The Evolution of the Cover Time

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, Lovász and Vu [23] yielded a (log log n) polynomial time approximation. We refine the upper bound of [23], 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-Rényi random graph G(n, c/n) in the supercritical regime with c > 1 fixed, is asymptotic to φ(c)n log n, where φ(c) → 1 as c ↓ 1. However, our new bound implies that the cover time for the critical Erdős-Rényi 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. For the graphs we consider, our results show that the blanket time, introduced by Winkler and Zuckerman [42], 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. Department of mathematics, University of British Columbia, Research partially supported by NSERC (Canada) and the Peter Wall Institute of Advanced Studies. Email: [email protected] Department of statistics, University of California at Berkeley, partially supported by Microsoft Research. Email: [email protected] Microsoft Research. Email: [email protected] Microsoft Research. Email: [email protected]