Up- and downgrading the 1-median in a network⋆

While classical location problems deal with finding optimal locations for facilities, the task of the corresponding upgrading (downgrading) version is to change the underlying network within certain bounds such that the optimal objective value that can be obtained in the modified network is as good (bad) as possible. In this paper we allow to change the vertex weights within given bounds such that a linear budget constraint is satisfied. For the upgrading 1-median problem an O(n) time algorithm is suggested. The downgrading 1-median problem is shown to be solvable in polynomial time. For the special case of a tree a concavity property leads to an O(n log n) time algorithm.