Maximum Matchings in Geometric Intersection Graphs

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 Ohspace{0.33325pt}(rho ^{3omega /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 ntimes 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 Ohspace{0.33325pt}(Psi ^6log ^{11}hspace{-0.55542pt}n + Psi ^{12 omega } n^{omega /2}) time with high probability.