Covering a graph with independent walks

TL;DR

This paper analyzes how multiple independent Markov chains can efficiently cover all vertices of a graph, establishing bounds on the expected cover time when multiple chains are used simultaneously.

Contribution

It provides new bounds on the expected cover time for multiple independent chains, showing how it scales with the number of chains and spectral properties of the transition matrix.

Findings

Expected cover time decreases roughly proportionally to 1/k with k chains.

Bounds are sharp when the number of chains is up to the ratio of cover time to relaxation time.

The product of the number of chains and the maximum expected cover time is proportional to the single-chain cover time.

Abstract

Let $P$ be an irreducible and reversible transition matrix on a finite state space $V$ with invariant distribution $\pi$. We let $k$ chains start by choosing independent locations distributed according to $\pi$ and then they evolve independently according to $P$. Let $\tau_{\mathrm{cov}}(k)$ be the first time that every vertex of $V$ has been visited at least once by at least one chain and let $t_{\rm{cov}}(k)=\mathbb{E}[\tau_{\mathrm{cov}}(k)]$ with $t_{\rm{cov}}=t_{\rm{cov}}(1)$. We prove that $t_{\rm{cov}}(k)\lesssim t_{\rm{cov}}/k$. When $k\leq t_{\mathrm{cov}}/t_{\rm{rel}}$, where $t_{\rm{rel}}$ is the inverse of the spectral gap, we show that this bound is sharp. For $k\leq t_{\mathrm{cov}}/t_{\rm{mix}}$ with $t_{\rm{mix}}$ the total variation mixing time of $(P+I)/2$ we prove that $k \cdot \max_{x_1,\ldots,x_k}\mathbb{E}_{x_1,\ldots,x_k}[\tau_{\rm{cov}}(k)] \asymp t_{\rm{cov}}$.