Capturing the Drunk Robber on a Graph

TL;DR

This paper proves that a cop can catch a randomly moving robber on any connected graph in expected time close to the number of vertices, showing the bound is nearly optimal and not dependent on graph diameter.

Contribution

It establishes a near-optimal upper bound on capture time for a cop chasing a random-walking robber on any connected graph, advancing understanding of pursuit-evasion dynamics.

Findings

Expected capture time is at most n + o(n) for any connected graph.

There exist graphs where capture time is at least n - o(n).

Capture time cannot be bounded solely by the graph's diameter.

Abstract

We show that the expected time for a smart "cop" to catch a drunk "robber" on an $n$-vertex graph is at most $n + {\rm o}(n)$. More precisely, let $G$ be a simple, connected, undirected graph with distinguished points $u$ and $v$ among its $n$ vertices. A cop begins at $u$ and a robber at $v$; they move alternately from vertex to adjacent vertex. The robber moves randomly, according to a simple random walk on $G$; the cop sees all and moves as she wishes, with the object of "capturing" the robber---that is, occupying the same vertex---in least expected time. We show that the cop succeeds in expected time no more than $n + {\rm o}(n)$. Since there are graphs in which capture time is at least $n - o(n)$, this is roughly best possible. We note also that no function of the diameter can be a bound on capture time.