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.

翻译：我们针对网络上的连续巡逻博弈问题给出了解决方案。在这个零和博弈中，攻击者选择一个时间和地点对网络进行固定时长的攻击，而巡逻者则在网络上巡逻，旨在以最大概率拦截攻击。我们的主要结果证明了一个关于树上最优巡逻策略的最新猜想。该猜想断言，一种被称为E-巡逻策略的特定巡逻策略对所有树网络都是最优的。此前该猜想仅在有限的特殊情况下成立。E-巡逻策略的优势在于计算和实现简便。我们通过给出攻击者的$\varepsilon$-最优策略来证明该猜想，这些策略为博弈值提供了上界，且该上界可以任意接近E-巡逻策略给出的下界。此外，我们还解决了完全网络在某些情况下的巡逻博弈问题。