Maximum Matchings in Geometric Intersection Graphs

TL;DR

This work addresses computing maximum matchings in intersection graphs of planar geometric objects by introducing a density parameter $\rho$ and combining algebraic methods with geometric separators. The authors prove a randomized algorithm running in $O\left(\rho^{3\omega/2} n^{\omega/2}\right)$ time (with high probability) for graphs in $\mathbb{G}_\rho$, and extend to subgraphs with a given geometric representation. A core contribution is a sparsification framework that reduces general instances to bounded-density ones for several families (disks, translates of convex bodies, axis-parallel shapes, and congruent balls), yielding concrete running times such as $O\left(\Psi^6 n \log^{4} n + \Psi^{12\omega} n^{\omega/2}\right)$ for disks of radius ratio $\Psi$ and $O\left(n^{\omega/2}\right)$ for translates of a fixed convex shape. The results harness a blend of Tutte-matrix–based algebraic techniques (Mucha–Sankowski) and small separator trees, enabling efficient maximum matching in a broad class of geometric graphs and offering several tight or near-tight cases across dimensions and object families.

Abstract

Let $G$ be an intersection graph of $n$ geometric objects in the plane. We show that a maximum matching in $G$ can be found in $O(ρ^{3ω/2}n^{ω/2})$ time with high probability, where $ρ$ is the density of the geometric objects and $ω>2$ is a constant such that $n \times n$ matrices can be multiplied in $O(n^ω)$ time. The same result holds for any subgraph of $G$, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators. We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in $O(n^{ω/2})$ time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in $[1, Ψ]$ can be found in $O(Ψ^6\log^{11} n + Ψ^{12 ω} n^{ω/2})$ time with high probability.

Figures (2)

• Figure 1: Splitting one single vertex of $Z$.
• Figure 2: Algorithm Sparsify-one-edge

Theorems & Definitions (42)

• Lemma 1
• proof
• Theorem 2: Lemma 2.21 in Har-PeledQ17
• Lemma 3
• proof
• Theorem 4
• Lemma 5
• proof
• proof : Proof of Theorem \ref{['thm:dissection']}
• Lemma 6