Proceedings 41st Annual Symposium on Foundations of Computer Science (2000)
Redondo Beach, California
Nov. 12, 2000 to Nov. 14, 2000
J. Kahn , Dept. of Math., Rutgers Univ., New Brunswick, NJ, USA
J.H. Kim , 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
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/).
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
J. Kahn, L. Lovasz, V. Vu and J. Kim, "The cover time, the blanket time, and the Matthews bound," Proceedings 41st Annual Symposium on Foundations of Computer Science(FOCS), Redondo Beach, California, 2000, pp. 467.