A near-linear constant-factor approximation for euclidean bipartite matching?

Abstract

In the Euclidean bipartite matching problem, we are given a set R of "red" points and a set B of "blue" points in ℝ d where |R|= |B| = n, and we want to pair up each red point with a distinct blue point so that the sum of distances between the paired points is minimized. We present an approximation algorithm that given any parameter 0 < ε < 1 runs in O(n1+ε) expected time and returns a matching whose expected cost is within a multiplicative factor O(log(1/ε)) of the optimal. The dimension d is considered to be a fixed constant.