Maximum Matchings in Geometric Intersection Graphs.

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 $O\left({\rho }^{3\omega /2}{n}^{\omega /2}\right)$ 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 $O\left(n^{\omega }\right)$ 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 $O\left(n^{\omega /2}\right)$ 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.