In this paper we consider the problem of scheduling on computing platforms composed of several independent organizations, known as the Multi-Organization Scheduling Problem (MOSP). Each organization provides both resources and tasks and follows its own objectives. We are interested in the best way to minimize the makespan on the entire platform when the organizations behave in a selfish way. We study the complexity of the MOSP problem with two different local objectives-makespan and average completion time-and show that MOSP is NP-Hard in both cases. We formally define a selfishness notion, by means of restrictions on the schedules. We prove that selfish behavior imposes a lower bound of 2 on the approximation ratio for the global makespan. We present various approximation algorithms of ratio 2 which validate selfishness restrictions. These algorithms are experimentally evaluated through simulation, exhibiting good average performances.

Nowadays, with the abundance of diverse air interfaces in the same operating area, advanced Radio Resource Management (RRM) is vital to take advantage of the available system resources. In such a scenario, a mobile user will be able to connect concurrently to different wireless access networks. In this paper, we consider the downlink of a hybrid network with two broadband Radio Access Technologies (RAT): WiMAX [1] and WiFi [2]. Two approaches are proposed to load balance the traffic of every user between the two available RATs: an individual approach where mobile users selfishly strive to improve their performance and a global approach where resource allocation is made in a way to satisfy all mobile users. We devise for the individual approach a fully distributed resource management scheme portrayed as a non-cooperative game. We characterize the Nash equilibriums of the proposed RRM game and put forward a decentralized algorithm based on replicator dynamics to achieve those equilibriums. In the global approach, resources are assigned by the system in order to enhance global performances. For the two approaches, we show that after convergence, each user is connected to a single RAT which avoids costly traffic splitting between available RATs.Key-words: Non-cooperative game theory, non-linear optimisation, WiMAX, WiFi, 4G networks.Approches Individuelle et globale pour la gestion des ressources radio dans un réseau large-bande hybride Résumé : Actuellement, avec la diversification des interfaces radio présentes dans une même zone géographique, il devient vital de mettre en place des techniques avancées de gestion de ressources radio afin de profiter des ressources disponibles. Dans un tel scénario, un utilisateur mobile peut se connecter simultanémentà différents réseaux d'accès sans-fil. Dans ce papier, nous considérons la voie descendante d'un réseau hybride avec deux technologies d'accès radio large-bande disponibles: WiMAX [1] et WiFi [2]. Deux approches sont proposées pouréquilibrer le trafic de chaque utilisateur sur les deux technologies d'accès: une approche individuelle où les mobiles cherchentà améliorer leurs performances d'une manièreégoste et une approche globale où l'allocation de ressources est faite de façonà satisfaire tous les utilisateurs mobiles. Nous introduisons pour l'approche individuelle un schéma de gestion distribuée des ressources formulée en un jeu non-coopératif. Nous caractérisons leséquilibres de Nash du jeu proposé et proposons un algorithme distribué pour atteindre ceséquilibres. Pour l'approche globale, les ressources sont allouées par le système dans le but d'améliorer les performances globales. Pour les deux approches, nous démontrons qu'après convergence, chaque utilisateur se trouve connectéà une seule technologie d'accès sans-fil, ce qui permet d'éviter un partage de trafic coûteux entre les technologies disponibles.

The computational power of networks of finitely many anonymous resourcelimited mobile agents has been investigated in several recent papers. In particular, the population protocol model, introduced in [1], consists of a population

This paper deals with b-colorings of a graph G, that is, proper colorings in which for each color c, there exists at least one vertex colored by c such that its neighbors are colored by each other color. The b-chromatic number b(G) of a graph G is the maximum number of colors for which G has a b-coloring. It is easy to see that every graph G has a b-coloring using (G) colors.We say that G is b-continuous iff for each k, (G) k b(G), there exists a b-coloring with k colors. It is well known that not all graphs are b-continuous. We call b-spectrum S b (G) of G to be the set of integers k for which there is a b-coloring of G by k colors. We show that for any finite integer set I, there exists a graph whose b-spectrum is I and we investigate the complexity of the problem of deciding whether a graph G is b-continuous, even if b-colorings using (G) and b(G) colors are given.

