## The Freeze-Tag Problem: How to Wake Up a Swarm of Robots (2002)

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Venue: | In Proc. 13th ACM-SIAM Sympos. Discrete Algorithms |

Citations: | 32 - 6 self |

### BibTeX

@INPROCEEDINGS{Arkin02thefreeze-tag,

author = {Esther M. Arkin and Michael A. Bender and Sandor P. Fekete and Joseph S. B. Mitchell and Martin Skutella},

title = {The Freeze-Tag Problem: How to Wake Up a Swarm of Robots},

booktitle = {In Proc. 13th ACM-SIAM Sympos. Discrete Algorithms},

year = {2002},

pages = {568--577}

}

### Years of Citing Articles

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### Abstract

An optimization problem that naturally arises in the study of "swarm robotics" is to wake up a set of "asleep" robots, starting with only one "awake" robot. One robot can only awaken another when they are in the same location. As soon as a robot is awake, it assists in waking up other robots. The goal is to compute an optimal awakening schedule such that all robots are awake by time t , for the smallest possible value of t . We consider both scenarios on graphs and in geometric environments. In the graph setting, robots sleep at vertices and there is a length function on the edges. An awake robot can travel from vertex to vertex along edges, and the length of an edge determines the time it takes to travel from one vertex to the other. While this problem bears some resemblance to problems from various areas in combinatorial optimization such as routing, broadcasting, scheduling and covering, its algorithmic characteristics are surprisingly different. 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 algorithmic results. We present a simple on-line algorithm that is O(log)-competitive for graphs with maximum degree . Other results include algorithms that require substantially more sophistication and development of new techniques: (1) The natural greedy strategy on star graphs has a worst-case performance of 7/3, which is tight. (2) There exists a PTAS for star graphs. (3) For the problem Dept. of Appl. Math. and Statistics, SUNY Stony Brook, NY 11794-3600, festie, jsbmg@ams.sunysb.edu. y Dept. of Computer Science, SUNY St...

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Citation Context ...s hard. Theorem 2.6 The FTP is strongly NP-hard, even for the special case of weighted stars with one (asleep) robot at each leaf. Proof: Our reduction is from NUMERICAL 3-DIMENSIONAL MATCHING (N3DM) =-=[6]-=-. Details can be found in the Appendix. 2 2.3 A PTAS. We give a PTAS for the FTP on weighted stars with one awake robot at the central node, v 0 , and an equal number q of sleeping robots at each leaf... |

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Citation Context ...awakening tree, T , having makespan t. Then, for any µ > 0, there exists a pseudo-balanced awakening tree, Tb, of makespan tb ≤ (1 + µ)t. Proof. First we perform a heavy-path decomposition (e.g., see =-=[34]-=-) of the tree T . For each node v, let d(v) be the number of descendents. Consider a node u with children v1, . . . , vk. The edge (u, vi) is heavy if vi has more descendents than any other child of u... |

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Citation Context ... of prior work on the dissemination of data in a graph. Most closely related to the FTP are the minimum broadcast time problem, the multicast problem, and the related minimum gossip time problem. See =-=[7]-=- for a survey; see [1, 10] for recent approximation results. However, that there are significant differences: While the broadcast problem can be solved in polynomial time in tree networks, the FTP tur... |

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Citation Context ...h as trees. Some of our results are specific to star metrics and ultrametrics, which arise as an important tool in obtaining approximation algorithms in more general metric spaces, as shown, e.g., in =-=[2, 3, 9]-=-. We also study a geometric variant of the problem where the robots are located at points of a geometric space and travel times are given by geometric distances. It turns out that geometry helps subst... |

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Citation Context ...ialtime approximation scheme. Similar techniques have been applied for other classes of problems before, e. g., in the construction of approximation schemes for machine scheduling problems, e.g., see =-=[-=-8]. However, the new challenge for the problem at hand is to cope with the awakened robots at short edges whose number can increase geometrically over time. Let T t be a good lower bound on the make... |

