The Freeze-Tag Problem: How to Wake Up a Swarm of Robots

Abstract

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:• 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.• We give a simple O(log )-competitive online algorithm for graphs with maximum degree and locally bounded edge weights. • We give a PTAS, running in nearly linear time, for geometrically embedded instances. Introduction.The following problem naturally arises in the study of swarm robotics. Consider a set of n robots, modeled as points in some metric space (e.g., vertices of an edge-weighted graph). Initially, there is one awake or active robot and all other robots are asleep, that is, in a stand-by mode. Our objective is to "wake up" all of the robots as quickly as possible. In order for an active robot to awaken a sleeping robot, the awake robot must travel to the location of the slumbering robot. Once awake, this new robot is available to assist in rousing other robots. The objective is to minimize the makespan, that is, the time when the last robot awakens.This awakening problem is reminiscent of the children's game of "freeze-tag", in which the person who is "it" tags other players to "freeze" them. A player remains "frozen" until an unfrozen player (who is not "it") rescues the frozen player by tagging him and thus unfreezing him. Our problem arises when there are a large number n of frozen players, and one (not "it") unfrozen player, whose goal it is to unfreeze the rest of the players as quickly as possible. (We do not take into consideration the effect of the person who is "it", who is likely running around and re-freezing the players that become defrosted!) As soon as a player becomes unfrozen, he is available to assist in helping other frozen players, so there is a cascading effect. Due to the similarity with this child's game, ...

The Online Freeze-Tag Problem

“…The Freeze-tag problem has received some attention recently [1,2,7]: given a set of n robots located at points in some metric space. Initially, there is one or more awake or active robots and the other robots are asleep, i.e., in a stand-by mode.…”

“…Once awake, this new robot is available to assist in awakening other sleeping robots. This problem was dubbed the Freeze-tag problem by the authors of [1,2,7].…”

“…The online step dependent Freeze-tag problem (online SDFT(k) for short) has k awake robots from the start and a request is released only when the previous request has been activated or served and the objective is to minimize the total distance travelled by all the robots. The online time dependent Freeze-tag problem (online TDFT for short), closely models the Freeze-tag problem defined in [1] but in an online setting. The requests are released independently of how many of them have already been served and our objective is to minimize the makespan.…”

“…The Freeze-tag problem has many applications such as data distribution, network design, broadcasting, routing and scheduling [1]. Another feature is that SDFT(k) and TDFT exhibit some elements of the k-server problem and the Traveling Salesman Problem, respectively.…”

“…Furthermore, in TDFT there are many salesmen working in parallel since whenever a salesman visit a city, he recruits a new salesman to help visit other cities. The time dependent Freeze-tag problem was first introduced by Arkin et al [1]. They showed that in the offline case, minimizing the time when the last robot awakens, even simple versions of the problem (e.g.…”