Approximate Localization Theorems for Optimal Facility Placement

The problem is that of locating a flow-receiving facility in a region, so as to minimize the weighted sum of distances between sources and facility. It is shown here that if a subregion S both generates “sufficiently much” of the region’s total flow, and admits entry via specified “gate” points without “too much” circuity, then (a) S contains at least one “near-optimal” location for the facility, and (b) no strictly optimal location can lie “too far” from S.