Improved approximation of the minimum cover time

Feige and Rabinovich, in [FR], gave a deterministic O(log n) 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 [FR], and show that the original algorithm gives a O(log n) 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).