Exact and Approximation Algorithms for Many-To-Many Point Matching in the Plane

Abstract

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(n 3 ) time. In a more restricted setting where all the points are on a line, the problem can be solved in O(n log n) time [Colannino, Damian, Hurtado, Langerman, Meijer, Ramaswami, Souvaine, Toussaint; Graphs Comb., 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(n 2 • poly(log n)) time exact algorithm and an O(n 3/2 • poly(log n)) time (1 + ϵ)-approximation in the planar case. Our results affirmatively address an open problem posed in [Colannino et al., Graphs Comb., 2007].