## Section: New Results

### Network Design and Management

Participants : Jean-Claude Bermond, Christelle Caillouet, David Coudert, Frédéric Giroire, Frédéric Havet, Nicolas Huin, Alvinice Kodjo, Fatima Zahra Moataz, Joanna Moulierac, Nicolas Nisse, Stéphane Pérennes.

#### Wireless Networks

##### Dimensioning Microwave Wireless Networks

In [47] , we aim at dimensioning fixed broadband microwave wireless networks under unreliable channel conditions. As the transport capacity of microwave links is prone to variations due to, e.g., weather conditions, such a dimensioning requires special attention. It can be formulated as the determination of the minimum cost bandwidth assignment of the links in the network for which traffic requirements can be met with high probability, while taking into account that transport link capacities vary depending on channel conditions. The proposed optimization model represents a major step forward since we consider dynamic routing. Experimental results show that the resulting solutions can save up to 45% of the bandwidth cost compared to the case where a bandwidth over-provisioning policy is uniformly applied to all links in the network planning. Comparisons with previous work also show that we can solve much larger instances in significantly shorter computing times, with a comparable level of reliability.

##### Data Gathering and Personalized Broadcasting in Radio Grids with Interference

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, two calls interfere in a step, if one sender is at distance at most $dI$ from the other receiver. 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 [24] , 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 $dI\in \{0,1,2\}$. In particular, we present an algorithm that achieves the optimal makespan up to an additive constant 2 when $dI=0$. 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 $dI=0$, 4 when $dI=2$, and 3 when both $dI=1$ 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.

#### Elastic Optical Networks

##### On Spectrum Assignment in Elastic Optical Tree-Networks

To face the explosion of the Internet traffic, a new generation of optical networks is being developed; the Elastic optical Networks (EONs). The aim with EONs is to use the optical spectrum efficiently and flexibly. The benefit of the flexibility is, however, accompanied by more difficulty in the resource allocation problems. In [54] , [51] , [14] , we study the problem of Spectrum Allocation in Elastic Optical Tree-Networks. In trees, even though the routing is fixed, the spectrum allocation is NP-hard. We survey the complexity and approximability results that have been established for the SA in trees and prove new results for stars and binary trees.

#### Fault Tolerance

##### 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 [59] , we extend these notions to characterize new cases in which these optimization problems can be solved in polynomial time. We also investigate 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.

In [22] , we investigate the $k$-diverse routing problem which is to find a set of $k$ pairwise SRLG-disjoint paths between a given pair of end nodes of the network. This problem has been proven NP-complete in general and some polynomial instances have been characterized. We consider more specifically the case where the SRLGs are localized and satisfy the star property. We first provide counterexamples to the polynomial time algorithm proposed by X. Luo and B. Wang (DRCN'05) for computing a pair of SRLG-disjoint paths in networks with SRLGs satisfying the star property, and then prove that this problem is in fact NP-complete. We then characterize instances that can be solved in polynomial time or are fixed parameter tractable, in particular when the number of SRLGs is constant, the maximum degree of the vertices is at most 4, and when the network is a directed acyclic graph. Finally we consider the problem of finding the maximum number of SRLG-disjoint paths in networks with SRLGs satisfying the star property. We prove that this problem is NP-hard to approximate within $O\left(\right|V{|}^{1-\epsilon})$ for any $0<\epsilon <1$, where $V$ is the set of nodes in the network. Then, we provide exact and approximation algorithms for relevant subcases.

##### Design of Fault-tolerant On-board Networks with Variable Switch Sizes

In [29] , we focus on designing networks that are capable, in the presence of faulty output ports, of rerouting input signals to operational output ports. Since the components of a satellite cannot be repaired, redundant amplifiers are added, and the interconnection network satisfies the following fault tolerance property: the network connects the set of input ports with the set of output ports, and for any set of at most $k$ output port failures, there exists a set of edge-disjoint paths connecting the input ports to the operational output ports. Since each switching device is expensive, these interconnection networks are constructed using the fewest possible switches, or at least a number of switches close to the minimum value. The networks are controlled centrally from Earth. Each time an amplifier in use develops a fault, the controller sends messages to the switches to change their settings, so as to ensure that the inputs remain connected to functioning amplifiers.

Current switches have four ports. Obviously, the larger the number of ports, the more expensive will be the switches, but then fewer will be required. So the cost of such a network involves a trade-off between the total number of switches and their unit cost. In order to determine the minimum-cost network, we give some bounds on the minimum number $\mathcal{N}(n,k,r)$ of $2r$-port switches in interconnection networks with $n$ inputs and $n+k$ outputs.

We first show $\mathcal{N}(n,k,r)\le \u2308\frac{k+2}{2r-2}\u2309\u2308\frac{n}{2}\u2309$. When $r\ge k/2$, we prove a better upper bound: $\mathcal{N}(n,k,r)\le \frac{r-2+k/2}{{r}^{2}-2r+k/2}n+O\left(1\right)$. Next, we establish some lower bounds. We show that if $k\ge r$, then $\mathcal{N}(n,k,r)\ge \frac{3n+k}{2r}$. We improve this bound when $k\ge 2r$: $\mathcal{N}(n,k,r)\ge \frac{3n+2k/3-r/2}{2r-2+\frac{3r}{\lfloor \frac{k}{r}\rfloor}}$. Finally, we determine $\mathcal{N}(n,k,r)$ up to additive constants for $k\le 6$.

