Coati is a joint team between INRIA Sophia Antipolis - Méditerranée and the I3S laboratory (Informatique Signaux et Systèmes de Sophia Antipolis) which
itself belongs to CNRS (Centre National de la Recherche Scientifique) and Univ. Côte d'Azur.
Its research fields are Algorithmics, Discrete Mathematics, and Combinatorial Optimization, with applications mainly in telecommunication networks.

The main objectives of the Coati project-team are to design networks and communication algorithms. In order to meet these objectives, the team studies various theoretical problems in Discrete Mathematics, Graph Theory, Algorithmics, and Operations Research and develops applied techniques and tools, especially for Combinatorial Optimization and Computer Simulation. In particular, Coati used in the last years both these theoretical and applied tools for the design of various networks, such as SDN (software defined networks), WDM, wireless (radio), satellite, and peer-to-peer networks. This research has been done within various industrial and international collaborations.

Coati also investigates other application areas such as bio-informatics and transportation networks.

The research done in Coati results in the production of prototypes and more advanced software, and in the contribution to large open source software such as Sagemath.

Since its creation in 2013, the objectives of Coati are to conduct fundamental research in Discrete Mathematics, Graph Theory, Digraph Theory, Algorithms and Operations Research, and to use these tools for studying specific network optimization problems. Notice that we are mostly interested in telecommunications networks. However, our expertise can be applied to solve many other problems in various areas (transport, biology, resource allocation, social sciences, smart-grids, speleology, etc.) and we collaborate with teams of these other domains. Coati also contributes to the development of software components in order to validate proposed algorithms and to boost their dissemination.

The research program of Coati is therefore structured as follows.

We also investigate optimization problems in other application fields (see Section 7.4) such as structural biology, transportation networks, economy, sociology, etc. For instance, we collaborate in Structural Biology with the Inria project-team ABS (Algorithms Biology Structure) from Sophia Antipolis. In the area of intelligent transport systems, we collaborate with the SMEs BeNomad and Instant-System on routing problems in multi-modal transportation systems. We also collaborate with GREDEG (research center in economics, law, and management) and the SKEMA business school on the analysis of the impact of competitive funding on the evolution of scientific networks.

On the one side, these collaborations benefit to the considered domains via the dissemination of our tools. On the other side, they give rise to new problems of interest for our community, and help us to improve our knowledge and to test our algorithms on specific instances.

Note also that beside our research activity, we are deeply involved in the dissemination of our domain towards a general public.

Coati is mostly interested in telecommunications networks but also in the network structure appearing in social, molecular and transportation networks.

We focus on the design and management of heterogeneous physical and logical networks. The project has kept working on the design of backbone networks (optical networks, radio networks, IP networks). However, the fields of Software Defined Networks and Network Function Virtualization are growing in importance in our studies. In all these networks, we study routing algorithms and the evolution of the routing in case of any kind of topological modifications (maintenance operations, failures, capacity variations, etc.).

Our combinatorial tools may be well applied to solve many other problems in various areas (transport, biology, resource allocation, chemistry, smart-grids, speleology, etc.) and we collaborate with experts of some of these domains.

For instance, we collaborate with project-team ABS (Algorithms Biology Structure) from Sophia Antipolis on problems from Structural Biology (co-supervision of a PhD student). In the area of transportation networks, we collaborate with SMEs Benomad and Instant-System on dynamic car-pooling combined with multi-modal transportation systems in the context of ANR project Multimod started in January 2018. We collaborate with SME MillionRoads since October 2019 on the modeling and exploration of the HumanRoads database that gathers more than 100 million curriculums (studies and career paths of persons). Last, we have started a collaboration with GREDEG (Groupe de Recherche en Droit, Economie et Gestion, Université Côte d'Azur) and the SKEMA business school on the analysis of the impact of competitive funding on the evolution of scientific collaboration networks.

Let us desribe new/updated software.

Modeling of a database linking users to their studies and careers in the form of a graph. Algorithms for graphs associated with the queries made (of the type: number of users who have completed a given curriculum, distribution of careers following a given curriculum, distribution of curriculums preceding a given career, etc.). Scaling for a database of >100 million users.

In addition, Neo4j implemetations of various algorithms tested on the HumanRoads data.

SageMath is a free mathematics software system written in Python and combining a large number of mathematical libraries under a common interface.

INRIA teams contribute in different ways to the software collection. COATI adds new graph algorithms along with their documentations and the improvement of underlying data structures. LFANT contributes through libraries such as ARB and PARI/GP, and directly through SageMath code for algebras and ring and field extensions.

