Optimally solving a transportation problem using Voronoi diagrams

TL;DR

This paper presents a geometric approach to solving a variant of the transportation problem by using weighted Voronoi diagrams, providing a unique optimal partition under broad conditions.

Contribution

It introduces a novel geometric method using weighted Voronoi diagrams to find optimal solutions for a transportation problem variant, differing from traditional analytic techniques.

Findings

Optimal solution obtained by intersecting the set with a weighted Voronoi diagram.

Solution is unique up to measure-zero sets.

Method applies under broad technical assumptions.

Abstract

We consider the following variant of the Monge-Kantorovich transportation problem. Let S be a finite set of point sites in d dimensions. A bounded set C in d-dimensional space is to be distributed among the sites p in S such that (i) each p receives a subset C_p of prescribed volume, and (ii) the average distance of all points of C from their respective sites p is minimized. In our model, volume is quantified by some measure, 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 with suitable additive weights. Moreover, this optimum partition is unique up to sets of measure zero. Unlike the deep analytic methods of classical transportation theory, our proof is based directly on geometric arguments.