## Section: New Results

### Network Design and Management

Participants : Jean-Claude Bermond, David Coudert, Frédéric Giroire, Frédéric Havet, Alvinice Kodjo, Aurélien Lancin, Bi Li, Fatima Zahra Moataz, Christelle Molle-Caillouet, Joanna Moulierac, Nicolas Nisse, Stéphane Pérennes, Truong Khoa Phan.

More information on several results presented in this section may be found in the PhD thesis of A. Kodjo [13] , B. Li [15] and T. K. Phan [18] .

#### Optimization in backbone networks

##### Shared Risk Link Group

The notion of Shared Risk Link Groups (SRLG) captures survivability issues when a set of links of a network may fail simultaneously. The theory of survivable network design relies on basic combinatorial objects that are rather easy to compute in the classical graph models: shortest paths, minimum cuts, or pairs of disjoint paths. In the SRLG context, the optimization criterion for these objects is no longer the number of edges they use, but the number of SRLGs involved. Unfortunately, computing these combinatorial objects is NP-hard and hard to approximate with this objective in general. Nevertheless some objects can be computed in polynomial time when the SRLGs satisfy certain structural properties of locality which correspond to practical ones, namely the star property (all links affected by a given SRLG are incident to a unique node) and the span 1 property (the links affected by a given SRLG form a connected component of the network). The star property is defined in a multi-colored model where a link can be affected by several SRLGs while the span property is defined only in a mono-colored model where a link can be affected by at most one SRLG. In [52] , we extend these notions to characterize new cases in which these optimization problems can be solved in polynomial time or are fixed parameter tractable. We also investigate on the computational impact of the transformation from the multi-colored model to the mono-colored one. Experimental results are presented to validate the proposed algorithms and principles.

##### Dynamic Routing and Spectrum Assignment in Optical Networks

Elastic Optical Networks (EONs) promises a better utilization of the spectrum in optical networks. In fact, as the optical transmission spectrum is carved into fixed-length bands in the traditional WDM networks, small bit rates are over-provisioned and very high bit rates do not fit. EONs are moving away from this fixed-grid and allow the spectrum to be divided flexibly: each request is allocated exactly the resources it needs. In [34] , we present two exact algorithms to route and allocate spectrum to a new request in an EON using only Non-Disruptive Defragmentation (Push-Pull). In the first algorithm, we find the shortest routing path for the new request (i.e., the shortest path from source to destination where contiguous spectrum to satisfy the request can be freed) and then find the position that gives the overall minimum delay on that path. In the second algorithm, we find at the same time a routing path and a position in the spectrum, that minimize the delay of insertion (over all other paths and positions). Both algorithms are polynomial in the size of the network, its bandwidth and the number of provisioned requests.

#### Microwave Backhaul networks

##### Chance-Constrained Optimization of Reliable Backhaul networks

In [25] , we extend our former investigation on conceiving reliable fixed point-to-point wireless networks under outage probability constraints. We consider the problem of determining the minimum cost bandwidth assignment of a network, while guaranteeing a reliability level of the solution. If the optimal bandwidth assignment and routing of traffic demands are accomplished, the reliability criterion requires that network flows remain feasible with high probability, regarding that the performance of microwave links is prone to variations due to external factors, e.g., weather. We introduce a *chance-constrained programming* approach to tackle this problem and we present reformulations to standard integer linear programming models, including a budget-constrained formulation. To improve the solving performance, we propose new valid inequalities and a primal heuristic. Computational results present a performance analysis of the valid inequalities and the heuristic. Further, the outperformance of the novel model compared to more traditional approaches is documented.

##### Robust optimization in multi-operators microwave backhaul networks

In [41] , we consider the problem of sharing the infrastructure of a backhaul network for routing. We investigate on the revenue maximization problem for the physical network operator (PNO) when subject to stochastic traffic requirements of multiple virtual network operators (VNO) and prescribed service level agreements (SLA). We use robust optimization to study the tradeoff between revenue maximization and the allowed level of uncertainty in the traffic demands. This mixed integer linear programming model takes into account end-to-end traffic delays as example of quality-of-service requirement in a SLA. To show the effectiveness of our model, we present a study on the price of robustness, i.e. the additional price to pay in order to obtain a feasible solution for the robust scheme, on realistic scenarios.

#### Energy efficiency

##### Robust Optimization for Energy-aware Routing with Redundancy Elimination

Many studies in literature have shown that energy-aware routing (EAR) can significantly reduce energy consumption for backbone networks. Also, as an arising concern in networking research area, the protocol-independent traffic redundancy elimination (RE) technique helps to reduce (a.k.a compress) traffic load on backbone network. In [35] , [50] , we present an extended model of the classical multi-commodity flow problem with compressible flows. Our model is robust with fluctuation of traffic demand and compression rate. In details, we allow any set of a predefined size of traffic flows to deviate simultaneously from their nominal volumes or compression rates. As an applicable example, we use this model to combine redundancy elimination and energy-aware routing to increase energy efficiency for a backbone network. Using this extra knowledge on the dynamics of the traffic pattern, we are able to significantly increase energy efficiency for the network. We formally define the problem and model it as a Mixed Integer Linear Program (MILP). We then propose an efficient heuristic algorithm that is suitable for large networks. Simulation results with real traffic traces on Abilene, Geant and Germany50 networks show that our approach allows for 16−28% extra energy savings with respect to the classical EAR model.

