Matching point sets with respect to the Earth mover's distance

Let A and B be two sets of m resp. n weighted points in the plane, with m ≤ n. We present (1 + 6) and (2+6)-approximation algorithms for the minimum Euclidean Earth Mover’s Distance between A and B under translations and rigid motions respectively. In the general case where the sets have unequal total weights the algorithms run in O((nm/6) log(n/6)) time for translations and O((nm/6) log(n/6)) time for rigid motions. When the sets have equal total weights, the respective running times decrease to O((n/6) log(n/6)) and O((nm/6) log(n/6)). We also show how to compute a (1 + 6) and (2 + 6)approximation of the minimum cost Euclidean bipartite matching under translations and rigid motions in O((n/6) log n) and O((n/6) log n) time respectively.