Idawi is a middleware for the development and experimentation of distributed algorithms. It boasts a very general and flexible multi-hop component-oriented model that makes it applicable in many contexts such as parallel and distributed computing, cloud, Internet of Things (IOT), P2P networks. Idawi components can be deployed anywhere a SSH connection is possible. They exhibit services which communicate with each other via explicit messaging. Messages can be sent synchronously or asynchronously, and can be handled in either a procedural (with the optional use of futures) or reactive (event-driven) fashion. In the latter case, multi-threading is used to maximize both the speed and the number of components in the system. Idawi message transport is done via TCP, UDP, SSH or shared-memory.

Idawi is a synthesis of past developments of the COATI Research group in the field of graph algorithms for big graphs, and it is designed to be useful to the current and future Research project of COATI and KAIROS team-projects.

Network design is a very wide subject which concerns all kinds of networks. In telecommunications, networks can be either physical (backbone, access, wireless, ...) or virtual (logical). 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 several paths between their end-nodes. The set of paths is chosen according to the technology, the protocol or the QoS constraints.

The last years have been very lively for networks with the rises of several new paradigms like Software Defined Networks (SDN) and Network Function Virtualization (NVF), of new technologies like 5G, Elastic Optical Networks or LoRa, and of new usages like Internet of Things, 5G, High quality video streaming. All these changes have brought or renewed a large number of algorithmic and optimization problems for the design and management of networks. In this context, our work has mainly focused on the study of three types of problems:

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 operations research and discrete mathematics (some of them developed in our teams, see Sections 7.2 and 7.3). This approach is shared by a number of other teams worldwide, e.g. UFC and UNIFOR (Fortaleza, Brazil), Concordia Univ. (Montréal, Canada), Univ. Adolfo Ibañez (Santiago, Chile), Univ. Oran (Algeria), with which we have a direct collaboration.

Recent advances in networks such as Software Defined Networking (SDN) and Network Function Virtualization (NFV) are changing the way network operators deploy and manage Internet services. On the one hand, SDN introduces a logically centralized controller with a global view of the network state. On the other hand, NFV enables the complete decoupling of network functions from proprietary appliances and runs them as software applications on general purpose servers. In such a way, network operators can dynamically deploy Virtual Network Functions (VNFs). SDN and NFV, both separately, bring to network operators new opportunities for reducing costs, enhancing network flexibility and scalability, and shortening the time-to-market of new applications and services. Moreover, the centralized routing model of SDN jointly with the possibility of instantiating VNFs on demand may open the way for an even more efficient operation and resource management of networks. For instance, an SDN/NFV-enabled network may simplify the Service Function Chain (SFC) deployment and provisioning by making the process easier and cheaper. We addressed several questions in this context.

In collaboration with Brigitte Jaumard (Concordia University, Montréal, Canada), we consider in 37, 36 the problem of network slice reconfiguration without interruption. A network slice can be seen as a virtual network embedded on the physical topology, with some VNFs placed in specific nodes. As an example, a simplified network slice could be an SFC (as in 37). Reconfiguring from time to time network slices allows to reduce the network operational costs and to increase the number of slices that can be managed within the network. However, it impacts users' Quality of Service during the reconfiguration step. To solve this issue, we study solutions implementing a make-before-break scheme. We propose new models and scalable algorithms (relying on column generation techniques in 36) that solve large data instances in few seconds.

In collaboration with Chidung Lac (Orange labs), we investigate in 53 on the protection against failures of VNF deployments. We use a representation of the VNFs as a set of VNF components (VNFCs). These VNFCs are typically designed with a redundancy scheme and need to be deployed against failures of, e.g., compute servers, and must be such deployed with respect a particular resiliency mechanism for protection purposes. Therefore, choosing an efficient mapping of VNFCs to the compute servers is a challenging problem in the optimization of the software-defined, virtualization-based next generation of networks. In 53, we model the problem of reliable VNFCs placement under anti-affinity constraints using several optimization techniques. A novel approach based on an extension of bin packing is proposed. We perform a comprehensive evaluation in terms of performance under real-world ISP networks along with synthetic traces. We show that our methods can calculate rapidly efficient solutions for large instances.