##### Optimizing IGP Link Weights for Energy-efficiency in a Changing World

Recently, due to the increasing power consumption and worldwide gas emissions in ICT (Information and Communication Technology), energy efficient ways to design and operate backbone networks are becoming a new concern for network operators. Since these networks are usually overprovisioned and since traffic load has a small influence on power consumption of network equipments, the most common approach to save energy is to put unused line cards that drive links between neighbouring routers into sleep mode. To guarantee QoS, all traffic demands should be routed without violating capacity constraints and the network should keep its connectivity. From the perspective of traffic engineering, we argue that stability in routing configuration also plays an important role in QoS. In details, frequent changes in network configuration (link weights, slept and activated links) to adapt with traffic fluctuation in daily time cause network oscillations. We propose in [62] a novel optimization method to adjust the link weights of Open Shortest Path First (OSPF) protocol while limiting the changes in network configurations when multi-period traffic matrices are considered. We formally define the problem and model it as Mixed Integer Linear Program (MILP). We then propose an efficient heuristic algorithm that is suitable for large networks. Simulation results with real traffic traces on three different networks show that our approach achieves high energy saving while keeping the networks in stable state (less changes in network configuration).

##### Grid spanners with low forwarding index for energy efficient networks

A routing $R$ of a connected graph $G$ is a collection that contains simple paths connecting every ordered pair of vertices in $G$. The edge-forwarding index with respect to $R$ (or simply the forwarding index with respect to $R$) $\pi (G,R)$ of $G$ is the maximum number of paths in $R$ passing through any edge of $G$. The forwarding index $\pi \left(G\right)$ of $G$ is the minimum $\pi (G,R)$ over all routings $R$’s of $G$. This parameter has been studied for different graph classes. Motivated by energy efficiency, we look in [57] , for different numbers of edges, at the best spanning graphs of a square grid, namely those with a low forwarding index.

#### Software-Defined Networks

##### Rule Placement in Software-Defined Networks for Energy-aware Routing

Software-defined Networks (SDN), in particular OpenFlow, is a new networking paradigm enabling innovation through network programmability. Over past few years, many applications have been built using SDN such as server load balancing, virtual-machine migration, traffic engineering and access control. We focus on using SDN for energy-aware routing (EAR). SDN can collect traffic matrix and then computes routing solutions satisfying QoS while being minimal in energy consumption (with minimal number of active links). However, prior works on EAR have assumed that the table of OpenFlow switch can hold an infinite number of rules. In practice, this assumption does not hold since the flow table is implemented with Ternary Content Addressable Memory (TCAM) which is expensive and power-hungry. In [39] , [56] , we propose an optimization method to minimize energy consumption for a backbone network while respecting capacity constraints on links and rule space constraints on routers. In details, we present an exact formulation using Integer Linear Program (ILP) and introduce efficient greedy heuristic algorithm. Based on simulations, we show that using this smart rule space allocation, it is possible to save almost as much power consumption as the classical EAR approach.

##### Compressing Two-dimensional Routing Tables with Order

A communication in a network is a pair of nodes $(s,t)$. The node $s$ is called the source source and $t$ the destination. A communication set is a set of distinct communications, i.e. two communications might have the same source or the same destination, but they cannot have both same source and same destination. A routing of a communication $(s,t)$ is a path in the network from $s$ to $t$. A routing of a communication set is a union of routings of its communications. At each node, there is a set $X$ of communications whose routing path goes through this node. The node needs to be able to find for each communication $(s,t)$ in $X$, the port that the routing path of $(s,t)$ uses to leave it. An easy way of doing it is to store the list of all triples $(s,t,k)$, where $(s,t)\in X$ and $k$ is the port used by the $(s,t)$-path to leave the node. Such triples are called communication triples. However, such a list might be very large. Motivated by routing in telecommunication network using Software Defined Network Technologies, we consider in [55] the problem of compacting this list using aggregation rules. Indeed, SDN routers use specific memory which is expensive and of small capacity. Hence, in addition, we can use some additional triples, called *-triples. As an example, a $t$-destination triple $(\text{*},t,p)$, means that every communication with destination t leaves on port $p$. We carry out in this work a study of the problem complexity, providing results of NP-completeness, of Fixed-Parameter Tractability and approximation algorithms.

#### Data gathering in radio networks

In the gathering problem, a particular node in a graph, the base station, aims at receiving messages from some nodes in the graph. At each step, a node can send one message to one of its neighbors (such an action is called a call ). However, a node cannot send and receive a message during the same step. Moreover, the communication is subject to interference constraints ; more precisely we consider a binary interference model where two calls interfere in a step, if the sender of one call is at distance at most ${d}_{I}$ from the receiver of the other call. Given a graph with a base station and a set of nodes having some messages, the goal of the gathering problem is to compute a schedule of calls for the base station to receive all messages as fast as possible, i.e., minimizing the number of steps (called makespan). The gathering problem is equivalent to the personalized broadcasting problem where the base station has to send messages to some nodes in the graph, with same transmission constraints. In [23] , we focus on the gathering and personalized broadcasting problem in grids. Moreover, we consider the non-buffering model: when a node receives a message at some step, it must transmit it during the next step. In this setting, though the problem of determining the complexity of computing the optimal makespan in a grid is still open, we present linear (in the number of messages) algorithms that compute schedules for gathering with ${d}_{I}\in \{0,1,2\}$. In particular, we present an algorithm that achieves the optimal makespan up to an additive constant 2 when ${d}_{I}=0$. If no messages are “close” to the axes (the base station being the origin), our algorithms achieve the optimal makespan up to an additive constant 1 when ${d}_{I}=0$, 4 when ${d}_{I}=2$, and 3 when both ${d}_{I}=1$ and the base station is in a corner. Note that, the approximation algorithms that we present also provide approximation up to a ratio 2 for the gathering with buffering. All our results are proved in terms of personalized broadcasting.