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We prove that the expected time for a random walk to cover all n vertices of a graph is at least (1 + o(1))n ln n. 1 Introduction Let G be a simple connected undirected graph on n vertices and m edges. We consider random walks on G, where at each step the random walk moves to a vertex chosen at random with uniform probability from the neighbors of the current vertex. Let E v [G] denote the cover time, the expected number of steps that it takes a walk that starts at v to visit all vertices of G. It is a well known conjecture (see for example [1, 6, 13, 16]) that for connected graphs on n vertices, minG min v E v [G] (1 + o(1))n ln n, where o(1) denotes a (possibly negative) term that tends to 0 an n tends to 1. We prove this conjecture. This lower bound is best possible up to low order terms, as demonstrated by the complete graph on n vertices. The complete graph is not extremal with respect to min v E v [G], and faster cover times are possible by considering graphs in which a short ...

The cover time is the expected time it takes a simple random walk to cover all vertices of a graph. It arises in numerous questions related to the behaviour of random walks on graphs. 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 PT IME . 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. 1 Introduction An n state Markov chain is a discrete-time process defined by an n \Theta n stochastic matrix P = fp ij g , called the transition matrix. Entry p ij specifies the probability that the Markov chain at state i moves at the next step to state j . For our purposes, it is convenie...

subset at the time when the graph is almost covered is believed to be "fractal" (see the Notes on Chapter 7). 1 We are ultimately interested in random walks on unweighted graphs, but some of the arguments have as their natural setting either reversible Markov chains or general Markov chains, so we sometimes switch to those settings. Results are almost all stated for discrete-time walks, but we occasionally work with continuized chains in the proofs, or to avoid distracting complications in statements of results. Results often can be simplified or sharpened under extra symmetry conditions, but such results and examples are deferred until Chapter 7. xxx contents of chapter 1 The spanning tree argument Except for Theorem 1, we consider in this section random walk on an n- vertex unweighted graph. Results can be stated in terms of the number of edges jE j of the graph, but to aid comparison with results involving minimal or maxi

We present a deterministic algorithm that given a tree T with n vertices, a starting vertex v and a slackness parameter ǫ> 0, estimates within an additive error of ǫ 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/ǫ, and hence remains polynomial in n also for ǫ = 1/n O(1). We also show how the algorithm can be extended to estimate the expected cover (without return) time on trees. 1