Maximum Matchings in Geometric Intersection Graphs

Abstract

AbstractLet G be an intersection graph of n geometric objects in the plane. We show that a maximum matching in G can be found in $$O\hspace{0.33325pt}(\rho ^{3\omega /2}n^{\omega /2})$$ O ( ρ 3 ω / 2 n ω / 2 ) time with high probability, where $$\rho $$ ρ is the density of the geometric objects and $$\omega >2$$ ω > 2 is a constant such that $$n\times n$$ n × n matrices can be multiplied in $$O(n^\omega )$$ 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 $$O(n^{\omega /2})$$ 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 $$[1, \Psi ]$$ [ 1 , Ψ ] can be found in $$O\hspace{0.33325pt}(\Psi ^6\log ^{11}\hspace{-0.55542pt}n + \Psi ^{12 \omega } n^{\omega /2})$$ O ( Ψ 6 log 11 n + Ψ 12 ω n ω / 2 ) time with high probability.