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

TL;DR

This paper introduces a deterministic algorithm for geometric bipartite matching that achieves near-linear time complexity and guarantees an approximation within a factor of (1+ε) of the optimal, improving over previous deterministic methods.

Contribution

It presents the first deterministic near-linear time algorithm for ε-approximate geometric bipartite matching in fixed dimensions, utilizing a novel tree cover approach.

Findings

Runs in $n(rac{1}{ ext{ε}} ext{log} n)^{O(d)}$ time

Achieves a (1+ε)-approximate perfect matching

Improves deterministic algorithms over prior $ ext{Ω}(n^{3/2})$ time methods

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.