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 ct_i(G)/dct_i(G), with ct_i(G) and dct_i(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 graphs. We also give an infinite family of graphs showing that iCOD can be arbitrarily close to any value in [2,infinty). Finally, we briefly examine one more CiR variant, in which the robber is invisible and "infinitely fast"; we argue that this variant is significantly different from the Graph Search game, despite several similarities between the two games.

Some remarks on cops and drunk robbers

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.