A Graph Patrol Problem with Random Attack Times

Abstract

This paper presents a patrol problem, where a patroller traverses a graph through edges to detect potential attacks at nodes. To design a patrol policy, the patroller needs to take into account not only the graph structure, but also the different attack time distributions, as well as different costs incurred due to successful attacks, at different nodes. We consider both random attackers and strategic attackers. A random attacker chooses which node to attack according to a probability distribution known to the patroller. A strategic attacker plays a two-person zero-sum game with the patroller. For each case, we give an exact linear program to compute the optimal solution. Because the linear programs quickly become computationally intractable as the problem size grows, we develop index-based heuristics. In the random-attacker case, our heuristic is optimal when there are two nodes, and in a suitably chosen asymptotic regime. In the strategic-attacker case, our heuristic is optimal when there are two nodes if the attack times are deterministic taking integer values. In our numerical experiments, our heuristic typically achieves within 1% of optimality with computation time orders of magnitude less than what is required to compute the optimal policy.

Optimal patrol to uncover threats in time when detection is imperfect

“…Specifically, our work extends that of Lin et al [15] by allowing the possibility of overlooking. In other words, when the patroller inspects a node, he will detect an ongoing attack with a probability that depends on the node.…”

“…To decide where to go next, it is no longer sufficient for the patroller to only keep track of the last time he inspected each of the nodes. Thus the methods reported in Lin et al [15] no longer apply.…”

“…In this article, we use a graph to model the patrol area, with one patroller traversing the edges in the graph to detect potential attacks at nodes, similar to the framework in Alpern et al [3] and Lin et al [15]. There are two prominent features of our model: (1) the time it takes to attack a node is a random variable whose probability distribution depends on the node and (2) the patroller may overlook an ongoing attack at the inspected node.…”

Distributed Online Patrolling with Multi-agent Teams of Sentinels and Searchers

“…Of closest relevance to this paper are formulations of the dynamic vehicle routing problem in relation to algorithmic queuing theory, such as in [3,8], in which events requiring servicing appear in the environment stochastically, such as random arrivals of intruders in a protected area, requiring one or more agents to prioritize and visit these locations in an online manner. Alternate formulations consider patrol sequences under different assumptions for intruder arrivals, such as cases where intrusion sites are determined according to known probability distributions or by assuming adversarial intruders requiring game-theoretic design of patrols [9]. Commonly used objectives in such patrol problem formulations include minimizing the average or worst case revisit rate to return to a given location, which has correspondence to measures of service rates and wait time in queuing theory models [7,12].…”

Optimal search and ambush for a hider who can escape the search region

“…The two monographs on search games are Garnaev (2000) and Alpern and Gal (2003). Related work on patrolling is given in Lin et al (2013) and Zoroa (2009). Connections with inspection and smuggling games are mentioned in Section 5, on a discrete version of our game.…”