The Power of Two Choices for Random Walks

We apply the power-of-two-choices paradigm to a random walk on a graph: rather than moving to a uniform random neighbour at each step, a controller is allowed to choose from two independent uniform random neighbours. We prove that this allows the controller to significantly accelerate the hitting and cover times in several natural graph classes. In particular, we show that the cover time becomes linear in the number $n$ of vertices on discrete tori and bounded degree trees, of order $\mathcal{O}(n\log \log n)$ on bounded degree expanders, and of order $\mathcal{O}(n(\log \log n{)}^2)$ on the Erd\H{o}s-R'{e}nyi random graph in a certain sparsely connected regime. We also consider the algorithmic question of computing an optimal strategy, and prove a dichotomy in efficiency between computing strategies for hitting and cover times.