Generalized loop-erased random walks and approximate reachability

In this paper we extend the loop-erased random walk (LERW) to the directed hypergraph setting. We then generalize Wilson’s algorithm for uniform sampling of spanning trees to directed hypergraphs. In several special cases, this algorithm perfectly samples spanning hypertrees in expected polynomial time. Our main application is to the reachability problem, also known as the directed all-terminal network reliability problem. This classical problem is known to be #P -complete, hence is most likely intractable [BP2]. We show that in the case of bi-directed graphs, a conjectured polynomial bound for the expected running time of the generalized Wilson algorithm implies a FPRAS for approximating reachability.