2013 IEEE 54th Annual Symposium on Foundations of Computer Science (2000)

Redondo Beach, California

Nov. 12, 2000 to Nov. 14, 2000

ISSN: 0272-5428

ISBN: 0-7695-0850-2

pp: 467

J. Kahn , Dept. of Math., Rutgers Univ., New Brunswick, NJ, USA

L. Lovasz , Dept. of Math., Rutgers Univ., New Brunswick, NJ, USA

V.H. Vu , Dept. of Math., Rutgers Univ., New Brunswick, NJ, USA

J.H. Kim , Dept. of Math., Rutgers Univ., New Brunswick, NJ, USA

ABSTRACT

We prove upper and lower bounds and give an approximation algorithm for the cover time of the random walk on a graph. We introduce a parameter M motivated by the well-known Matthews bounds (P. Matthews, 1988) on the cover time, C, and prove that M/2>C= O(M(lnlnn)/sup 2/). We give a deterministic-polynomial time algorithm to approximate M within a factor of 2; this then approximates C within a factor of O((lnlnn)/sup 2/), improving the previous bound O(lnn) due to Matthews. The blanket time B was introduced by P. Winkler and D. Zuckerman (1996): it is the expectation of the first time when all vertices are visited within a constant factor of the number of times suggested by the stationary distribution. Obviously C/spl les/B. Winkler and Zuckerman conjectured B=O(C) and proved B=O(Clnn). Our bounds above are also valid for the blanket time, and so it follows that B=O(C(lnlnn)/sup 2/).

INDEX TERMS

graph theory; computational complexity; theorem proving; approximation theory; deterministic algorithms; cover time; blanket time; Matthews bound; random walk; Matthews bounds; deterministic-polynomial time algorithm; stationary distribution; approximation algorithm; graph

CITATION

J. Kahn,
L. Lovasz,
V.H. Vu,
J.H. Kim,
"The cover time, the blanket time, and the Matthews bound",

*2013 IEEE 54th Annual Symposium on Foundations of Computer Science*, vol. 00, no. , pp. 467, 2000, doi:10.1109/SFCS.2000.892134