Some remarks on cops and drunk robbers

TL;DR

This paper investigates the impact of a randomly moving (drunk) robber on graph pursuit games, analyzing the expected capture times, developing algorithms for search strategies, and exploring variations like invisible robbers.

Contribution

It introduces the concept of the 'cost of drunkenness' in pursuit games, providing bounds and algorithms for near-optimal cop strategies against a random-walking robber.

Findings

Expected capture time ratios between optimal and drunk robber scenarios

Algorithms for near-optimal cop search schedules

Differences identified in invisible robber game dynamics

Abstract

The cops and robbers game has been extensively studied under the assumption of optimal play by both the cops and the robbers. In this paper we study the problem in which cops are chasing a drunk robber (that is, a robber who performs a random walk) on a graph. Our main goal is to characterize the "cost of drunkenness." Specifically, we study the ratio of expected capture times for the optimal version and the drunk robber one. We also examine the algorithmic side of the problem; that is, how to compute near-optimal search schedules for the cops. Finally, we present a preliminary investigation of the invisible robber game and point out differences between this game and graph search.