Inverse single facility location problem on a tree with balancing on the distance of server to clients

We introduce a case of inverse single facility location problem on a tree where by minimum modifying in the length of edges, the difference of distances between the farthest and nearest clients to a given facility is minimized. Two cases are considered: bounded and unbounded nonnegative edge lengths. In the unbounded case, we show the problem can be reduced to solve the problem on a star graph. Then an \begin{document}$ O(nlogn) $\end{document} algorithm is developed to find the optimal solution. For the bounded edge lengths case an algorithm with time complexity \begin{document}$ O(n^2) $\end{document} is presented.