A Near-linear Time ε-Approximation Algorithm for Geometric Bipartite Matching

Sharath Raghvendra, Pankaj K. Agarwal

Abstract: For point sets A, B ⊂ Rd, ∣A∣ = ∣B∣ = n, and for a parameter ε > 0, we present a Monte Carlo algorithm that computes, in O(npoly(log n, 1/ε)) time, an ε-approximate perfect matching of A and B under any Lp-norm with high probability; the previously best-known algorithm takes Ω(n3/2) time. We approximate the Lp-norm using a distance function, d(⋅, ⋅) based on a randomly shifted quad-tree. The algorithm iteratively generates an approximate minimum-cost augmenting path under d(⋅, ⋅) in time proportional, within a polylogarithmic factor, to the length of the path. We show that the total length of the augmenting paths generated by the algorithm is O(n/ε)log n), implying that the running time of our algorithm is O(npoly(log n, 1/ε)).