Patrolling a Border

Abstract

Patrolling games were recently introduced by Alpern, Morton and Papadaki to model the problem of protecting the nodes of a network from an attack. Time is discrete and in each time unit the Patroller can stay at the same node or move to an adjacent node. The Attacker chooses when to attack and which node to attack, and needs m consecutive time units to carry it out. The Attacker wins if the Patroller does not visit the chosen node while it is being attacked; otherwise the Patroller wins. This paper studies the patrolling game where the network is a line graph of n nodes, which models the problem of guarding a channel or protecting a border from infiltration. We solve the patrolling game for any values of m and n, providing an optimal Patroller strategy, an optimal Attacker strategy and the value of the game (optimal probability that the attack is intercepted).

Optimizing periodic patrols against short attacks on the line and other networks

“…The case of a unit attack duration m = 1 is covered by the field of geometric games as defined by Ruckle (1983), so we here consider the next smallest duration m = 2, which is the only case thus far susceptible to analysis. We note that the easier version of non-periodic patrolling games is able to handle line graphs for larger values of m, as recently solved by Papadaki et al (2016). It is likely that the techniques introduced here will be extended to larger attack durations in the future, but clearly additional ideas will be required.…”

“…Multiple patrollers, when only some portions of the boundary need to be protected, are considered by Collins et al (2013), who show how the problem can be divided up. Papadaki et al (2016) consider the discrete border patrol problem, where the infiltration can only be accomplished at certain points of the border (perhaps mountain passes). When patrollers are restricted to periodic patrols, as here, the analysis of the continuous problem (with elements such as turning radius included) has been analyzed by Chung et al (2011).…”

“…We show that if the Patroller's period is even, the game can be solved on any graph by finding the fractional covering number and fractional independence number of the graph. We also give a complete solution to the periodic patrolling game on line graphs of arbitrary size, extending the work of Papadaki et al (2016) to the periodic domain. This models the patrolling problem on a border or channel, which is related to a classical problem of operational research going back to Morse and Kimball (1951).…”

Timely exposure of a secret project: Which activities to monitor?

“…Atkinson and Wein (2010) examine how a government should allocate its resources over the inspection of terror and criminal networks to exploit the finding of Smith, Damphousse, and Roberts (2006) that, prior to an attack, terrorists frequently participate in crimes such as theft or procuring explosives. Other articles address problems such as predicting the number of undetected terror threats (Kaplan, 2010), estimating the duration of a terrorist plot (Kaplan, 2012a), locating terrorists (Alpern & Lidbetter, 2013;Atkinson, Kress, & Lange, 2016), processing intelligence (Dimitrov, Kress, & Nevo, 2016;Lin, Kress, & Szechtman, 2009), patrolling an area (Lin, Atkinson, & Glazebrook, 2014;Papadaki, Alpern, Lidbetter, & Morton, 2016;Szechtman, Kress, Lin, & Cfir, 2008), and predicting the goal of a suspected terrorist (Tsitsiklis & Xu, 2018). In particular, Atkinson et al (2016) consider a searcher who, based on a stream of unreliable intelligence about a target's location, needs to decide whether to engage or to wait for more information.…”

Spy-game on graphs: Complexity and simple topologies

“…Eternal Domination and Spy game are also related to Patrolling games where a team of patrollers must move in a graph such that every vertex must never be unoccupied during more than d consecutive steps where d is a fixed parameter [4,19]. In particular, since at each step, no vertex is at distance more than d from some patroller, the minimum size of a team for the Patrolling game provides an upper bound on the minimum number of guards required for controlling the spy at distance d, whatever be its speed.…”