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...

Abstract The following scheduling problem naturally arises in thestudy of swarm robotics. Consider a set of n robots, mod-eled as points in some metric space (e.g., vertices of an edge-weighted graph). Initially, there is one awake oractive robot and all other robots are asleep, that is, in a stand-by mode. Our objective is to "wake up " all ofthe robots as quickly as possible. In order for an active robot to awaken a sleeping robot, the awake robotmust travel to the location of the slumbering robot. Once awake, this new robot is available to assist in rousing otherrobots. The objective is to minimize the makespan, that is, the time when the last robot awakens. This problemis the Freeze-Tag Problem (FTP) because it resembles the child's game of freeze-tag. The FTP is a scheduling problem that arises naturallyas a hybrid of problems from the areas of broadcasting,

Summary. Dispersing swarms of robots to cover an unknown, potentially hostile area is useful to setup a sensor network for surveillance. Previous research assumes relative locations (distance and bearing) of neighboring robots are available to each robot through sensors. Many robots are too small to carry sensors capable of providing this information. We use wireless signal intensity as a rough approximation of distance to assist a large swarm of small robots in dispersion. Simulation experiments indicate that a swarm can effectively disperse through the use of wireless signal intensities without knowing the relative locations of neighboring robots. 1

We give approximation algorithms and inapproximability results for a class of movement problems. In general, these problems involve planning the coordinated motion of a large collection of objects (representing anything from a robot swarm or firefighter team to map labels or network messages) to achieve a global property of the network while minimizing the maximum or average movement. In particular, we consider the goals of achieving connectivity (undirected and directed), achieving connectivity between a given pair of vertices, achieving independence (a dispersion problem), and achieving a perfect matching (with applications to multicasting). This general family of movement problems encompass an intriguing range of graph and geometric algorithms, with several real-world applications and a surprising range of approximability. In some cases, we obtain tight approximation and inapproximability results using direct techniques (without use of PCP), assuming just that P != NP.

Summary. We examine how the choice of the movement algorithm can affect the success of a swarm of simple mobile robots attempting to disperse themselves in an unknown environment. We assume there is no central control, and the robots have limited processing power, simple sensors, and no active communication. We evaluate different movement algorithms based on the percentage of the environment that the group of robots succeeds in observing. 1

We present the design and implementation of a multi-agent simulation that models deployment and coverage of sensors performing collaborative target detection. The focus is on sensor networks with enough sensors that humans cannot individually manage each. Experiments evaluated both known and novel deployment algorithms, and considered effects of the sensor type, number of sensors deployed, presence of obstacles, and mobility of the sensors. A particular focus was barrier (traversal) coverage which has many military applications but which has been less studied than other sensor placement problems; experiments showed that good algorithms for it are different than those good for area monitoring. This work provides both useful data for guiding sensor deployment and a valuable testbed for planning of sensor networks. 1.

Abstract. A large group of autonomous, mobile entities e.g. robots initially placed at some arbitrary node of the graph has to jointly visit all nodes (not necessarily all edges) and finally return to the initial position. The graph is not known in advance (an online setting) and robots have to traverse an edge in order to discover new parts (edges) of the graph. The team can locally exchange information, using wireless communication devices. We compare cost of the online and optimal offline algorithm which knows the graph beforehand (competitive ratio). If the cost is the total time of exploraiton, we prove the lower bound of Ω(log k / log log k) for competitive ratio of any deterministic algorithm (using global communication). This significantly improves the best known constant lower bound. For the cost being the maximal number of edges traversed by a robot (the energy) we present an improved (4 − 2/k)-competitive online algorithm for trees. 1

We illustrate algorithms for dispersing a swarm of primitive robots in a two-dimensional unknown environment R. Each robot in the swarm is equipped with a very simple sensor that is able to query the contents of neighboring locations to test the presence of other robots or obstacles. Based on the sensor readings, at each time step a robot may decide to take a discrete step to a neighboring (grid) point. The objective is to minimize the makespan, that is, the time to fill R with robots, one per grid point inside R. We focus here on the case of a discrete environment, composed of unit squares (pixels) that are induced by the integer grid within a polygonal domain R. There is at most one robot per pixel and robots move horizontally or vertically at unit speed. Robots enter R by means of k ≥ 1 door pixels on the boundary of R, each of which acts as an infinite source of robots. Robots are primitive finite automata, only having local communication, local sensors, and a constant-sized memory...

The research areas of mobile robotic sensors lie in the intersection of two major fields of investigations carried out by quite distinct communities of researchers: autonomous robots and mobile sensor networks. Robotic sensors are micro-robots capable of locomotion and sensing. Like the sensors in wireless sensor networks, they are myopic: their sensing range is limited. Unlike the sensors in wireless sensor networks, robotic sensors are silent: they have no direct communication capabilities. This means that synchronization, interaction, and communication of information among the robotic sensors can be achieved solely by means of their sensing capability, usually called vision. In this Chapter, we review the results of the investigations on the computability and complexity aspects of systems formed by these myopic and silent mobile sensors.

Abstract. We study an extensive class of movement minimization problems which arise from many practical scenarios but so far have little theoretical study. In general, these problems involve planning the coordinated motion of a collection of agents (representing robots, people, map labels, network messages, etc.) to achieve a global property in the network while minimizing the maximum or average movement (expended energy). The only previous theoretical results about this class of problems are about approximation, and mainly negative: many movement problems of interest have polynomial inapproximability. Given that the number of mobile agents is typically much smaller than the complexity of the environment, we turn to fixed-parameter tractability. We characterize the boundary between tractable and intractable movement problems in a very general set up: it turns out the complexity of the problem fundamentally depends on the treewidth of the minimal configurations. Thus the complexity of a particular problem can be determined by answering a purely combinatorial question. Using our general tools, we determine the complexity of several concrete problems and fortunately show that many movement problems of interest can be solved efficiently. 1