We present polynomial time results for computing a minimum separation between two regions in a planar weighted subdivision. Our results are based on a (more general) theorem that characterizes a class of functions for which optimal solutions arise on the boundary of the feasible domain. A direct consequence of this theorem is that a minimum separation goes through a vertex of the weighted subdivision. We also consider extensions and present results for the 3-D case and for a more general case of the 2-D separation problem, in which the separation (link) has associated an ϵ-width. Our results are the first nontrivial upper bounds for these problems. We also discuss simple approximation algorithms for the 2-D case and present a prune-and-search approach that can be used with either the continuous or the approximate solutions to speed up the computation. We have implemented a variant of the two region minimum separation algorithm based on the prune-and-search scheme.