Exact Subquadratic Algorithm for Many-to-Many Matching on Planar Point Sets with Integer Coordinates
arXiv:2604.16921v1 Announce Type: new
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.
