[Paper Review] Cover times, blanket times, and majorizing measures

Jian Ding, James R. Lee+1 more|arXiv (Cornell University)|Apr 25, 2010

Markov Chains and Monte Carlo Methods27 references4 citations

TL;DR

This paper establishes a precise connection between the cover time of a graph and the expected maximum of its Gaussian free field, showing that the cover time is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field scaled by the number of edges. This result enables a deterministic polynomial-time algorithm that approximates the cover time within a constant factor and resolves the blanket time conjecture by proving that blanket and cover times are within a constant factor of each other.

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).

Motivation & Objective

• To establish a tight connection between graph cover times and Gaussian processes via majorizing measures.
• To resolve the long-standing open question of whether a deterministic polynomial-time algorithm can approximate the cover time within a constant factor.
• To prove the blanket time conjecture of Winkler and Zuckerman, showing that blanket and cover times are within a constant factor for all graphs.
• To provide a computationally efficient method for approximating the cover time using the geometry of the Gaussian free field and majorizing measures.

Proposed method

• Uses Talagrand’s theory of majorizing measures to bound the expected maximum of the Gaussian free field on a graph.
• Establishes that the cover time is equivalent to |E| times the square of the expected maximum of the Gaussian free field, up to universal constants.
• Applies a randomized algorithm that computes a matrix approximation of the graph Laplacian pseudoinverse to simulate the Gaussian process efficiently.
• Employs a coupling argument to relate local times of random walks to the Gaussian free field, enabling bounds on cover times.
• Uses the γ₂ functional from majorizing measures to derive tight bounds on the expected maximum of the Gaussian process.
• Leverages the resistance metric and tree-like structures in the Gaussian free field to construct a sub-process that mimics cover time dynamics.

Experimental results

Research questions

• RQ1 Can the cover time of any graph be approximated within a constant factor by a deterministic polynomial-time algorithm?
• RQ2 Is the blanket time, defined as the time when all vertices have been visited within a δ-fraction of their stationary measure, within a constant factor of the cover time for all graphs and δ ∈ (0,1)?
• RQ3 What is the precise relationship between the cover time and the expected maximum of the Gaussian free field on a graph?
• RQ4 To what extent does the asymptotic equivalence t_cov(G) ∼ |E|·(𝔼[max η_v])² hold across different families of graphs?

Key findings

• The cover time t_cov(G) is equivalent, up to universal constants, to |E| times the square of the expected maximum of the Gaussian free field on G.
• A deterministic polynomial-time algorithm exists that approximates the cover time within a constant factor, resolving an open question of Aldous and Fill (1994).
• The blanket time conjecture of Winkler and Zuckerman is confirmed: t_bl(G,δ) ≤ C·t_cov(G) for some universal constant C and all δ ∈ (0,1).
• The previous best approximation factor of O((log log n)^2) for both cover and blanket times is improved to O(1), independent of n.
• The asymptotic formula t_cov(G_n) ∼ |E(G_n)|·(𝔼[max η_v])² holds for both complete graphs and regular trees, confirming its generality in key examples.
• The standard deviation of the cover time is bounded by the maximal hitting time, and the normalized cover time τ_cov − t_cov / t_hit exhibits exponential concentration.