Maximum Matchings in Geometric Intersection Graphs

TL;DR

This paper presents a new algorithm for finding maximum matchings in geometric intersection graphs with improved time complexity, leveraging algebraic and separator-based methods, applicable to various geometric objects.

Contribution

The paper introduces a novel approach combining algebraic techniques and geometric separators to efficiently compute maximum matchings in intersection graphs.

Findings

Maximum matching can be computed in $O( ho^{3 ext{omega}/2}n^{ ext{omega}/2})$ time.

Reductions to bounded density cases enable faster algorithms for specific geometric objects.

Algorithms are applicable to translates of convex objects and planar disks with radii in $[1, ext{ extPsi}]$.

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 $O(\rho^{3\omega/2}n^{\omega/2})$ time with high probability, where $\rho$ is the density of the geometric objects and $\omega>2$ is a constant such that $n \times n$ matrices can be multiplied in $O(n^\omega)$ 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 $O(n^{\omega/2})$ time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in $[1, \Psi]$ can be found in $O(\Psi^6\log^{11} n + \Psi^{12 \omega} n^{\omega/2})$ time with high probability.