To Satisfy Impatient Web surfers is Hard

Description

Prefetching is a basic mechanism for faster data access and efficient computing. An important issue in prefetching is the tradeoff between the amount of network's resources wasted by the prefetching and the gain of time. For instance, in the Web, browsers may download documents in advance while a Web surfer is surfing on the Web. Since the Web surfer follows the hyperlinks in an unpredictable way, the choice of the Web pages to be prefetched must be computed online. The question is then to determine the minimum amount of resources used by prefetching that ensures that all documents accessed by the Web surfer have previously been loaded in the cache. We model this problem as a two-players game similar to Cops and Robber Games in graphs. The first player, a {\it fugitive}, starts on a marked vertex of a (di)graph $G$. The second player, an {\it observer}, marks $k \geq 1$ vertices, then the fugitive moves along one edge/arc of $G$ to a new vertex, then the observer marks $k$ vertices, etc. The observer wins if he prevents the fugitive to reach an unmarked vertex. The fugitive wins otherwise, i.e., if she succeed to enter an unmarked vertex. The {\it surveillance number} of a (di)graph is the minimum $k \geq 1$ allowing the observer to win against any strategy of the fugitive. We study the computational complexity of the game. We show that deciding whether the surveillance number of a chordal graph equals $2$ is NP-hard. Deciding if the surveillance number of a DAG equals $4$ is PSPACE-complete. Moreover, computing the surveillance number is NP-hard in split graphs. On the other hand, we provide polynomial time algorithms computing surveillance numbers of trees and interval graphs. Moreover, in the case of trees, we establish a combinatorial characterization, related to isoperimetry, of the surveillance number.

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