#### Reducing Networks' Energy Consumption

Due to the increasing impact of ICT (Information and Communication Technology) on power consumption and worldwide gas emissions, energy efficient ways to design and operate backbone networks are becoming a new concern for network operators. Recently, energy-aware routing (EAR) has gained an increasing popularity in the networking research community. The idea is that traffic demands are redirected over a subset of the network links, allowing other links to sleep to save energy. We studied variant of this problems.

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

In [31] , we propose GreenRE – a new EAR model with the support of data redundancy elimination (RE). This technique, enabled within routers, can virtually increase the capacity of network links. Based on real experiments on a Orange Labs platform, we show that performing RE increases the energy consumption for routers. Therefore, it is important to determine which routers should enable RE and which links to put into sleep mode so that the power consumption of the network is minimized. We model the problem as a Mixed Integer Linear Program and propose greedy heuristic algorithms for large networks. Simulations on several network topologies show that the GreenRE model can gain further 37% of energy savings compared to the classical EAR model. In [27] , we introduce an extended model of the classical multi-commodity flow problem with compressible flows which is also robust with fluctuation of traffic demand and compression rate. An heuristic built on this model allows for 16-28% extra energy saving.

##### Optimizing IGP Link Weights for Energy-efficiency in Multi-period Traffic Matrices

To guarantee QoS while implementing EAR, 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. In [35] , we propose 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.

##### Energy Efficient Content Distribution

Recently, there is a trend to introduce content caches as an inherent capacity of network equipment, with the objective of improving the efficiency of content distribution and reducing the network congestion. In [18] , we study the impact of using in-network caches and content delivery network (CDN) cooperation on an EAR. We formulate this problem as Energy Efficient Content Distribution, we propose an integer linear program (ILP) and a heuristic algorithm to solve it. The objective of this problem is to find a feasible routing, so that the total energy consumption of the network is minimized while the constraints given by the demands and the link capacity are satisfied. We exhibit for which the range of parameters (size of caches, popularity of content, demand intensity, etc.) it is useful to use caches. Experimental results show that by placing a cache on each backbone router to store the most popular content, along with well choosing the best content provider server for each demand to a CDN, we can save about 20% of power in average of all the backbone networks considered.

#### Routing Theory and Forwarding Index

Motivated by finding the best set of links that should be on for energy efficiency, we study the problem of determining the minimum forwarding index of a graph. The (edge) forwarding index of a graph is the minimum, over all possible routings of all the demands, of the maximum load of an edge. This metric is of a great interest since it captures the notion of global congestion in a precise way: the lesser the forwarding-index, the lesser the congestion. This parameter has been studied for different graph classes in the literature. In [42] , we determine, for different numbers of edges, the best spanning graphs of a square grid, namely those with a low forwarding index. In [61] , [43] , we study the following design question: Given a number $e$ of edges and a number $n$ of vertices, what is the least congested graph that we can construct? and what forwarding-index can we achieve? We answer here these questions for different families of graphs: general graphs, graphs with bounded degree, sparse graphs with a small number of edges by providing constructions, most of them asymptotically optimal. Doing so, we partially answer the practical problem that initially motivated our work: If an operator wants to power only $e$ links of its network, in order to reduce the energy consumption (or wiring cost) of its networks, what should be those links and what performance can be expected?

*On the complexity of equal shortest path routing.*

Additionally, we studied the complexity of configuring the OSPF-ECMP (for Open Shortest Path First-Equal Cost Multiple Path) protocol. In [32] , we show that the problem of maximizing even a single commodity flow for the OSPF-ECMP protocol cannot be approximated within any constant factor ratio. Besides this main theorem, we derive some positive results which include polynomial-time approximations and an exponential-time exact algorithm.

#### Routing in Software Defined Networks (SDN)

Software Defined Networking (SDN) is gaining momentum with the support of major manufacturers. While it brings flexibility in the management of flows within the data center fabric, this flexibility comes at the cost of smaller routing table capacities. In [50] , we investigate compression techniques to reduce the forwarding information base (FIB) of SDN switches. We validate our algorithm, called MINNIE, on a real testbed able to emulate a 20 switches fat tree architecture. We demonstrate that even with a small number of clients, the limit in terms of number of rules is reached if no compression is performed, increasing the delay of all new incoming flows. MINNIE, on the other hand, reduces drastically the number of rules that need to be stored with a limited impact on the packet loss rate. We also evaluate the actual switching and reconfiguration times and the delay introduced by the communications with the controller. In parallel, we considered the algorithmic problem of compressing bidimensional routings table with priorities on the rules. We carry out in [40] a study of the problem complexity, providing results of NP-completeness, of Fixed-Parameter Tractability and approximation algorithms. In [44] , we then propose green routing schemes performing simultaneously the selection of the routes, the compression of the routing tables, and decide to put in sleep mode unused links. These algorithms are tested on networks from the SNDLib library.

#### Video Streaming

##### Study of Repair Protocols for Live Video Streaming Distributed Systems

In [41] , we study distributed systems for live video streaming. These systems can be of two types: structured and un-structured. In an unstructured system, the diffusion is done opportunistically. The advantage is that it handles churn, that is the arrival and departure of users, which is very high in live streaming systems, in a smooth way. On the opposite, in a structured system, the diffusion of the video is done using explicit diffusion trees. The advantage is that the diffusion is very efficient, but the structure is broken by the churn. In this paper, we propose simple distributed repair protocols to maintain, under churn, the diffusion tree of a structured streaming system. We study these protocols using formal analysis and simulation. In particular, we provide an estimation of the system metrics, bandwidth usage, delay, or number of interruptions of the streaming. Our work shows that structured streaming systems can be efficient and resistant to churn.