Exact and Approximation Algorithms for Many-To-Many Point Matching in the Plane

TL;DR

This paper presents improved algorithms for many-to-many point matching in the plane, achieving quadratic time for exact solutions and near-linear time for approximate solutions, advancing the state of the art in computational geometry.

Contribution

It introduces the first sub-cubic exact algorithm and a near-linear approximation algorithm for planar many-to-many point matching.

Findings

Exact algorithm runs in O(n^2 poly(log n)) time.

Approximation algorithm achieves (1+ε)-approximation in O(n^{3/2} poly(log n)) time.

Addresses an open problem in planar point matching complexity.

Abstract

Given two sets $S$ and $T$ of points in the plane, of total size $n$, a {many-to-many} matching between $S$ and $T$ is a set of pairs $(p,q)$ such that $p\in S$, $q\in T$ and for each $r\in S\cup T$, $r$ appears in at least one such pair. The {cost of a pair} $(p,q)$ is the (Euclidean) distance between $p$ and $q$. In the {minimum-cost many-to-many matching} problem, the goal is to compute a many-to-many matching such that the sum of the costs of the pairs is minimized. This problem is a restricted version of minimum-weight edge cover in a bipartite graph, and hence can be solved in $O(n^3)$ time. In a more restricted setting where all the points are on a line, the problem can be solved in $O(n\log n)$ time [Colannino, Damian, Hurtado, Langerman, Meijer, Ramaswami, Souvaine, Toussaint; Graphs Comb., 2007]. However, no progress has been made in the general planar case in improving the cubic time bound. In this paper, we obtain an $O(n^2\cdot poly(\log n))$ time exact algorithm and an $O( n^{3/2}\cdot poly(\log n))$ time $(1+\epsilon)$-approximation in the planar case. Our results affirmatively address an open problem posed in [Colannino et al., Graphs Comb., 2007].