Min-Max延迟多机器人巡逻问题的循环解
On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem
摘要：
我们考虑以下监视问题：给定一组$P$的$n$网站在度量空间和一组$k$机器人具有相同的最大速度，计算巡逻时间表的机器人的最小延迟。在这里，巡逻时间表为每个机器人指定了要访问的无限序列的站点（以给定的顺序），并且时间表的延迟$L$是任何站点的最大延迟，其中站点$s$的延迟是连续访问$s$之间的时间间隔长度的上确界。当$k=1$的问题是等价的旅行商问题（TSP），因此它是NP-难的。我们有两个主要结果。我们认为循环的解决方案，其中的网站必须划分成$\ell$组，为一些~ $\ell \leq k$，每个组被分配一个子集的机器人移动沿旅行推销员旅游的组在彼此之间的距离相等。我们的第一个主要结果是，近似的循环解决方案的类的最佳延迟可以减少到近似的最佳旅行推销员旅游的一些输入，只有一个$1+\varepsilon$因子损失的近似因子和$O\left(\left( k/\varepsilon \right)^k\right)$因子损失的运行时，任何$\varepsilon >0$。我们的第二个主要结果表明，一个最佳的循环解决方案是一个$2(1-1/k)$ -近似的整体最佳的解决方案。注意，对于$k=2$，这意味着最优循环解总体上是最优的。结果会产生许多后果。对于欧几里德版本的问题，例如，结合我们的结果与已知的结果在欧几里德TSP，产生一个PTAS近似的最佳循环解决方案，它产生一个$(2(1-1/k)+\varepsilon)$ -近似的最佳无限制的解决方案。如果上述猜想成立，那么我们的算法实际上是欧几里德环境下一般问题的PTAS。
英文摘要：
We consider the following surveillance problem: Given a set $P$ of $n$ sites in a metric space and a set of $k$ robots with the same maximum speed, compute a patrol schedule of minimum latency for the robots. Here a patrol schedule specifies for each robot an infinite sequence of sites to visit (in the given order) and the latency $L$ of a schedule is the maximum latency of any site, where the latency of a site $s$ is the supremum of the lengths of the time intervals between consecutive visits to $s$. When $k=1$ the problem is equivalent to the travelling salesman problem (TSP) and thus it is NP-hard. We have two main results. We consider cyclic solutions in which the set of sites must be partitioned into $\ell$ groups, for some~$\ell \leq k$, and each group is assigned a subset of the robots that move along the travelling salesman tour of the group at equal distance from each other. Our first main result is that approximating the optimal latency of the class of cyclic solutions can be reduced to approximating the optimal travelling salesman tour on some input, with only a $1+\varepsilon$ factor loss in the approximation factor and an $O\left(\left( k/\varepsilon \right)^k\right)$ factor loss in the runtime, for any $\varepsilon >0$. Our second main result shows that an optimal cyclic solution is a $2(1-1/k)$-approximation of the overall optimal solution. Note that for $k=2$ this implies that an optimal cyclic solution is optimal overall. The results have a number of consequences. For the Euclidean version of the problem, for instance, combining our results with known results on Euclidean TSP, yields a PTAS for approximating an optimal cyclic solution, and it yields a $(2(1-1/k)+\varepsilon)$-approximation of the optimal unrestricted solution. If the conjecture mentioned above is true, then our algorithm is actually a PTAS for the general problem in the Euclidean setting.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
