Section: New Results
Backbone Networks
Participants : Jean-Claude Bermond, Nathann Cohen, David Coudert, Frédéric Giroire, Dorian Mazauric, Gianpiero Monaco, Joanna Moulierac, Napoleão Nepomuceno, Nicolas Nisse, Stéphane Pérennes, Hervé Rivano, Ignasi Sau-Valls.
Network design is a very wide subject that concerns all kinds of networks. For telecommunications networks it can be either physical networks (backbone, access, wireless, ...) or virtual (logical) ones. The objective is to design a network able to route a (given, estimated, dynamic, ...) traffic under some constraints (e.g. capacity) and with some quality of service (QoS) requirements. Usually the traffic is expressed as a family of requests with parameters attached to them. In order to satisfy these requests, we need to find one (or many) path(s) between their end nodes. The set of paths is chosen according to the technology, the protocol or the QoS constraints. For instance, optical backbones use the WDM technology to take better advantage of the capacity of the optical fibers often already installed. This is achieved through the multiplexing of several wavelength channels onto the same fiber. In that case a resource allocation is an optical channel, which consists of a path and a wavelength assigned on each link along the path, and is called a lightpath . If wavelength translation is performed in optical switching, then to each channel may be assigned different wavelengths on each link along the path; otherwise the wavelength continuity constraint must be satisfied on all links along the path. Of course, two lightpaths sharing a link must use different wavelengths on that link. The design can be done at the conception of the network (i.e. when conceiving a virtual network in MPLS where we have to establish virtual paths) or to adapt the network to changes (failures, new link, updates of routers, variation of traffic, ...). Finally there are various optimization criteria which differ according to the point of view: for a network user they are related to his/her satisfaction (minimizing delays, increasing available bandwidth, ...), while for a network operator, economics criteria like minimizing deployment and operating costs are more important.
This very wide topic is considered by a lot of academic and industrial teams in the world. Our approach is to attack these problems with tools from Discrete Mathematics and to consider mainly telecommunications networks. This approach is shared by other teams in Europe, most of them being part of European projects IST FET AEOLUS (where Mascotte is leader of sub-project SP2 Resource management ) and COST 293 Graal (where Mascotte is leader of working group WG-A broadband and optical networks ). Outside Europe, many teams have also this approach and sometimes we have direct collaborations with them: Vancouver (EA RESEAUXCOM), Montréal, Fortaleza (EA EWIN),...
Traffic Grooming
In a WDM network, routing a connection request consists in assigning it a route in the physical network and a wavelength. When each request uses at most 1/C of the bandwidth of the wavelength, we say that the grooming factor is C . That means that on a given edge of the network we can groom at most C requests on the same wavelength. With this constraint the objective can be either to minimize the number of wavelengths (related to the transmission cost) or minimize the number of Add/Drop Multiplexers (ADM) used in the network (related to the cost of the nodes).
We have first addressed the problem of traffic grooming in WDM rings or paths with All-to-All uniform unitary traffic. The goal is to minimize the total number of ADMs required. We have shown that this problem corresponds to a partition of the edges of the complete graph into subgraphs, where each subgraph has at most C edges (where C is the grooming factor) and where the total number of vertices has to be minimized. Using tools of graph and design theory, we optimally solved the problem for practical values and infinite congruence classes of values for a given C . We give optimal constructions on unidirectional rings when C
N(N-1)/6 and when C = 3, 4, 5, 6, 12 , on paths when C = 2 , and bidirectional rings when C = 1, 2, 3 and C = k(k + 1)/2 (k1 ) for infinite congruence classes [102] , and propose an approximate construction for all-to-all traffic on unidirectional rings for any value of C , and on bidirectional rings for C = 2, 3 [102] . We also showed how to improve lower bounds by using refined counting techniques, and how to determine the maximum number of connections which can be established in a path of size N or in a DAG.
Then, we have also studied the all-to-all traffic grooming on unidirectional rings with grooming factor C and with the extra constraints that the traffic between a subset of vertices must be served with grooming factor C' . We provided optimal constructions for C = 4 and C' = 1, 2, 3 [96] .
Furthermore, we refined the complexity analysis of the problem for general traffic requirement and established the first in-approximability result on traffic grooming using a study of the complexity (including parametrized complexity) [24] . We have provided an approximation algorithm for ring and path networks with approximation factor of O(n1/3log2n) , independent of the grooming factor. Moreover, we have proposed an a priori placement of ADMs in unidirectional WDM rings allowing to satisfy any set of requests with bounded degree d (a node is source or destination of at most d requests) [75] .
