冻结标签问题：如何唤醒一群机器人
The Freeze-Tag Problem: How to Wake Up a Swarm of Robots
摘要：
在群机器人研究中自然出现的一个优化问题是冻结标签问题（FTP），即如何唤醒一组“睡着”的机器人，让一个被唤醒的机器人移动到他们的位置。一旦一个机器人醒来，它可以帮助唤醒其他沉睡的机器人。目标是让所有机器人尽早醒来。虽然FTP与组合优化中的路由、广播、调度和覆盖等领域的问题有一些相似之处，但其算法特征却惊人地不同。我们考虑了图上和几何环境中的两种情况，在图中，机器人睡在顶点上，边上有一个长度函数。清醒的机器人沿着边缘移动，时间取决于边缘的长度。对于大多数场景，我们考虑问题的离线版本，其中每个清醒的机器人都知道所有其他机器人的位置。我们证明了这个问题是NP-困难的，即使是特殊的情况下的星图。我们还建立了逼近的困难度，证明了即使对于有界度的图，也很难得到优于5/3的逼近因子，并给出了一些积极的算法结果，包括：（1）证明了星图上的自然贪婪策略具有7/3的紧最坏情况性能，并给出了星图的多项式时间逼近方案（PTAS）. (2)我们给出了一个简单的O（log D）-竞争的在线算法的最大度D和局部有界的边权重的图。(3)我们给出了一个PTAS，运行在近线性时间，几何嵌入的实例。
英文摘要：
An optimization problem that naturally arises in the study of swarm robotics is the Freeze-Tag Problem (FTP) of how to awaken a set of ``asleep'' robots, by having an awakened robot move to their locations. Once a robot is awake, it can assist in awakening other slumbering robots.The objective is to have all robots awake as early as possible. While the FTP bears some resemblance to problems from areas in combinatorial optimization such as routing, broadcasting, scheduling, and covering, its algorithmic characteristics are surprisingly different. We consider both scenarios on graphs and in geometric environments.In graphs, robots sleep at vertices and there is a length function on the edges. Awake robots travel along edges, with time depending on edge length. For most scenarios, we consider the offline version of the problem, in which each awake robot knows the position of all other robots. We prove that the problem is NP-hard, even for the special case of star graphs. We also establish hardness of approximation, showing that it is NP-hard to obtain an approximation factor better than 5/3, even for graphs of bounded degree.These lower bounds are complemented with several positive algorithmic results, including: (1) We show that the natural greedy strategy on star graphs has a tight worst-case performance of 7/3 and give a polynomial-time approximation scheme (PTAS) for star graphs. (2) We give a simple O(log D)-competitive online algorithm for graphs with maximum degree D and locally bounded edge weights. (3) We give a PTAS, running in nearly linear time, for geometrically embedded instances.
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