Cops and Invisible Robbers: the Cost of Drunkenness

We examine a version of the Cops and Robber (CR) game in which the robber is invisible, i.e., the cops do not know his location until they capture him. Apparently this game (CiR) has received little attention in the CR literature. We examine two variants: in the first the robber is adversarial (he actively tries to avoid capture); in the second he is drunk (he performs a random walk). Our goal in this paper is to study the invisible Cost of Drunkenness (iCOD), which is defined as the ratio cti(G)/dcti(G), with cti(G) and dcti(G) being the expected capture times in the adversarial and drunk CiR variants, respectively. We show that these capture times are well defined, using game theory for the adversarial case and partially observable Markov decision processes (POMDP) for the drunk case. We give exact asymptotic values of iCOD for several special graph families such as d-regular trees, give some bounds for grids, and provide general upper and lower bounds for general classes of graph...

• Type: Journal
• Year: 2012
• Where: CORR
• Authors: Athanasios Kehagias, Dieter Mitsche, Pawel Pralat
• Keywords: Classes Of Graphs, CORR 2012, Education, Graph Families, Graph Search