In collaboration with Chidung Lac (Orange labs), Damien Saucez and Thierry Turletti (DIANA), and Issam Tahiri, Ruslan Sadykov and François Vanderbeck (RealOpt), we consider in 41 a path-based protection scheme with the global rerouting strategy in which, for each failure situation, there may be a new routing of all the demands. Our optimization task is to minimize the needed amount of bandwidth. After discussing the hardness of the problem, we develop two scalable mathematical models that we handle using both Column Generation and Benders Decomposition techniques. Through extensive simulations on real-world IP network topologies and on randomly generated instances, we show the effectiveness of our methods: they lead to savings of 40 to 48% of the bandwidth to be installed in a network to protect against failures compared to traditional schemes. Finally, our implementation in OpenDaylight demonstrates the feasibility of the approach. Its evaluation with Mininet shows that our solution provides sub-second recovery times, but the way it is implemented may greatly impact the amount of signaling traffic exchanged. In our evaluations, the recovery phase requires only a few tens of milliseconds for the fastest implementation, compared to a few hundreds of milliseconds for the slowest one.

Mininet is the most popular tool when it comes to evaluate SDN propositions. Mininet allows to emulate SDN networks on a single computer but shows its limitations with resource intensive experiments as the emulating host may become overloaded. To tackle this issue, we propose Distrinet 31, a distributed implementation of Mininet over multiple hosts, based on LXD/LXC, Ansible, and VXLAN tunnels. Distrinet uses the same API than Mininet, meaning that it is compatible with Mininet programs. It is generic and can deploy experiments on Linux clusters (e.g., Grid'5000), as well as on the Amazon EC2 cloud platform.

In 49 we propose and implement a new placement module for distributed emulation of SDN/NFV emulation. To handle the ever growing demand of resource intensive experiments distributed, network emulation tools such as Mininet and Maxinet have been proposed. They automatically allocate experimental resources. However, we have shown that resources are poorly allocated, leading to resource overloading and hence to dubious experimental results. This is why we propose and implement a new placement module for distributed emulation. Our algorithms take into account both link and node resources and minimize the number of physical hosts needed to carry out the emulation. Through extensive numerical evaluations, simulations, and actual experiments , we show that our placement methods outperform existing ones and allowing to re-establish trust in experimental results.

This work has been done in collaboration with Chidung Lac (Orange labs) and Damien Saucez and Thierry Turletti (DIANA).

Optical multilayer optimization periodically reorganizes layer 0-1-2 network elements to handle both existing and dynamic traffic requirements in the most efficient manner. This delays the need for adding new resources in order to cope with the evolution of the traffic, thus saving capex (capital expenditure).

In collaboration with Brigitte Jaumard and Huy Duong (Concordia Université, Montréal, Canada) and Armolavicius, Romualdas (Ciena), we focussed in 34 on Layer 2, i.e., on capacity reoptimization at the optical transport network (OTN) layer when routes (e.g., LSPs in MPLS networks) are making unnecessarily long detours to evade congestion. Reconfiguration into optimized routes can be achieved by re-defining the routes, one at a time, so that they use the vacant resources generated by the disappearance of services using part of a path that transits the congested section. To maintain the Quality of Service, it is desirable to operate under a Make-Before-Break (MBB) paradigm, with the minimum number of reroutings. The challenge is to determine the best rerouting order while minimizing the bandwidth requirement. We propose an exact and scalable optimization model for computing a minimum bandwidth rerouting scheme subject to MBB in the OTN layer of an optical network. Numerical results show that we can successfully apply it on networks with up to 30 nodes, a very significant improvement with respect to the state of the art. We also provide some reoptimization analysis in terms of the bandwidth requirement vs. the number of reroutings.

In collaboration with Brigitte Jaumard and Huy Duong (Concordia Université, Montréal, Canada), we revisited in 52 MBB rerouting with the objective of identifying the reroute sequence planning that minimizes the number of reroutes in order to minimize the resource usage. We propose a Dantzig-Wolfe decomposition mathematical model to solve this complex rerouting problem. We investigate how multiple or parallel rerouting reduces the overall minimum number of rerouting events (shortest makespan), and achieve the best resource usage. Numerical results bring interesting insights on that question and show a computational time reduction by about one order of magnitude over the state of the art.

Filterless optical networks use passive splitters and combiners with coherent optics, providing wavelength selection in the digital domain, while forming a passive fiber-tree topology between nodes. In collaboration with Brigitte Jaumard and Yan Wang (Concordia Université, Montréal, Canada), we investigate in 38 the optimal design of filterless optical networks while minimizing the number of required wavelengths. We propose a Dantzig-Wolfe decomposition model in which each subproblem aims to generate a potential filterless optical subnetwork, with a directed tree topology. The master problem then selects the best combination of subnetworks. Numerical experiments demonstrate significant performance improvement over previous work, reducing previous computational results by a factor of 2 to 10 depending on the size of the data instances.

