Partial-Matching RMS Distance Under Translation: Combinatorics and Algorithms

We consider the problem of minimizing the RMS distance (sum of squared distances between pairs of points) under translation between two point sets A and B, in the plane, with m= | B| ≪ n= | A| , in the partial-matching setup, in which each point in B is matched to a distinct point in A. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision D_{B , A} of the plane and derive improved bounds on its complexity. Specifically, we show that this complexity is O(n²m^3.5(eln m+ e) ^m) , so it is only quadratic in |A|. These results lead to the best known algorithm for finding a translation for which the partial-matching RMS distance between the point sets is minimized. In addition, we show how to compute a local minimum of the partial-matching RMS distance under translation, in polynomial time.