Sparse reliable graph backbones

Authors:
Shiri Chechik;Yuval Emek;Boaz Patt-Shamir;David Peleg
Affiliations:
Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel;Computer Engineering and Networks Laboratory, ETH Zurich, CH-8092 Zurich, Switzerland;School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel;Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
Venue:
Information and Computation
Year:
2012

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Abstract

Given a connected graph G and a failure probability p(e) for each edge e in G, the reliability of G is the probability that G remains connected when each edge e is removed independently with probability p(e). In this paper it is shown that every n-vertex graph contains a sparse backbone, i.e., a spanning subgraph with O(nlogn) edges whose reliability is at least (1-n^-^@W^(^1^)) times that of G. Moreover, for any pair of vertices s, t in G, the (s,t)-reliability of the backbone, namely, the probability that s and t remain connected, is also at least (1-n^-^@W^(^1^)) times that of G. Our proof is based on a polynomial time randomized algorithm for constructing the backbone. In addition, it is shown that the constructed backbone has nearly the same Tutte polynomial as the original graph (in the quarter-plane x=1, y1), and hence the graph and its backbone share many additional features encoded by the Tutte polynomial.