Sparse Reliable Graph Backbones

Given a connected graph G and a failure probability pe 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 pe. In this paper it is shown that every n-vertex graph contains a sparse backbone, i.e., a spanning subgraph with O(n log n) edges whose reliability is at least (1−n−Ω(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−Ω(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, y > 1), and hence the graph and its backbone share many additional features encoded by the Tutte polynomial.

• Type: Conference
• Year: 2010
• Where: ICALP
• Authors: Shiri Chechik, Yuval Emek, Boaz Patt-Shamir, David Peleg
• Keywords: Connected Graph, Failure Probability Pe, ICALP 2010, Probability Pe, Programming Languages