Inverse 1-Median Problem on Block Graphs with Variable Vertex Weights

This paper addresses the problem of modifying the vertex weights of a block graph at minimum total cost so that a prespecified vertex becomes a 1-median of the perturbed graph. We call this problem the inverse 1-median problem on block graphs with variable vertex weights. For block graphs with equal edge lengths in each block, we can formulate the problem as a univariate optimization problem. By the convexity of the objective function, the local optimizer is also the global one. Therefore, we use the convexity to develop an O ( M log M ) algorithm that solves the problem on block graphs with M vertices.