This paper is concerned with all-optical networks using deflection routing and time division multiplexing. Slotted networks make use of the synchronous arrival of the packets to the routers to minimize locally the number of deflections. In this paper, we show that the difference in performances between slotted and unslotted networks is mainly due to the fact that unslotted networks cannot easily perform such local optimization. We also show that minimizing locally the number of deflections in unslotted networks gives rise to an NP-complete problem. To overcome this problem, we have designed a heuristic whose aim is to limit locally the number of deflections. We experimentally demonstrate that this heuristic enhances unslotted routing almost at the same performance level as slotted routing. As a consequence, we have shown that unslotted deflection routing can be implemented is a way which makes it a competitive alternative to slotted deflection routing for optical time division multiplexing deflection networks.

Abstract. We study the optimal linear arrangement (OLA) problem on interval graphs. Several linear layout problems that are NP-hard on general graphs are solvable in polynomial time on interval graphs. We prove that, quite surprisingly, optimal linear arrangement of interval graphs is NP-hard. The same result holds for permutation graphs. We present a lower bound and a simple and fast 2-approximation algorithm based on any interval model of the input graph.

Carrier-grade networks comprise several layers where different protocols coexist. Nowadays, most of these networks have different control planes to manage routing on different layers, leading to a suboptimal use of the network resources and to additional operational costs. However, some routers are able to encapsulate, decapsulate, and convert protocols, and act as a liaison between these layers. A unified control plane would be useful to optimize the use of the network resources and to automate the routing configurations. Software-Defined Networking based architectures, offer an opportunity to design such a control plane. One of the most important problems to deal with in this design is the path computation process. Classical path computation algorithms cannot resolve the problem as they do not take into account encapsulations and conversions of protocols. In this paper, we propose algorithms to solve this problem, and we study several cases. If there is no bandwidth constraint, we propose a polynomial algorithm that compute the optimal path. We also give lower and upper bounds on the optimal path length. On the other hand, we show that the problem is NPhard if there is a bandwidth constraint (or other Quality of Service parameters), even if there is only two protocols and in a symmetric graph. We study the complexity and the scalability of our algorithms and evaluate their performances on real and random topologies. The results show that they are faster than the previous ones proposed in the literature. These algorithms can also have important applications in automatic tunneling.

In a network, a tunnel is a part of a path where a protocol is encapsulated in another one. A tunnel starts with an encapsulation and ends with the corresponding decapsulation. Several tunnels can be nested at some stage, forming a protocol stack. Tunneling is very important nowadays and it is involved in several tasks: IPv4/IPv6 transition, VPNs, security (IPsec, onion routing), etc. However, tunnel establishment is mainly performed manually or by script, which present obvious scalability issues. Some works attempt to automate a part of the process (e.g., TSP, ISATAP, etc.). However, the determination of the tunnel(s) endpoints is not fully automated, especially in the case of an arbitrary number of nested tunnels. The lack of routing protocols performing automatic tunneling is due to the unavailability of path computation algorithms taking into account encapsulations and decapsulations. There is a polynomial centralized algorithm to perform the task. However, to the best of our knowledge, no fully distributed path computation algorithm is known. Here, we propose the first fully distributed algorithm for path computation with automatic tunneling, i.e., taking into account encapsulation, decapsulation and conversion of protocols. Our algorithm is a generalization of the distributed Bellman-Ford algorithm, where the distance vector is replaced by a protocol stack vector. This allows to know how to route a packet with some protocol stack. We prove that the messages size of our algorithm is polynomial, even if the shortest path can be of exponential length. We also prove that the algorithm converges after a polynomial number of steps in a synchronized setting. We adapt our algorithm into a proto-protocol for routing with automatic tunneling and we show its efficiency through simulations.