Finding the k most vital edges with respect to minimum spanning tree

For a connected, undirected and weighted graph G = (V, E), the problem of rinding the k most vital edges of G with respect to minimum spanning tree is to find k edges in G whose removal will cause greatest weight increase in the minimum spanning tree of the remaining graph. This problem is known to be NP-hard for arbitrary k. In this paper, we first describe a simple exact algorithm for this problem, based on the approach of edge replacement in the minimum spanning tree of G. Next we present polynomial-time randomized algorithms that produce optimal and approximate solutions to this problem. For \V\ = n and \E\ = m, our algorithm producing optimal solution has a time complexity of O(mn) with probability of success at least e^-√2k/k-2, which is 0.90 for k ≥ 200 and asymptotically 1 when k goes to infinity. The algorithm producing approximate solution runs in O(mn+nk² log k) time with probability of success at least 1 - 1/4 (2/n)^k/2-2, which is 0.998 for k ≥ 10, and produces solution within factor 2 to the optimal one. Finally we show that both of our randomized algorithms can be easily parallelized. On a CREW PRAM, the first algorithm runs in O(n) time using n² processors, and the second algorithm runs in O(log² n) time using mn/ log n processors and hence is RNC.