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

TL;DR

This work breaks the long-standing 3-approximation barrier for Bipartization on disk graphs by giving the first $(3-\alpha)$-approximation algorithm, with an explicit guaranteed bound $\le 2.99993033741$ in expectation. The approach reduces to the $K_4$-free case via a maximal $K_4$ packing, leverages a degeneracy bound of $11$ (hence $d\le 22$) for the reduced graph, and combines three OCT strategies (S1,S2,S3) to ensure at least one yields a sub-3 ratio; the analysis carefully relates triangle packings to the OCT size via parameters $a$ and $b$. The algorithm is robust (does not require a geometric input realization) and extends naturally to pseudo-disk graphs, with generalizations to a broad class of hereditary, triangle-conflicting vertex-deletion problems, plus a derandomization path. The results provide a principled, practical framework for improving approximation guarantees on geometric intersection graphs and open avenues for further applications beyond Bipartization, including Planarization&Bipartition variants.

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 $(3-α)$-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.

Theorems & Definitions (34)

• Theorem 1.1: folklore
• proof
• Theorem 1.1
• Theorem 1.1
• Lemma 3.1
• proof
• Lemma 3.2: Marathe1995SimpleHF
• Lemma 3.3
• proof
• Lemma 3.4