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.
When one seeks to estimate the cover time from a given vertex (rather than from the worst possible vertex), the known bounds deteriorate. The deterministic algorithms known @cite_0 @cite_10 pay an extra @math factor in the approximation ratio compared to the approximation ratios known from worst possible vertex. For the special case of trees, some upper bounds are presented in @cite_3 .
@cite_0 @cite_10 @cite_3
1989885196 1597165385 2116953739 2057596055
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. We first devise moderately exponential exact algorithms for maxk-vertex cover, with time-complexity exponential in n but with polynomial space-complexity by developing a branch and reduce method based upon the measure-and-conquer technique. We then prove that, there exists an exact algorithm for maxk-vertex cover with complexity bounded above by the maximum among ck and γτ, for some γ<2, where τ is the cardinality of a minimum vertex cover of G (note that @math k @math with respect to parameter k unless @math ), using polynomial space. We finally study approximation of maxk-vertex cover by moderately exponential algorithms. The general goal of the issue of moderately exponential approximation is to catch-up on polynomial inapproximability, by providing algorithms achieving, with worst-case running times importantly smaller than those needed for exact computation, approximation ratios unachievable in polynomial time. We give a nearly optimal sublinear-time algorithm for approximating the size of a minimum vertex cover in a graph G. The algorithm may query the degree deg(v) of any vertex v of its choice, and for each 1 ≤ i ≤ deg(v), it may ask for the ith neighbor of v. Letting VCopt(G) denote the minimum size of vertex cover in G, the algorithm outputs, with high constant success probability, an estimate [EQUATION] such that [EQUATION], where e is a given additive approximation parameter. We refer to such an estimate as a (2, e)-estimate. The query complexity and running time of the algorithm are O([EQUATION] · poly(1 e)), where d denotes the average vertex degree in the graph. The best previously known sublinear algorithm, of (STOC 2009), has query complexity and running time O(d4 e2), where d is the maximum degree in the graph. Given the lower bound of Ω(d) (for constant e) for obtaining such an estimate (with any constant multiplicative factor) due to Parnas and Ron (TCS 2007), our result is nearly optimal. In the case that the graph is dense, that is, the number of edges is Θ(n2), we consider another model, in which the algorithm may ask, for any pair of vertices u and v, whether there is an edge between u and v. We show how to adapt the algorithm that uses neighbor queries to this model and obtain an algorithm that outputs a (2, e)-estimate of the size of a minimum vertex cover whose query complexity and running time are O(n) · poly(1 e). The cover time of a graph is a celebrated example of a parameter that is easy to approximate using a randomized algorithm, but for which no constant factor deterministic polynomial time approximation is known. A breakthrough due to Kahn, Kim, Lovasz and Vu [25] yielded a (log logn)2 polynomial time approximation. We refine the upper bound of [25], and show that the resulting bound is sharp and explicitly computable in random graphs. Cooper and Frieze showed that the cover time of the largest component of the Erdős–Renyi random graph G(n, c n) in the supercritical regime with c > 1 fixed, is asymptotic to ϕ(c)nlog2n, where ϕ(c) → 1 as c ↓ 1. However, our new bound implies that the cover time for the critical Erdős–Renyi random graph G(n, 1 n) has order n, and shows how the cover time evolves from the critical window to the supercritical phase. Our general estimate also yields the order of the cover time for a variety of other concrete graphs, including critical percolation clusters on the Hamming hypercube 0, 1 n, on high-girth expanders, and on tori ℤdn for fixed large d. This approach also gives a simpler proof of a result of Aldous [2] that the cover time of a uniform labelled tree on k vertices is of order k3 2. For the graphs we consider, our results show that the blanket time, introduced by Winkler and Zuckerman [45], is within a constant factor of the cover time. Finally, we prove that for any connected graph, adding an edge can increase the cover time by at most a factor of 4.