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Citation Context ... class of weighted stars. In the field of robotics, several related algorithmic problems have been studied for controlling swarms of robots to perform various tasks, including environment exploration =-=[1,2,10,19,23,36,38]-=-, robot formation [9, 29–31], searching [37], and recruitment [35]. Ant behaviors have inspired algorithms for multi-agent problems such as searching and covering; see, e.g., [35–37]. Multi-robot form... |

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Citation Context ... class of weighted stars. In the field of robotics, several related algorithmic problems have been studied for controlling swarms of robots to perform various tasks, including environment exploration =-=[1,2,10,19,23,36,38]-=-, robot formation [9, 29–31], searching [37], and recruitment [35]. Ant behaviors have inspired algorithms for multi-agent problems such as searching and covering; see, e.g., [35–37]. Multi-robot form... |

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Citation Context ...dissemination of data in a graph. Most closely related to the FTP are the minimum broadcast time problem, the multicast problem, and the related minimum gossip time problem. See [7] for a survey; see =-=[1, 10]-=- for recent approximation results. However, that there are significant differences: While the broadcast problem can be solved in polynomial time in tree networks, the FTP turns out to be NP-hard alrea... |

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Citation Context ...dissemination of data in a graph. Most closely related to the FTP are the minimum broadcast time problem, the multicast problem, and the related minimum gossip time problem. See [7] for a survey; see =-=[1, 10]-=- for recent approximation results. However, that there are significant differences: While the broadcast problem can be solved in polynomial time in tree networks, the FTP turns out to be NP-hard alrea... |

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Citation Context ...ening tree, T , having makespan t. Then, for any > 0, there exists a pseudo-balanced awakening tree, T b , of makespan t b (1 + )t. Proof: First we perform a heavy path decomposition (see, e.g., [11=-=]-=-) of the tree T . For each node v, let d(v) be the number of descendents. Consider a node u with children v 1 ; : : : ; v k . The edge (u; v i ) is heavy if v i has more descendents than any other chi... |

35 |
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Citation Context ... of prior work on the dissemination of data in a graph. Most closely related to the FTP are the minimum broadcast time problem, the multicast problem, and the related minimum gossip time problem. See =-=[21]-=- for a survey; see [7,26] for recent approximation results. However, the proximity required in the FTP leads to significant differences: While the broadcast problem can be solved in polynomial time in... |

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Citation Context ... class of weighted stars. In the field of robotics, several related algorithmic problems have been studied for controlling swarms of robots to perform various tasks, including environment exploration =-=[1,2,10,19,23,36,38]-=-, robot formation [9, 29–31], searching [37], and recruitment [35]. Ant behaviors have inspired algorithms for multi-agent problems such as searching and covering; see, e.g., [35–37]. Multi-robot form... |

33 |
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Citation Context ...in order to distribute important information. The two-robot scenario with initial positions unknown to both players is the problem of rendezvous search that has received quite a bit of attention, see =-=[4, 27]-=- and the upcoming book by Alpern and Gal [3] for an overview. In subsequent work on the FTP, Sztainberg, Arkin, Bender, and Mitchell [32,33] analyze and implement heuristics for the FTP. They show tha... |

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Citation Context ...in order to distribute important information. The two-robot scenario with initial positions unknown to both players is the problem of rendezvous search that has received quite a bit of attention, see =-=[4, 27]-=- and the upcoming book by Alpern and Gal [3] for an overview. In subsequent work on the FTP, Sztainberg, Arkin, Bender, and Mitchell [32,33] analyze and implement heuristics for the FTP. They show tha... |

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Citation Context ...oblems have been studied for controlling swarms of robots to perform various tasks, including environment exploration [1,2,10,19,23,36,38], robot formation [9, 29–31], searching [37], and recruitment =-=[35]-=-. Ant behaviors have inspired algorithms for multi-agent problems such as searching and covering; see, e.g., [35–37]. Multi-robot formation in continuous and grid environments has been studied recentl... |

