Covering minimum spanning trees of random subgraphs

Authors:
Michel X. Goemans;Jan Vondrák
Affiliations:
MIT, Cambridge, MA;-
Venue:
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Year:
2004

Citing 7
Cited 0

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Abstract

We consider the problem of covering the minimum spanning tree (MST) of a random subgraph of G by a sparse set of edges, with high probability. The two random models that we consider are subgraphs induced by a random subset of vertices, each vertex included independently with probability p, and subgraphs generated as a random subset of edges, each edge with probability p.Let n be the number of vertices in G. We show that in both cases, there is a covering set Q of cardinality O(n logb n) where b = 1/(1 -- p) (and p is possibly a function of n) and this is asymptotically optimal. More generally, we show a similar bound on the covering set in a matroid, which contains the minimum-weight basis of a random subset with high probability. Also, we give a randomized algorithm which calls an MST subroutine only a polylogarithmic number of times, and finds the covering set with high probability.