具有平移目标的动态边界守卫问题
A Dynamic Boundary Guarding Problem with Translating Targets
摘要：
我们引入了一个问题，其中的服务车辆试图保护动态到达的移动目标的最后期限（边界）。环境是一个矩形，截止日期是它的一条边。目标随着时间的推移不断到达截止日期对面的边缘，并以固定的速度向截止日期移动。飞行器的目标是最大限度地提高在截止日期之前捕获的目标的比例。我们考虑两种情况下，当服务车辆比目标快，和，当服务车辆比目标慢。在第一种情况下，我们开发了一种新的车辆政策的基础上计算最长路径的有向无环图。我们给出了该策略的捕获分数的下界，并证明了当目标到达边缘和截止时间之间的距离变得很大时，该策略是最优的。我们提出的数值结果表明，接近最佳性能远离这个限制制度。在第二种情况下，当目标速度小于车辆速度时，我们提出了一种基于平移最小哈密顿路径服务分数的策略，在目标速度低、到达率高的限制下，该策略的捕获分数与最优值相差一个小的常数因子。
英文摘要：
We introduce a problem in which a service vehicle seeks to guard a deadline (boundary) from dynamically arriving mobile targets. The environment is a rectangle and the deadline is one of its edges. Targets arrive continuously over time on the edge opposite the deadline, and move towards the deadline at a fixed speed. The goal for the vehicle is to maximize the fraction of targets that are captured before reaching the deadline. We consider two cases; when the service vehicle is faster than the targets, and; when the service vehicle is slower than the targets. In the first case we develop a novel vehicle policy based on computing longest paths in a directed acyclic graph. We give a lower bound on the capture fraction of the policy and show that the policy is optimal when the distance between the target arrival edge and deadline becomes very large. We present numerical results which suggest near optimal performance away from this limiting regime. In the second case, when the targets are slower than the vehicle, we propose a policy based on servicing fractions of the translational minimum Hamiltonian path. In the limit of low target speed and high arrival rate, the capture fraction of this policy is within a small constant factor of the optimal.
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