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

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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−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+nk2logk) time with probability of success at least 1−14(2n)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(log2n) time using mn/logn processors and hence is RNC.

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Acta Informatica

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36

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5

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Computation Theory and Mathematics

Computer Software

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