Cover times, blanket times, and majorizing measures

TL;DR

This paper establishes a deep connection between graph cover times, Gaussian processes, and majorizing measures, providing new algorithms and resolving longstanding conjectures in graph theory.

Contribution

It demonstrates that cover times are equivalent to Gaussian free field maxima, offers a polynomial-time approximation algorithm, and confirms blanket time conjectures.

Findings

Cover time is proportional to the square of Gaussian free field maximum.

Provides a polynomial-time algorithm approximating cover time within a constant factor.

Resolves blanket time conjectures, showing blanket and cover times are within a constant factor.

Abstract

We exhibit a strong connection between cover times of graphs, Gaussian processes, and Talagrand's theory of majorizing measures. In particular, we show that the cover time of any graph $G$ is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field on $G$, scaled by the number of edges in $G$. This allows us to resolve a number of open questions. We give a deterministic polynomial-time algorithm that computes the cover time to within an O(1) factor for any graph, answering a question of Aldous and Fill (1994). We also positively resolve the blanket time conjectures of Winkler and Zuckerman (1996), showing that for any graph, the blanket and cover times are within an O(1) factor. The best previous approximation factor for both these problems was $O((\log \log n)^2)$ for $n$-vertex graphs, due to Kahn, Kim, Lovasz, and Vu (2000).