Abstract

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 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)logn).

Keywords: Random walks; Markov chains; Cover time; Approximation algorithms