-disk graphs, improving the longstanding 3-approximation ratio to a better constant, without needing geometric input.

Contribution

It introduces the first improvement over the 3-approximation for Bipartization on disk graphs, achieving a ratio of (3-α) and extending to pseudo-disk graphs.

Findings

Achieved a (3-α)-approximation ratio for Bipartization on disk graphs.

Extended the algorithm to pseudo-disk graphs.

The algorithm is robust and does not require geometric realization.

Abstract

In a disk graph, every vertex corresponds to a disk in $\mathbb{R}^2$ 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 $(3-\alpha)$-approximation algorithm for Bipartization on disk graphs, for some constant $\alpha>0$. 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.