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 $R,B\subset [\mathrm{\Delta }{]}^2$ with $\mathrm{\mid }R\mathrm{\mid }+\mathrm{\mid }B\mathrm{\mid }=n$, 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 $\tilde{O}(n^{1.5}\log \mathrm{\Delta })$ time algorithm for point sets in $[\mathrm{\Delta }{]}^2$. To the best of our knowledge, this is the first subquadratic exact algorithm for planar many-to-many matching under bounded integer coordinates.