Optimizing budget allocation in graphs

In a classical facility location problem we consider a graph G with fixed weights on the edges of G. The goal is then to find an optimal positioning for a set of facilities on the graph with respect to some objective function. We consider a new model for facility location problems, where the weights on the graph edges are not fixed, but rather should be assigned. The goal is to find the valid assignment for which the resulting weighted graph optimizes the facility location objective function. We present algorithms for finding the optimal budget allocation for the center point problem and for the median point problem on trees. Our algorithms work in linear time, both for the case that a candidate vertex is given as part of the input, and for the case where finding a vertex that optimizes the solution is part of the problem. We also present an O(log2(n)) approximation algorithm for the center point problem over general metric spaces.