A Faster Algorithm for the Inverse Spanning Tree Problem

In this paper, we consider the inverse spanning tree problem. Given an undi0 Ž 0 0 . rected graph G s N , A with n nodes, m arcs, an arc cost vector c, and a spanning tree T 0, the inverse spanning tree problem is to perturb the arc cost vector c to a vector d so that T 0 is a minimum spanning tree with respect to the < < < < cost vector d and the cost of perturbation given by d y c s Ý d y c is Ž i, j.g A i j i j minimum. We show that the dual of the inverse spanning tree problem is a Ž bipartite node weighted matching problem on a specially structured graph which . Ž .Ž . we call the path graph that contains m nodes and as many as m y n q 1 n y 1 Ž . s O nm arcs. We first transform the bipartite node weighted matching problem into a specially structured minimum cost flow problem and use its special structure Ž 3. to develop an O n algorithm. We next use its special structure more effectively Ž 2 . Ž 3. and develop an O n log n time algorithm. This improves the previous O n time Ž . algorithm due to Sokkalingam et al. 1999, Oper. Res. 47, 291]298 . Q 2000