Optimal Patrolling Strategies for Trees and Complete Networks

TL;DR

This paper proves the optimality of the E-patrolling strategy for trees in a continuous patrolling game, providing a practical approach for network security and extending solutions to complete networks.

Contribution

It confirms the conjecture that E-patrolling is optimal for all tree networks and introduces methods to approximate optimal strategies for the attacker.

Findings

E-patrolling strategy is proven optimal for all trees.

Constructed ε-optimal attacker strategies approaching the game value.

Extended analysis to some complete networks.

Abstract

We present solutions to a continuous patrolling game played on network. In this zero-sum game, an Attacker chooses a time and place to attack a network for a fixed amount of time. A Patroller patrols the network with the aim of intercepting the attack with maximum probability. Our main result is the proof of a recent conjecture on the optimal patrolling strategy for trees. The conjecture asserts that a particular patrolling strategy called the E-patrolling strategy is optimal for all tree networks. The conjecture was previously known to be true in a limited class of special cases. The E-patrolling strategy has the advantage of being straightforward to calculate and implement. We prove the conjecture by presenting $\varepsilon$-optimal strategies for the Attacker which provide upper bounds for the value of the game that come arbitrarily close to the lower bound provided by the E-patrolling strategy. We also solve the patrolling game in some cases for complete networks.