Hitting time, access time and optimal transport on graphs

Abstract

Given a discrete source distribution $\mu$ and discrete target distribution $\nu$ on a common finite state space $\mathcal{X}$, we are tasked with transporting $\mu$ to $\nu$ using a given discrete-time Markov chain $X$ with the quickest possible time on average. We define the optimal transport time $H(\mu,\nu)$ as stopping rule of $X$ that gives the minimial expected transport time. This is also known as the access time from $\mu$ to $\nu$ of $X$ in [L. Lov\'{a}sz and P. Winkler. Efficient Stopping Rules for Markov Chains. Proceedings of the Twenty-seventh Annual ACM Symposium on Theory of Computing (STOC '95) 76-82.]. We study bounds of $H(\mu,\nu)$ in various special graphs, which are expressed in terms of the mean hitting times of $X$ as well as parameters of $\mu$ and $\nu$ such as their moments. Among the Markov chains that we study, random walks on complete graphs is a good choice for transport as $H(\mu,\nu)$ grows linearly in $n$, the size of the state space, while that of the winning streak Markov chain exhibits exponential dependence in $n$.