Combinatorial algorithms for solving the restricted bounded inverse optimal value problem on minimum spanning tree under weighted [formula omitted] norm

Abstract

We consider the restricted bounded inverse optimal value problem on minimum spanning tree. In a connected undirected network G=(V,E,w), we are given a spanning tree T0, a weight vector w, a lower bound vector l, an upper bound vector u, a cost vector c and a value K. We aim to obtain a new weight vector w̄ satisfying the lower and upper bounds such that T0 is a minimum spanning tree under the vector w̄ with weight K. We aim to minimize the modification cost maxei∈Eci|w̄i−wi| under weighted l∞ norm. We first analyze some properties of feasible and optimal solutions of the problem and develop a strongly polynomial time algorithm with running time O(m2n), where m=|E|,n=|V|. Then we reduce the time complexity to O(m2logn) by devising a more complex algorithm using a binary search method. Thirdly, we apply the first algorithm to the problem under unit l∞ norm, where ci=1, and obtain an O(mn) time algorithm. Finally, we give some examples to demonstrate the algorithms and present some numerical experiments to show the effectiveness of these algorithms.