An Efficient Algorithm for Computing Network Reliability in Small Treewidth

We consider the classic problem of Network Reliability. A network is given together with a source vertex, one or more target vertices, and probabilities assigned to each of the edges. Each edge appears in the network with its associated probability and the problem is to determine the probability of having at least one source-to-target path. This problem is known to be NP-hard. We present a linear-time fixed-parameter algorithm based on a parameter called treewidth, which is a measure of tree-likeness of graphs. Network Reliability was already known to be solvable in polynomial time for bounded treewidth, but there were no concrete algorithms and the known methods used complicated structures and were not easy to implement. We provide a significantly simpler and more intuitive algorithm that is much easier to implement. We also report on an implementation of our algorithm and establish the applicability of our approach by providing experimental results on the graphs of subway and transit systems of several major cities, such as London and Tokyo. To the best of our knowledge, this is the first exact algorithm for Network Reliability that can scale to handle real-world instances of the problem.Comment: 14 page

Closed geodesics and holonomies for Kleinian manifolds

For a rank one Lie group G and a Zariski dense and geometrically finite subgroup $\Gamma$ of G, we establish equidistribution of holonomy classes about closed geodesics for the associated locally symmetric space. Our result is given in a quantitative form for real hyperbolic geometrically finite manifolds whose critical exponents are big enough. In the case when G=PSL(2, C), our results can be interpreted as the equidistribution of eigenvalues of $\Gamma$ in the complex plane. When $\Gamma$ is a lattice, this result was proved by Sarnak and Wakayama in 1999.Comment: 28 pages, Minor corrections in the main term of the effective versions of Theorem 1.2, 1.3 and 5.1 are made from the printed version (GAFA,Vol 24 (2014) 1608-1636