2-balanced flows and the inverse 1-median problem in the Chebyshev space

In this paper, we consider the 1-median problem in RdRd with the Chebyshev-norm. We give an optimality criterion for this problem which enables us to solve the following inverse location problem by a combinatorial algorithm in polynomial time: Given nn points P1,…,Pn∈RdP1,…,Pn∈Rd with non-negative weights wiwi and a point P0P0 the task is to find new non-negative weights w̃i such that P0P0 is a 1-median with respect to the new weights and ‖w−w̃‖1 is minimized. In fact, this problem reduces to a 2-balanced flow problem for which an optimal solution can be obtained by solving a fractional bb-matching problem.