LoRa is a low-power and long range radio communication technology designed for low-power Internet of Things devices. These devices are often deployed in remote areas where the end-to-end connectivity provided through one or more gateways may be limited. In collaboration with Dimitrios Zorbas, Khaled Abdelfadeel Hassan and Dirk Pesch (Tyndall National Institute, Irland), we examine in 43 the case where the gateway is not available at all times. As a consequence, the sensing data need to be buffered locally and transmitted as soon as a gateway becomes available. However, due to the Aloha-style transmission policy of current LoRa-based standards, such as the LoRaWAN, delivering a large number of packets in a short period of time by a large number of nodes becomes impossible. To avoid bursts of collisions and expedite data collection, we propose a time-slotted transmission scheduling mechanism.We formulate the data scheduling optimisation problem, taking into account LoRa characteristics, and compare its performance to low complexity heuristics. Moreover, we conduct a set of simulations to show the benefits of synchronous communications on the data collection time and the network performance.

In collaboration with Martin Heusse and Franck Rousseau (Drakkar, LIG, Grenoble), we propose in 57 an optimization model for single-cell LoRaWAN planning which computes the limit range of each spreading factor (SF) in order to maximize the minimum packet delivery ratio (PDR) of every node in the network. It allows to balance the opposite effects of attenuation and collision of the transmissions and guarantee fairness among the nodes.

The use of autonomous unmanned aerial vehicles (UAVs) or drones has emerged to efficiently collect data from mobile sensors when there is no infrastructure available. The drones can form a flying ad-hoc network through which the sensors can send their data to a base station at any time.

In 50, we address the data collection problem using the minimum number of unmanned aerial vehicles (UAVs) in a disaster management scenario where mobile sensors are investigating the devastated area. Critical information needs to be quickly gathered for processing by the rescue team, so the use of UAVs in this situation is of great interest. We propose an optimal model for computing the trajectories of the UAVs while guaranteeing the total coverage of the ground mobile sensors and connectivity among the UAVs with a central base station dedicated to data processing. Our model is based on a decomposition model and is solved effectively using column generation. We show that we can provide a plan for deploying the UAVs minimizing the total traveled distance.

Live streaming traffic represents an increasing part of the global IP traffic. Hybrid CDN-P2P architectures have been proposed as a way to build scalable systems with a good Quality of Experience (QoE) for users, in particular, using the WebRTC technology which enables real-time communication between browsers and, thus, facilitates the deployment of such systems. An important challenge to ensure the efficiency of P2P systems is the optimization of peer selection. Most existing systems address this problem using simple heuristics, e.g. favor peers in the same ISP or geographical region. In collaboration with Soufiane Roubia (EasyBroadcast) and Guillaume Urvoy-Keller (I3S), we analyzed in 54 9 months of operation logs of a hybrid CDN-P2P system and demonstrate the sub-optimality of those classical strategies. Over those 9 months, over 18 million peers downloaded over 2 billion video chunks. We propose learning-based methods that enable the tracker to perform adaptive peer selection. Furthermore, we demonstrate that our best models, which turn out to be the neural network models can (i) improve the throughput by 22.7%, 14.5%, and 6.8% (reaching 89%, 20.4%, and 24.3% for low bandwidth peers) over random peer, same ISP, and geographical selection methods, respectively (ii) reduce by 18.6%, 18.3%, and 16% the P2P messaging delay and (iii) decrease by 29.9%, 29.5%, and 21.2% the chunk loss rate (video chunks not received before the timeout that triggers CDN downloads), respectively.

Many companies and organizations are moving their applications from on-premises data centers to the cloud. The cloud infrastructures can potentially provide an infinite amount of computation (e.g., Elastic Compute) and storage (e.g., Simple Service Storage). In addition, all cloud providers propose different offers: IaaS, PaaS, and SaaS. The demo 96, done in collaboration with Chidung Lac (Orange labs) and Thierry Turletti (DIANA), focuses on the IaaS services, presenting a simple tool to measure the network delay in a virtual infrastructure built entirely in the cloud. These measurements are useful for organizations that are moving current applications to, or creating new applications in, the cloud, but have requirements on the maximum, or average, network delay that these applications can tolerate. We present CloudTrace, a simple CLI tool that creates regional and multiregional experiments to measure delay, using Amazon AWS.

