The Minmax Regret Reverse 1-Median Problem on Trees with Uncertain Vertex Weights

Abstract

The classical reverse 1-median problem on trees is to adjust the edge lengths within a budget so as to reduce the 1-median function at a predetermined vertex as much as possible. This paper concerns the corresponding problem with uncertain vertex weights presented by linear functions. Moreover, we use the minmax regret criterion to measure the maximum loss of a feasible solution with respect to the worst-case scenario. The regarding problem is called the minmax regret reverse 1-median problem on trees. We first partition the set of scenarios into parts such that the optimal solution of the corresponding reverse 1-median problem does not change in each part. Then the problem can be reformulated as the minimization of a quadratic number of affine linear functions. We finally develop a greedy algorithm that solves the problem in [Formula: see text] time where [Formula: see text] is the number of vertices in the underlying tree.