Computational aspects of the inverse single facility location problem on trees under lk-norm

Abstract

We consider in this paper the inverse versions of the two popular problems in location theory, say the 1-median and the 1-center problems on trees. The cost for modifying vertex weights is measured under lk-norm for a positive integer k and hence the cost function is nonlinear. For the inverse 1-median problem, we develop a linear time algorithm based on the optimal solution of the induced unconstrained problem. For the inverse 1-center problem, we prove that the problem can be decomposed into linearly many subproblems and the objective function in each subproblem is piecewise-convex. Furthermore, we also discuss an O(nlog⁡n) time algorithm for the inverse 1-center problem under l2-norm based on the convexity of the cost function of each subproblem.