We present a deterministic algorithm that given a tree T with n vertices, a starting vertex v and a slackness parameter epsilon > 0, estimates within an additive error of epsilon the cover and return time, namely, the expected time it takes a simple random walk that starts at v to visit all vertices of T and return to v. The running time of our algorithm is polynomial in n epsilon, and hence remains polynomial in n also for epsilon = 1 n^ O(1) . We also show how the algorithm can be extended to estimate the expected cover (without return) time on trees.
The cover time is the expected time it takes a random walk to cover all vertices of a graph. Despite the fact that it can be approximated with arbitrary precision by a simple polynomial time Monte-Carlo algorithm which simulates the random walk, it is not known whether the cover time of a graph can be computed in deterministic polynomial time. In the present paper we establish a deterministic polynomial time algorithm that, for any graph and any starting vertex, approximates the cover time within polylogarithmic factors. More generally, our algorithm approximates the cover time for arbitrary reversible Markov chains. The new aspect of our algorithm is that the starting vertex of the random walk may be arbitrary and is given as part of the input, whereas previous deterministic approximation algorithms for the cover time assume that the walk starts at the worst possible vertex. In passing, we show that the starting vertex can make a difference of up to a multiplicative factor of Θ(n3 2 √log n) in the cover time of an n-vertex graph. Feige and Rabinovich, in [Feige and Rabinovich, Rand. Struct. Algorithms 23(1) (2003) 1-22], gave a deterministic O(log4n) approximation for the time it takes a random walk to cover a given graph starting at a given vertex. This approximation algorithm was shown to work for arbitrary reversible Markov chains. We build on the results of [Feige and Rabinovich, Rand. Struct. Algorithms 23(1) (2003) 1-22], and show that the original algorithm gives a O(log2n) approximation as it is, and that it can be modified to give a O(log n(log log n)2) approximation. Moreover, we show that given any c(n)-approximation algorithm for the maximum cover time (maximized over all initial vertices) of a reversible Markov chain, we can give a corresponding algorithm for the general cover time (of a random walk or reversible Markov chain) with approximation ratio O(c(n) log n). We consider the cover time E>sub sub sub sub sub sub sub sub sub sub 2n^2. This improves the leadingconstant in previously known upper bounds. also provide upper bouhnds on E^+_uG, the expected timeto cover G and return to u