Exact Subquadratic Algorithm for Many-to-Many Matching on Planar Point Sets with Integer Coordinates

TL;DR

This paper introduces the first subquadratic exact algorithm for many-to-many matching on planar point sets with integer coordinates, significantly improving over previous quadratic-time solutions.

Contribution

The authors develop an exact algorithm with subquadratic complexity, specifically rac{n^{1.5} \, \log \Delta}{\text{time}}, for the planar many-to-many matching problem with bounded integer coordinates.

Findings

Achieves rac{n^{1.5} \, \log \Delta}{\text{time}} complexity.

First subquadratic exact algorithm for this problem.

Improves upon the previous rac{n^2}{\text{time}} algorithms.

Abstract

In this paper, we study the many-to-many matching problem on planar point sets with integer coordinates: Given two disjoint sets $R,B \subset [\Delta]^2$ with $|R|+|B|=n$, the goal is to select a set of edges between $R$ and $B$ so that every point is incident to at least one edge and the total Euclidean length is minimized. In the general case that $R$ and $B$ are point sets in the plane, the best-known algorithm for the many-to-many matching problem takes $\tilde{O}(n^2)$ time. We present an exact $\tilde{O}(n^{1.5} \log \Delta)$ time algorithm for point sets in $[\Delta]^2$. To the best of our knowledge, this is the first subquadratic exact algorithm for planar many-to-many matching under bounded integer coordinates.