Maximum Matchings in Geometric Intersection Graphs
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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/2n^ω/2) time with high probability, where ρ is the density of the geometric objects and ω>2 is a constant such that n × 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^6log^11 n + Ψ^12 ω n^ω/2) time with high probability.
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