摘要：Given two sets S and T of points in the plane, of total size n, a many-to-many matching between S and T is a set of pairs (p,q) such that p ∈ S, q ∈ T and for each r ∈ S ∪ T, r appears in at least one such pair. The cost of a pair (p,q) is the (Euclidean) distance between p and q. In the minimum-cost many-to-many matching problem, the goal is to compute a many-to-many matching such that the sum of the costs of the pairs is minimized. This problem is a restricted version of minimum-weight edge cover in a bipartite graph, and hence can be solved in O(n³) time. In a more restricted setting where all the points are on a line, the problem can be solved in O(nlog n) time [Justin Colannino et al., 2007]. However, no progress has been made in the general planar case in improving the cubic time bound. In this paper, we obtain an O(n²⋅ poly(log n)) time exact algorithm and an O(n^{3/2}⋅ poly(log n)) time (1+ε)-approximation in the planar case.
