Résumé: 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 N P-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(-)'rac'(k-2/k-2), which is 0.90 fork > 200 and asymptotically 1 when k goes to infinity. The algorithm producing approximate solution runs in O(mn+nk(2) 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 n(2) processors, and the second algorithm runs in O(log(2) n) time using mn/log n processors and hence is RNC.

Langue: Anglais