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

In this paper, we study the many-to-many matching problem on planar point sets with integer coordinates: Given two disjoint sets R,B ⊂ [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 Õ(n²) time. We present an exact Õ(n1.5 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.