Covering minimum spanning trees of random subgraphs

We consider the problem of finding a sparse set of edges containing the minimum spanning tree (MST) of a random subgraph of G 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 denote the number of vertices, choose p (0, 1) possibly depending on n and let b = 1/(1 - p). We show that in both random models, for any weighted graph G, there is a set of edges Q of cardinality O(n logb n) which contains the minimum spanning tree of a random subgraph of G with high probability. This result is asymptotically optimal. As a consequence, we also give a bound of O(kn) on the size of the union of all minimum spanning trees of G with some k vertices (or edges) removed. More generally, we show a bound of O(n logb n) on the size of a covering set in a matroid of rank n, which contains the minim...

• Type: Conference
• Year: 2004
• Where: SODA
• Authors: Michel X. Goemans, Jan Vondrák
• Keywords: Algorithms, Minimum Spanning Trees, Random, Random Subset, SODA 2004