With the continuous growing level of dynamicity, heterogeneity, and complexity of traffic data, anomaly detection remains one of the most critical tasks to ensure an efficient and flexible management of a network. Recently, driven by their empirical success in many domains, especially bioinformatics and computer vision, graph kernels have attracted increasing attention. Our work aims at investigating their discrimination power for detecting vulnerabilities and distilling traffic in the field of networking. In 59, we propose Nadege, a new graph-based learning framework which aims at preventing anomalies from disrupting the network while providing assistance for traffic monitoring.
Specifically, we design a graph kernel tailored for network profiling by leveraging propagation schemes which regularly adapt to contextual patterns.
Moreover, we provide provably efficient algorithms and consider both offline and online detection policies.
Finally, we demonstrate the potential of kernel-based models by conducting extensive experiments on a wide variety of network environments. Under different usage scenarios, Nadege significantly outperforms all baseline approaches.

In the last years, Coati has conducted an intense research effort on the algorithmic aspects of graph theory. We are mainly interested in designing efficient algorithms for large graphs and in understanding how structural properties of networks can help for this purpose. In general we try to find the most efficient algorithms, either exact algorithms or approximation ones, to solve various problems of graph theory, often with applications in telecommunication networks. We are involved in many international and national collaborations with academic and industrial partners.

We mainly focus on four topics: efficient computation of graph parameters, graph decompositions, combinatorial games in graphs and distributed computing.

The length of a tree-decomposition of a graph is the maximum distance between two vertices of a same bag of the decomposition. The treelength of a graph is the minimum length among its tree-decompositions. Treelength of graphs has been studied for its algorithmic applications in classical metric problems such as Traveling Salesman Problem or metric dimension of graphs and also, in compact routing in the context of distributed computing. Deciding whether the treelength of a general graph is at most 2 is NP-complete (graphs of treelength one are precisely the chordal graphs), and it is known that the treelength of a graph cannot be approximated up to a factor less than

In collaboration with Guillaume Ducoffe (Université Politehnica, Bucarest, Roumanie) and Simon Nivelle (INSPÉ Paris), we first fully describe in 51, 58 the treelength of melon graphs (set of pairwise internally disjoint paths linking two vertices), showing that, even in such a restricted graph class, the expression of the treelength is not trivial. Then, we show that treelength can be approximated up to a factor

The similarity between two paths can be measured according to the proportion of arcs they share.
We study the complexity of several variants of the problem of computing “dissimilar" paths (whose measure of similarity does not exceed a certain threshold) between two given vertices of a weighted directed graph. For four of the most studied measures in the literature, we give in 71, 55 a unified and simple proof of the fact that finding

If a graph locally irregular.
In collaboration with Nikolaos Melissinos (LAMSADE, Université Paris-Dauphine) and Theofilos Triomatis (School of Electrical Engineering, University of Liverpool),
we introduce and study in 84 the problem of identifying a largest induced subgraph of a given graph

Then, looking for more positive results, we turn towards computing the parameter

In the eternal domination game played on graphs, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices on their turn. The goal is to determine the eternal domination number

We have continued the study of the eternal domination game on strong grids. Cartesian grids have been vastly studied with tight bounds for small grids such as

In 39, we prove that, for all

The protection number of a tree is the length of the shortest path from the root to a leaf. It is interchangeably called the protection number of the root. We define the protection number of a vertex $v$ in tree

In collaboration with Fionn Mc Inerney (Technical University of Denmark), we have introduced in 75, 56, 44 the largest connected subgraph game played on an undirected graph

We have first proved that Bob can never win, and define a large class of graphs (called reflection graphs) in which the game is a draw. We have then showed that determining the outcome of the game is PSPACE-complete, even in bipartite graphs of small diameter, and that recognising reflection graphs is GI-hard. We have also proved that the game is a draw in paths if and only if the path is of even order or has at least 11 vertices, and that Alice wins in cycles if and only if the cycle is of odd length. Lastly, we have given an algorithm to determine the outcome of the game in cographs in linear time.

Inspired by the board game Kahuna, we have introduced and studied in 79, in collaboration with Fionn Mc Inerney (Technical University of Denmark), a new 2-player scoring game played on graphs called the vertex-capturing game. The game is played on a graph by two players, Alice and Bob, who take turns colouring an uncoloured edge of the graph. Alice plays first and colours edges red, while Bob colours edges blue. The game ends once all the edges have been coloured. A player captures a vertex if more than half of its incident edges are coloured by that player, and the player that captures the most vertices wins.

