Dividing a Territory Among Several Facilities

We consider the problem of dividing a geographic region into sub-regions so as to minimize the maximum workload of a collection of facilities over that region. We assume that the cost of servicing a demand point is a monomial function of the distance to its assigned facility and that demand points follow a continuous probability density. We show that, when our objective is to minimize the maximum workload of all facilities, the optimal partition consists of a collection of circular arcs that are induced by a multiplicatively weighted Voronoi diagram. When we require that all sub-regions have equal area, the optimal partition consists of a collection of hyperbolic or quartic curves. We show that, for both problems, the dual variables correspond to "prices" for a facility to serve a demand point, and our objective is to determine a set of prices such that the entire region is "purchased" by the facilities, i.e. that the market clears. This allows us to solve the partitioning problem quickly without discretizing the service region.