The inverse Fermat-Weber problem

Given n points in the plane with nonnegative weights, the inverse Fermat-Weber problem consists in changing the weights at minimum cost such that a prespecified point in the plane becomes the Euclidean 1-median. The cost is proportional to the increase or decrease of the corresponding weight. In case that the prespecified point does not coincide with one of the given n points, the inverse Fermat-Weber problem can be formulated as linear program. We derive a purely combinatorial algorithm which solves the inverse Fermat-Weber problem with unit cost in O(n logn) time. If the prespecified point coincides with one of the given n points, it is shown that the corresponding inverse problem can be written as convex problem and hence is solvable in polynomial time to any fixed precision. 1 Inverse and reverse location problems In recent years inverse and reverse optimization problems found an increased interest. In a reverse optimization problem, we are given a budget for modifying parameters of the problem. The goal is to modify parameters of the problem such that an objective function attains its best possible value subject to the given budget. The inverse optimization problem consists in changing parameters of the problem at minimum cost such that a prespecified solution becomes optimal. In one of the first papers on this subject, Burton and Graz University of Technology, Institute of Optimization and Discrete Mathematics, Steyrergasse 30, Graz, Austria. {burkard,galavi,gassner}@opt.math.tugraz.at University of Zabol, Department of Mathematics, Jahad Square, Zabol, Iran. mr [email protected], mohammad [email protected] This research has been supported by the Austrian Science Fund (FWF) Project P18918-N18.