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 finding 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 ^-k ²/2(m-n-1)-2 log _c ^k/k-4, c = 1+1/2 ^k/2, and the algorithm producing approximate solution runs in time O(mn+nk ² log k) and yields results 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 n ² processors, and the second algorithm runs in O(log ² n) time using mn/log n processors.