Cops and Invisible Robbers: The Cost of Drunkenness
Résumé
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 fi rst 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 defi ned 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 defi ned, 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 infi nite family of graphs showing that iCOD can be arbitrarily close to any value in [2,\infty). Finally, we briefly examine one more CiR variant, in which the robber is invisible and "infi nitely fast"; we argue that this variant is signi cantly di fferent from the Graph Search game, despite several similarities between the two games.
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