Optimizing budget allocation for center and median points

In typical graph minimization problems, we consider a graph G with fixed weights on the edges of G. The goal is then to find an optimal vertex or set of vertices with respect to some objective function, for example. We introduce a new framework for graph minimization problems, where the weights on the graph edges are not fixed, but rather must be assigned, and the weight is inversely proportional to the cost paid. The goal is to find a valid assignment for which the resulting weighted graph optimizes the 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 run in linear time, both for the case where 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 a hardness result for the center point problem on complete metric graphs, followed by an O(log²(n)) approximation algorithm in this setting.