In the standard Freeze-Tag Problem (FTP), an initially awake robot (the source) is in charge of waking up a swarm of sleeping robots by moving towards them, given that all the awake robots can participate in the awakening process. The goal is to minimize the makespan to wake up all robots assuming they move at unit speed. In this paper we introduce the Freeze-Tag-with-Return Problem (FTRP) variant, where the robots must eventually return to their initial positions. In the Euclidean plane with n sleeping robots lying on the unit disk centered at the initial position of the source, we show a non-trivial relationship between FTP and FTRP by proving that the difference between the optimal makespan of both problems never exceeds 1.959, and is at least 1.732 in the worst-case. We also present several upper and lower bounds on the optimal makespan. In particular, we show that if the sleeping robots are in convex positions, then the optimal makespan is at most 2 + 2√2, which is achieved by some instances. From an algorithmic point-of-view, we present single-exponential algorithms for general distance functions. In metric spaces, these algorithms are asymptotically optimal under the ETH, which we show via an NP-hardness reduction on unweighted graphs.

The Freeze-Tag Problem, introduced in Arkin et al. (SODA'02) consists of waking up a swarm of n robots, starting from a single active robot. In the basic geometric version, every robot is given coordinates in the plane. As soon as a robot is awakened, it can move towards inactive robots to wake them up. The goal is to minimize the makespan of the last robot, the makespan. Despite significant progress on the computational complexity of this problem and on approximation algorithms, the characterization of exact bounds on the makespan remains one of the main open questions. In this paper, we settle this question for the 𝓁₁-norm, showing that a makespan of at most 5r can always be achieved, where r is the maximum distance between the initial active robot and any sleeping robot. Moreover, a schedule achieving a makespan of at most 5r can be computed in time O(n). Both bounds, the time and the makespan are optimal. Our results also imply for the 𝓁₂-norm a new upper bound of 5√2r ≈ 7.07r on the makespan, improving the best known bound of (5+2√2+√5)r ≈ 10.06r. Along the way, we introduce new linear time wake-up strategies, that apply to any norm and show that an optimal bound on the makespan can always be achieved by a schedule computable in linear time.

A geometric graph G=(P,E) is a set of points in the plane and edges between pairs of points, where the weight of an edge is equal to the Euclidean distance between its two endpoints. In local routing we find a path through G from a source vertex s to a destination vertex t, using only knowledge of the current vertex, its incident edges, and the locations of s and t. We present an algorithm for local routing on the Delaunay triangulation, and show that it finds a path between a source vertex s and a target vertex t that is not longer than 3.56|st|, improving the previous bound of 5.9|st|.

Let ϵ be the complete Euclidean graph on a set of points embedded in the plane. Given a constant t ≥ 1, a spanning subgraph G of ϵ is said to be a t-spanner, or simply a spanner, if for any pair of vertices u, v in ϵ the distance between u and v in G is at most t times their distance in ϵ. A spanner is plane if its edges do not cross.

We consider the classical problem of broadcasting a large message at an optimal rate in a large scale distributed network under the multi-port communication model. In this context, we are interested in both building an overlay network and providing an explicit algorithm for scheduling the communications. From an optimization point of view, we aim both at maximizing the throughput (i.e., the rate at which nodes receive the message) and minimizing the degree of the participating nodes, i.e., the number of TCP connections they must handle simultaneously. The main novelties of our approach are the introduction of this degree constraint and the classification of the set of participating nodes into two parts: open nodes that stay in the open-Internet and “guarded” nodes that lie behind firewalls or NATs. Two guarded nodes cannot communicate directly, but rather need to use an open node as a gateway for transmitting a message. In the case without guarded nodes, we prove that it is possible to reach the optimal throughput, at the price of a quasi-optimal (up to a small additive increase) degree of the participating nodes. In presence of guarded nodes, our main contributions are a closed form formula for the optimal cyclic throughput and the proof that the optimal solution may require arbitrarily large degrees. In the acyclic case, we propose an algorithm that reaches the optimal throughput with low degree. Then, we prove a worst case ratio between the optimal acyclic and cyclic throughput and show through simulations that this ratio is on average very close to 1, what makes acyclic solutions efficient both in terms of throughput maximization and degree minimization.

