Deterministic, Near-Linear ε-Approximation Algorithm for Geometric Bipartite Matching

04/08/2022

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by   Pankaj K. Agarwal, et al.

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Given point sets A and B in ℝ^d where A and B have equal size n for some constant dimension d and a parameter ε>0, we present the first deterministic algorithm that computes, in n·(ε^-1log n)^O(d) time, a perfect matching between A and B whose cost is within a (1+ε) factor of the optimal under any ℓ_p-norm. Although a Monte-Carlo algorithm with a similar running time is proposed by Raghvendra and Agarwal [J. ACM 2020], the best-known deterministic ε-approximation algorithm takes Ω(n^3/2) time. Our algorithm constructs a (refinement of a) tree cover of ℝ^d, and we develop several new tools to apply a tree-cover based approach to compute an ε-approximate perfect matching.