Covering a graph with independent walks
Jonathan Hermon, Perla Sousi
TL;DR
This work analyzes the cover time when $k$ independent Markov chains with an irreducible, reversible transition matrix $P$ run on the same finite state space. It establishes a near-optimal speed-up: the expected time to cover all states scales as $t_{ ext{cov}}(k)\lesssim t_{ ext{cov}}/k$, with sharpness for $k$ up to a constant multiple of $t_{ ext{cov}}/t_{ ext{rel}}$, and linear speed-up up to $t_{ ext{cov}}/t_{ ext{mix}}$ in the regime $k\le t_{ ext{cov}}/t_{ ext{mix}}$. The authors introduce a one-parameter family of auxiliary chains $P^\lambda$ by adding a boundary state and relate their cover times to the original via Gaussian free fields (GFF) and isomorphism theorems, leveraging Sudakov-Fernique comparisons of effective resistances and blanket-time arguments to control excursions. A complementary lower-bound framework uses chains $K_\lambda$ to compare resistances and apply the GFF-based cover-time bounds, yielding near-tight bounds in the relevant parameter regimes. The work advances understanding of multi-walker cover times and links probabilistic cover phenomena to Gaussian and electrical-network techniques with potential algorithmic implications for parallel exploration.
Abstract
Let $P$ be an irreducible and reversible transition matrix on a finite state space $V$ with invariant distribution $π$. We let $k$ chains start by choosing independent locations distributed according to $π$ and then they evolve independently according to $P$. Let $τ_{\mathrm{cov}}(k)$ be the first time that every vertex of $V$ has been visited at least once by at least one chain and let $t_{\rm{cov}}(k)=\mathbb{E}[τ_{\mathrm{cov}}(k)]$ 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 $k \cdot \max_{x_1,\ldots,x_k}\mathbb{E}_{x_1,\ldots,x_k}[τ_{\rm{cov}}(k)] \asymp t_{\rm{cov}}$.