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 $\epsilon$-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.

我们提出了在网络上玩的连续巡逻游戏的解决方案。在这种零和游戏中，攻击者选择时间和地点在固定的时间内攻击网络。巡逻者在网络中巡逻，旨在以最大概率拦截攻击。我们的主要结果是证明了最近关于树木最佳巡逻策略的猜想。该猜想断言，一种称为“巡逻”的特定巡逻策略$乙$- 巡逻策略对于所有树状网络都是最佳的。此前已知该猜想在有限的特殊情况下是正确的。这$乙$-巡逻策略的优点是易于计算和实施。我们通过提出来证明这个猜想$\epsilon$- 攻击者的最佳策略，为游戏的价值提供上限，该上限任意接近由攻击者提供的下限$乙$- 巡逻策略。我们还在某些情况下解决了完整网络的巡逻博弈。