Using classical arguments from the field, we have first proved general properties of this game. Namely, we have proved that there is no graph in which Bob can win (if Alice plays optimally), while Alice can never capture more than 2 more vertices than Bob (if Bob plays optimally). Through dedicated arguments, we have then investigated more specific properties of the game, and have focused on its outcome when played in particular graph classes. Specifically, we have determined the outcome of the game in paths, cycles, complete bipartite graphs, and Cartesian grids, and have given partial results for trees and complete graphs.

Let infected. In collaboration with Fabricio Benevides (Federal University of Ceará, Brasil), we consider in 72 the process in which, at every step, each non-infected vertex with at least

Rikudo is a number-placement puzzle, where the player is asked to complete a Hamiltonian path on a hexagonal grid, given some clues (numbers already placed and edges of the path). In collaboration with Kévin Perrot (Aix Marseille Université), we prove in 88 that the game is complete for NP, even if the puzzle has no hole. When all odd numbers are placed it is in P, whereas it is still NP-hard when all numbers of the form

Since games and puzzles have been studied under a computational lens, researchers unearthed a rich landscape of complexity results showing deep connections between games and fundamental problems and models in computer science. Complexity of Games (CoG, steven3k.gitlab.io/isnphard-test/) is a compendium of complexity results on games and puzzles. It aims to serve as a reference guide for enthusiasts and researchers on the topic and is a collaborative and open source project that welcomes contributions from the community. Emanuele Natale is both a contributor and an administrator of CoG.

Algorithm Engineering is concerned with the design, analysis, implementation, tuning, and experimental evaluation of computer programs for solving algorithmic problems. It provides methodologies and tools for developing and engineering efficient algorithmic codes and aims at integrating and reinforcing traditional theoretical approaches for the design and analysis of algorithms and data structures. This approach is particularly suited when formal analysis pessimistically suggests bounds which are unlikely to appear on inputs of practical interest.

The

Then, in collaboration with Arthur Finkelstein (I3S and Instant-System), we have extended in 80 the Yen's and PNC algorithms to find the

Hyperbolicity is a graph parameter which indicates how much the shortest-path distance metric of a graph deviates from a tree metric. It is used in various fields such as networking, security, and bioinformatics for the classification of complex networks, the design of routing schemes, and the analysis of graph algorithms. Despite recent progress, computing the hyperbolicity of a graph remains challenging. Indeed, the best known algorithm has time complexity

In collaboration with André Nusser (MPII, Saarbrücken, Germany) and Laurent Viennot (GANG, Inria Paris), we designed a tool for enumerating all far-apart pairs of a graph by decreasing distances 81, a key component that was previously used to drastically reduce the computation time for hyperbolicity in practice. However, it required the computation of the distance matrix to sort all pairs of nodes by decreasing distance. We proposed a new data structure that avoids this memory bottleneck in practice and for the first time enables computing the hyperbolicity of several graphs with more than 100 000 nodes that were far out-of-reach using previous algorithms. As iterating over far-apart pairs in decreasing order without storing them explicitly is a very general tool, we believe that our approach might also be relevant to other problems.

We then proposed in 47 a new approach that uses a hierarchy of distance-

The C++ code of all our algorithms is available at 98.

In 42, we study the clustering of directed social graphs. The clustering coefficient has been introduced to capture the social phenomena that a friend of a friend tends to be my friend. This metric has been widely studied and has been shown to be of great interest to describe the characteristics of a social graph. But, the clustering coefficient is originally defined for a graph in which the links are undirected, such as friendship links (Facebook) or professional links (LinkedIn). For a graph in which links are directed from a source of information to a consumer of information, it is no more adequate. We show that former studies have missed much of the information contained in the directed part of such graphs. We introduce a new metric to measure the clustering of directed social graphs with interest links, namely the interest clustering coefficient. We compute it (exactly and using sampling methods) on a very large social graph, a Twitter snapshot with 505 million users and 23 billion links, as well as other various datasets. We additionally provide the values of the formerly introduced directed and undirected metrics, a first on such a large snapshot. We observe a higher value of the interest clustering coefficient than classic directed clustering coefficients, showing the importance of this metric. By studying the bidirectional edges of the Twitter graph, we also show that the interest clustering coefficient is more adequate to capture the interest part of the graph while classic ones are more adequate to capture the social part. We also introduce a new model able to build random networks with a high value of interest clustering coefficient. We finally discuss the interest of this new metric for link recommendation.

