The 1-Median Problem in with the Chebyshev-Norm and its Inverse Problem

In this paper, we consider the 1-median problem in R d with the Chebyshev-norm. We give an optimality criterion for this problem which enables us to solve the following inverse location problem in polynomial time: Given n points P 1 , … , P n with non-negative weights w i and a point P 0 the task is to find new non-negative weights w ˜ i such that P 0 is a 1-median with respect to the new weights and ‖ w − w ˜ ‖ 1 is minimized. In fact, this problem reduces to a balancing flow problem for which an optimal solution can be obtained in polynomial time.