Geometric Matching and Bottleneck Problems

TL;DR

This paper develops algorithms for maximum geometric matchings with supply and demand constraints, and for minimum bottleneck matchings, achieving near-linear and sub-quadratic time complexities in fixed dimensions.

Contribution

It introduces new techniques for computing maximum matchings and minimum bottleneck matchings in geometric settings with supply and demand constraints.

Findings

Maximum matching can be computed efficiently in fixed dimensions.

Minimum bottleneck matching times are near-linear for $L_ ext{infty}$ and $O(n^{4/3 + ext{small}})$ for $L_2$ in the plane.

Algorithms improve upon previous methods for geometric matching problems.

Abstract

Let $P$ be a set of at most $n$ points and let $R$ be a set of at most $n$ geometric ranges, such as for example disks or rectangles, where each $p \in P$ has an associated supply $s_{p} > 0$, and each $r \in R$ has an associated demand $d_{r} > 0$. A (many-to-many) matching is a set $\mathcal{A}$ of ordered triples $(p,r,a_{pr}) \in P \times R \times \mathbb{R}_{>0}$ such that $p \in r$ and the $a_{pr}$'s satisfy the constraints given by the supplies and demands. We show how to compute a maximum matching, that is, a matching maximizing $\sum_{(p,r,a_{pr}) \in \mathcal{A}} a_{pr}$. Using our techniques, we can also solve minimum bottleneck problems, such as computing a perfect matching between a set of $n$ red points $P$ and a set of $n$ blue points $Q$ that minimizes the length of the longest edge. For the $L_\infty$-metric, we can do this in time $O(n^{1+\varepsilon})$ in any fixed dimension, for the $L_2$-metric in the plane in time $O(n^{4/3 + \varepsilon})$, for any $\varepsilon > 0$.