The degree distributions of complex networks are usually considered to be power law. However, it is not the case for a large number of them. We thus propose in 60 a new model able to build random growing networks with (almost) any wanted degree distribution. The degree distribution can either be theoretical or extracted from a real-world network. The main idea is to invert the recurrence equation commonly used to compute the degree distribution in order to find a convenient attachment function for node connections-commonly chosen as linear. We compute this attachment function for some classical distributions, as the power-law, broken power-law, geometric and Poisson distributions. We also use the model on an undirected version of the Twitter network, for which the degree distribution has an unusual shape.

Numerous works have been proposed to generate random graphs preserving the same properties as real-life large scale networks. However, many real networks are better represented by hypergraphs. Few models for generating random hypergraphs exist and no general model allows to both preserve a power-law degree distribution and a high modularity indicating the presence of communities. In 85, we present a dynamic preferential attachment hypergraph model which features partition into communities. We prove that its degree distribution follows a power-law and we give theoretical lower bounds for its modularity. We compare its characteristics with a real-life co-authorship network and show that our model achieves good performances. We believe that our hypergraph model will be an interesting tool that may be used in many research domains in order to reflect better real-life phenomena.

In collaboration with Michał Lasoń (Wroclaw University of Science and Technology, Poland), we prove in 92 that a class of graphs with an excluded minor and with the maximum degree sublinear in the number of edges is maximally modular, that is, modularity tends to 1 as the number of edges tends to infinity.

In collaboration with George Giakkoupis (WIDE Team, IRISA, Rennes) and Andrea Clementi (Univ. of Rome 2 "Tor Vergata", Rome, Italy), we investigate in 46 a parallel version of the famous Lévy walk stochastic process, the most famous general model of animal movement.
More precisely, motivated by the Lévy foraging hypothesis – the premise that various animal species have adapted to follow Lévy walks to optimize their search efficiency – the authors study the parallel hitting time of Lévy walks on the infinite two-dimensional grid.
They consider

In collaboration with Hlafo Alfie Mimun and Matteo Quattropani (LUISS, Roma, Italy) and Sara Rizzo (GSSI, L'Aquila, Italy),
we analyze in 48 the binary-state (either

In collaboration with Isabella Ziccardi (UNIVAQ, L'Aquila, Italy), we study in 94 the behavior of the 3-Majority dynamics in presence of noise. Communication noise is a common feature in several real-world scenarios where systems of agents need to communicate in order to pursue some collective task. In particular, many biologically inspired systems that try to achieve agreements on some opinion must implement resilient dynamics that are not strongly affected by noisy communications. In this work, the authors study the popular 3-Majority dynamics, an opinion dynamics which has been proved to be an efficient protocol for the majority consensus problem, in which they introduce a simple feature of uniform communication noise, following 104. They prove that in the fully connected communication network of

The poster 95 by Francesco d'Amore and Emanuele Natale, in collaboration with Emilio Cruciani (University of Salzburg, Austria), has been presented at MOMI2021: Le Monde des Mathématiques Industrielles (Sophia Antipolis, France). Some recent results about opinion dynamics have been summarized: the community detection properties of the 2-Choices dynamics on core-periphery networks, and the phase transition of the Undecided-State dynamics in fully connected networks with uniform communication noise,

Coati works mainly on two important topics in graph theory, namely graph colouring and directed graphs (digraphs), as well as on the interaction between the two.

We are putting an effort on understanding better directed graphs and partitioning problems, and in particular colouring problems. We also try to better understand the many relations between orientations and colourings. We study various substructures and partitions in (di)graphs. For each of them, we aim at giving sufficient conditions that guarantee its existence and at determining the complexity of finding it.

In distinguishing labelling problems, the general goal is, given a graph, to label some of its elements so that some pairs of elements can be distinguished accordingly to some parameter computed from the labelling. Note that this description involves many parameters that can be played with, such as the set of elements to be labelled, the set of labels to be assigned, the set of elements to be distinguished, and the distinguishing parameter computed from the labelling. A notable example is the so-called 1-2-3 Conjecture, which asks whether almost all graphs can have their edges labelled with 1,2,3 so that every two adjacent vertices are distinguished accordingly to their sums of incident labels.

We have recently obtained a number of results, related both to the 1-2-3 Conjecture and related problems. These results stand both as notable progress towards some open questions, and as new problems of independent interest.