Moreover, we studied the traffic grooming on the path with online traffic and distributed routing algorithm. We have shown how to design the best possible virtual topologies (assignment of ADMs to wavelengths), independently of the routing algorithm and for any bounded degree traffic instances. We have in particular analyzed the performances of distributed greedy routing algorithms [101] .
We also studied the placement of optical add/drop multiplexers (OADM) that allows adding or dropping wavelengths from/to an optical fiber. For this problem, we provided a 4-approximation algorithm for general instances on unidirectional path, and improved approximation factors for particular instances [64] .
Finally, in [92] we survey the main results obtained on traffic grooming, including complexity and hardness results, optimal constructions, approximation algorithms, ILP formulations and heuristic algorithms.
Reconfiguration in WDM Networks
In production networks, traffic evolution, failures and maintenance operations force to adapt regularly the current configuration of the network (virtual topology, routing of connections). In this context, we have developed tools to switch connections one after the other from a pre-computed routing to another with limited service disruptions. We thus concentrated on the reoptimization phase of the network, or migration of the routing. We have modeled this problem as a scheduling problem in a dependency digraph that may contains cycles, the process number , and then established some similarities and differences with two other known problems: the pathwidth and a particular graph searching problem . Dependency cycles are broken through the use of temporary routes or temporary disruptions (called “agents” in the model) that have to be minimized. We have proved that the problem is NP-complete and difficult to approximate in general, characterized the classes of (di)graphs that can be processed with at most two agents, and proposed distributed algorithms for computing this parameter and other graph invariants in trees. We proposed a heuristic algorithm that performs better and faster than previous proposals, and investigated the problem with the extra constraints that some connections should never be interrupted (particular service level agreement) [58] , [88] . We also studied tradeoffs between the total number of interruptions and the maximum number of concurrent interruptions, proving in particular that the knowledge of one parameter does not help to optimize the other [104] .
We have extended our study to the case in which connections use only a fraction of the bandwidth of a link. In [59] , [108] we proved that deciding whether it exists a scheduling of the rerouting of connection requests without traffic interruption is NP-complete even if requests use the third of the bandwidth of a link.
All-Optical Label Switching, AOLS
All-Optical Label Switching (AOLS) is a promising technology that performs packet forwarding without any optical-electrical-optical conversions, thus speeding up the forwarding. However, the cost of this technology requires limiting the number of labels needed to ensure the forwarding when routing a set of requests using GMPLS technology. In particular, this prevents the usage of label swapping techniques.
We have studied the routing problem in this context using label stacking techniques. We have formalized the problem by associating to each routing strategy a logical hypergraph, called a hypergraph layout, whose hyperarcs are dipaths of the physical graph, called tunnels in GMPLS terminology. We defined a cost function for the hypergraph layout, depending on its total length plus its total hop count. Minimizing the cost of the design of an AOLS network can then be expressed as finding a minimum cost hypergraph layout. In [100] , [51] , we prove hardness results for the problem, namely for general directed networks we prove that it is NP-hard to find a Clogn -approximation, where C is a positive constant and n is the number of nodes of the network. For symmetric directed networks, we prove that the problem is APX-hard. These hardness results hold even if the traffic instance is a partial broadcast. On the other hand, we provide approximation algorithms, in particular an O(logn) -approximation for symmetric directed networks. We focussed on the case where the physical network is a directed path, providing a polynomial-time dynamic programming algorithm first for one source [50] , [98] , and then for a fixed number k of sources running in time O(nk + 2) [99] .
We have also considered label-stripping techniques that allow reducing further the overall number of labels but at the cost of increasing the stack size and so waste bandwidth. We proposed a heuristic algorithm performing tradeoffs between the stack size and the waste of bandwidth [85] .
Regenerator placement in WDM networks
The transmission of an optical signal in a fiber causes small phase shifting and power loss forcing to regenerate the signal after a certain distance (e.g. 1000km). We have investigated the problem of minimizing the number of locations to place the regenerators in a WDM network [63] in various settings: limited or unlimited number of regenerators per site, routing given or not, general and particular topologies. In each setting, we established the complexity and in-approximability of the problem, and provided approximation algorithms and exact polynomial time algorithms whenever possible.
