Optimally Solving a Transportation Problem Using Voronoi Diagrams

Abstract

In this paper we consider the following variant of the well-known Monge-Kantorovich transportation problem. Let S be a set of n point sites in ℝ^d. A bounded set C ⊂ ℝ^d is to be distributed among the sites p ∈ S such that (i), each p receives a subset C _p of prescribed volume and (ii), the average distance of all points z of C from their respective sites p is minimized. In our model, volume is quantified by a measure μ, and the distance between a site p and a point z is given by a function d _p (z). Under quite liberal technical assumptions on C and on the functions d _p (·) we show that a solution of minimum total cost can be obtained by intersecting with C the Voronoi diagram of the sites in S, based on the functions d _p (·) equipped with suitable additive weights. Moreover, this optimum partition is unique, up to subsets of C of measure zero. Unlike the deep analytic methods of classical transportation theory, our proof is based on direct geometric arguments.