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arXiv · 2204.03875

Deterministic, Near-Linear $\varepsilon$-Approximation Algorithm for Geometric Bipartite Matching

Pankaj K. Agarwal·Hsien-Chih Chang·Sharath Raghvendra·Allen Xiao
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Abstract

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

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Pankaj K. Agarwal, Hsien-Chih Chang, Sharath Raghvendra, Allen Xiao. 2022-04-08. Deterministic, Near-Linear $\varepsilon$-Approximation Algorithm for Geometric Bipartite Matching. https://arxiv.org/abs/2204.03875

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