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Citation Context ...h as trees. Some of our results are specific to star metrics and ultrametrics, which arise as an important tool in obtaining approximation algorithms in more general metric spaces, as shown, e.g., in =-=[2, 3, 9]-=-. We also study a geometric variant of the problem where the robots are located at points of a geometric space and travel times are given by geometric distances. It turns out that geometry helps subst... |

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Citation Context ...related algorithmic problems have been studied for controlling swarms of robots to perform various tasks, including environment exploration [1,2,10,19,23,36,38], robot formation [9, 29–31], searching =-=[37]-=-, and recruitment [35]. Ant behaviors have inspired algorithms for multi-agent problems such as searching and covering; see, e.g., [35–37]. Multi-robot formation in continuous and grid environments ha... |

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12 | Improved approximation algorithms for the freeze-tag problem
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Citation Context ...) for points in d dimensions. They also present experimental results on classes of randomly generated data, as well as on datasets from the TSPLIB repository [28]. Arkin, Bender, Ge, He, and Mitchell =-=[6]-=- give an O(1)-approximation algorithm for the FTP in unweighted graphs, in which there is one asleep robot at each node, and they show that this version of the FTP is NPhard. They generalize to the ca... |

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Citation Context ...O((L/d)log n + 1)-approximationFREEZE-TAG: WAKING UP A SWARM OF ROBOTS 3 algorithm, where L is the length of the longest edge and d is the diameter of the graph. Recently, Könemann, Levin, and Sinha =-=[24]-=- gave an O( √ log n)-approximation algorithm for the general FTP, in the context of bounded degree minimum diameter spanning tree problem. Thus, the authors answer in the affirmative an important open... |

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11 | Analysis of heuristics for the freeze-tag problem
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Citation Context ...vous search that has received quite a bit of attention, see [4, 27] and the upcoming book by Alpern and Gal [3] for an overview. In subsequent work on the FTP, Sztainberg, Arkin, Bender, and Mitchell =-=[32,33]-=- analyze and implement heuristics for the FTP. They show that the greedy strategy gives a tight approximation bound of Θ( √ log n) for the case of points in the plane and, more generally, Θ((log n) 1−... |

10 |
Approximating a metric by a small number of tree metrics
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(Show Context)
Citation Context ...h as trees. Some of our results are specic to star metrics and ultrametrics , which arise as an important tool in obtaining approximation algorithms in more general metric spaces, as shown, e.g., in [=-=2, 3, 7]-=-. We also study a geometric variant of the problem where the robots are located at points of a geometric space and travel times are given by geometric distances. It turns out that geometry helps subst... |

8 |
Rapid rumor rami Approximating the minimum broadcast time
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Citation Context ...dissemination of data in a graph. Most closely related to the FTP are the minimum broadcast time problem, the multicast problem, and the related minimum gossip time problem. See [5] for a survey; see =-=[1, -=-8] for recent approximation results. However, there are signicant dierences: While the broadcast problem can be solved in polynomial time in tree networks, the FTP turns out to be NP-hard already on t... |

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Citation Context ...aspects related to freeze tag, like considering algorithmic issues arising from the full game of freeze-tag in the presence of an adversary. This is somewhat related to the Competing Salesman Problem =-=[15]-=-, where two salesmen travel in a graph and try to visit vertices before the opponent does. As described in the introduction, there are many other related questions, and we do expect many more interest... |

3 | Online dispersion for swarms of robots - Hsiang, Arkin, et al. - 2003 |

3 |
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Citation Context ...vous search that has received quite a bit of attention, see [4, 27] and the upcoming book by Alpern and Gal [3] for an overview. In subsequent work on the FTP, Sztainberg, Arkin, Bender, and Mitchell =-=[32,33]-=- analyze and implement heuristics for the FTP. They show that the greedy strategy gives a tight approximation bound of Θ( √ log n) for the case of points in the plane and, more generally, Θ((log n) 1−... |

3 | abstractions to support many-robot systems - Sensor - 1992 |

2 | Cost-optimization of many-robot systems - Gage - 1994 |

1 |
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1 | anonymous mobile robots: Formation of geometric patterns - Distributed - 1999 |