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

Abstract.

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 arbitraryk. In this paper, we first describe a simple exact algorithm for this problem, based on t he 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 $e^{-\frac{\sqrt{2k}}{k-2}}$, which is 0.90 for $k\geq 200$ 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 $1-\frac{1}{4}(\frac{2}{n})^{k/2-2}$, which is 0.998 for $k\geq 10$, 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 $O(\log^2n)$ time using mn/logn processors and hence is RNC.