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

2026-04-18

AI Summary

This work addresses the many-to-many matching problem for red and blue point sets embedded in a $[Δ]^2$ grid in the plane, where each point must be incident to at least one edge and the total Euclidean distance of the matching is minimized. The paper presents the first exact algorithm for this problem that runs in subquadratic time, breaking the longstanding $\tilde{O}(n^2)$ complexity barrier of conventional approaches. By integrating geometric partitioning, dynamic programming, and efficient data structures, the proposed algorithm computes an optimal matching in $\tilde{O}(n^{1.5} \log Δ)$ time, offering a significant theoretical improvement over existing general-purpose methods.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In this paper, we study the many-to-many matching problem on planar point sets with integer coordinates: Given two disjoint sets $R,B \subset [Δ]^2$ 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 $\tilde{O}(n^{1.5} \log Δ)$ time algorithm for point sets in $[Δ]^2$. To the best of our knowledge, this is the first subquadratic exact algorithm for planar many-to-many matching under bounded integer coordinates.

Problem

• many-to-many matching
• planar point sets
• integer coordinates
• Euclidean length minimization
• subquadratic algorithm

Innovation

• many-to-many matching
• subquadratic algorithm
• planar point sets
• integer coordinates
• exact algorithm