Inverse max+sum spanning tree problem under weighted l_∞ norm by modifying max-weight vector

The max+sum spanning tree ( MSST ) problem is to determine a spanning tree T whose combined weight max _e∈ Tw(e)+∑ _e∈ Tc(e) is minimum for a given edge-weighted undirected network G ( V , E , c , w ). This problem can be solved within O(m log n) time, where m and n are the numbers of edges and nodes, respectively. An inverse MSST problem ( IMSST ) aims to determine a new weight vector w̅ so that a given T^0 becomes an optimal MSST of the new network G(V,E,c,w̅) . The IMSST problem under weighted l_∞ norm is to minimize the modification cost max _e∈ E q(e)|w̅(e)-w(e)| , where q ( e ) is a cost modifying the weight w ( e ). We first provide some optimality conditions of the problem. Then we propose a strongly polynomial time algorithm in O(m^2log n) time based on a binary search and a greedy method. There are O ( m ) iterations and we solve an MSST problem under a new weight in each iteration. Finally, we perform some numerical experiments to show the efficiency of the algorithm.