Malleable tasks are jobs that can be scheduled with preemptions on a varying number of resources. We focus on the special case of work-preserving malleable tasks, for which the area of the allocated resources does not depend on the allocation and is equal to the sequential processing time. Moreover, we assume that the number of resources allocated to each task at each time instant is limited. We consider both the clairvoyant and non-clairvoyant cases, and we focus on minimizing the weighted sum of completion times. In the weighted non-clairvoyant case, we propose an approximation algorithm whose ratio (2) is the same as in the unweighted non-clairvoyant case. In the clairvoyant case, we provide a normal form for the schedule of such malleable tasks, and prove that any valid schedule can be turned into this normal form, based only on the completion times of the tasks. We show that in these normal form schedules, the number of preemptions per task is bounded by 3 on average. At last, we analyze the performance of list schedules, and prove that optimal schedules are list schedules for a special case of homogeneous instances. We conjecture that there exists an optimal list schedule for all instances, which would greatly simplify the study of this problem. Finally, we explore the complexity of the problem restricted to homogeneous instances, which is still open despite its very simple expression.

We consider the classical problem of broadcasting a large message at an optimal rate in a large scale distributed network. The main novelty of our approach is that we consider that the set of participating nodes can be split into two parts: "green" nodes that stay in the open-Internet and "red" nodes that lie behind firewalls or NATs. Two red nodes cannot communicate directly, but rather need to use a green node as a gateway for transmitting a message. In this context, we are interested in both maximizing the throughput (i.e. the rate at which nodes receive the message) and minimizing the degree at the participating nodes, i.e. the number of TCP connections they must handle simultaneously. We consider both cyclic and a cyclic solutions for the flow graph. In the cyclic case, our main contributions are a closed form formula for the optimal cyclic throughput and the proof that the optimal solution may require arbitrarily large degrees. In the a cyclic case, we propose an algorithm to achieve the optimal throughput with low degree. Then, we prove a worst case ratio between the optimal a cyclic and cyclic throughput and show through simulations that this ratio is on average very close to 1, which makes a cyclic solutions efficient both in terms of throughput and of number of connections.

We consider a generalization of a classical optimization problem related to server and replica location problems in networks. More precisely, we suppose that a set of users distributed over a network wish to have access to a particular service proposed by a set of providers. The aim is then to distinguish a set of service providers able to offer a sufficient amount of resources in order to satisfy the requests of the clients. Moreover, a quality of service following some requirements in terms of latencies is desirable. A smart repartition of the servers in the network may also ensure good fault tolerance properties. We model this problem as a variant of Bin Packing, namely Bin Packing under Distance Constraint(BPDC) where the goal is to build a minimal number of bins(i.e. to choose a minimal number of servers) so that (i) each client is associated to exactly one server, (ii) the capacity of the server is large enough to satisfy the requests of its clients and (iii) the distance between two clients associated to the same server is minimized. We prove that this problem is hard to approximate even when using resource augmentation techniques : we compare the number of obtained bins when using polynomial time algorithms allowed to build bins of diameter at most beta.dmax, for beta > 1, to the optimal number of bins of diameter at most dmax. On the one hand, we prove that (i) if _ = (2âˆ'epsilon), BPDC is hard to approximate within any constant approximation ratio, for any epsilon > 0, and that (ii) BPDC is hard to approximate at a ratio lower than 3/2 even if resource augmentation is used. On the other hand, if beta = 2, we propose a polynomial time approximation algorithm for BPDC with approximation ratio 7/3 in the general case. We show how to turn an approximation algorithm for BPDC into an approximation algorithm for the non-uniform capacitated K-center problem and vice-versa. Then, we present a comparison of the quality of results for BPDC in the context of several Internet latency embedding tools such as Sequoia and Vivaldi, using datasets based on Planet Lab latency measurements.

In this paper, we discuss complexity issues for scheduling divisible workloads on heterogeneous systems under the bounded multi-port model. To our best knowledge, this paper is the first attempt to consider divisible load scheduling under a realistic communication model, where the master node can communicate simultaneously to several slaves, provided that bandwidth constraints are not exceeded. In this paper, we concentrate on one round distribution schemes, where a given node starts its processing only once all data has been received. Our main contributions are (i) the proof that processors start working immediately after receiving their work (ii) the study of the optimal schedule in the case of 2 processors and (iii) the proof that scheduling divisible load under the bounded multi-port model is NP-complete. This last result strongly differs from divisible load literature and represents the first NP-completeness result when latencies are not taken into account.