In collaboration with Hervé Hocquard, Dimitri Lajou and Éric Sopena (LaBRI, Université de Bordeaux),
we have investigated, through several works, the multiplicative version of the 1-2-3 Conjecture.
In that variant of the 1-2-3 Conjecture, adjacent vertices are required to be distinguished, through a labelling, by their products of incident labels.
The main conjecture here, is due to Skowronek-Kaziów, who conjectured in 2012 that labels

In 23, we have made many progress towards that Multiplicative 1-2-3 Conjecture, proving that the conjecture holds for 4-colourable graphs, and providing a result that is very close to what is actually conjectured. Later on, in 76, building upon that earlier study, we have come up with a full proof of the Multiplicative 1-2-3 Conjecture. This stands as one of the most important results of the field, in the recent years.

In 77, we have also initiated the study of a list version of the Multiplicative 1-2-3 Conjecture, which is a standard way to generalise colouring/labelling problems. In particular, we conjecture that any list of three labels should permit to design labellings distinguishing adjacent vertices by products. Towards that presumption, we have provided several results and bounds as support.

In a few more works, we have investigated several side aspects of the 1-2-3 Conjecture, resulting in the study of related variants.
Notably, interesting questions relate to the labels

For instance, in 26, with Fionn Mc Inerney (CISPA Helmholtz Center for Information Security, Saarbrücken, Germany)
and Kasper Lyngsie (Technical University of Denmark), we have investigated the generalisation of existing results with labels

With Bi Li and Binlong Li (from Northwestern Polytechnical University and Xidian University, Xi'an, China), we have investigated proper labellings achieving additional constraints, such as minimising the sum of assigned labels 21 or the maximum vertex sum 25, the main point being to understand proper labellings further. From these questions, with Fionn Mc Inerney, we have run into the question of designing proper 3-labellings minimising the number of assigned 3's 20, 45. This last question has led us to also consider, with Fionn Mc Inerney, so-called equitable proper 3-labellings in 19, for which we have provided new results.

In 63, 69, with Pierre Aboulker (DI ENS Paris) and Kolja Knauer and Clément Rambaud (Aix Marseille Université), we give bounds on the dichromatic number

In two works 17, 18 with Sandip Das, Soumen Nandi and Sagnik Sen (from various institutes in India) and Théo Pierron and Éric Sopena (LaBRI, Université de Bordeaux), we have pursued the study of the usual chromatic theory of graphs to the realm of decorated graphs. Namely, we have considered the analogue of the chromatic number for pushable graphs (oriented graphs in which vertices can be pushed at will, i.e., have the direction of their incident arcs reversed) and signed graphs (2-edge-coloured graphs in which vertices can be switched at will, i.e., have the polarity of their incident edges interchanged). We have mainly focused of graphs with bounded maximum degree. Notably, we have managed to determine the exact value of the analogues of the chromatic number for pushable graphs and signed graphs with maximum degree 3.

The chromatic number, which refers to the minimum number of colours required to colour the vertices of graphs properly, is one of the most central notions of the graph chromatic theory. Several of its aspects of interest have been investigated in the literature, including variants for modifications of proper colourings. These variants include, notably, the achromatic number of graphs, which is the maximum number of colours required to colour the vertices of graphs properly so that each possible combination of distinct colours is assigned along some edge. The behaviours of this parameter have led to many investigations of interest, bringing to light both similarities and discrepancies with the chromatic number.

73 takes place in a recent trend aiming at extending the chromatic theory of graphs to the realm of signed graphs, and, in particular, at investigating how classic results adapt to the signed context. Most of the works done in that line to date are with respect to two main generalisations of proper colourings of signed graphs, attributed to Zaslavsky and Guenin. Generalising the achromatic number to signed graphs was initiated recently by Lajou, his investigations being related to Guenin's colourings. With François Dross and Éric Sopena (LaBRI, Université de Bordeaux) and Nacim Oijid (LIRIS Lyon) we have pursued this line of research in 73, but with taking Zaslavsky's colourings as our notion of proper colourings. We have studied the general behaviour of our resulting variant of the achromatic number, mainly by investigating how known results on the classic achromatic number generalise to our context. Our results cover, notably, bounds, standard operations on graphs, and complexity aspects.

In a recent work 100 motivated by applications for time division multiple access (TDMA) scheduling problems in wireless networks, we have introduced the notion of

In 22, we have pursued these investigations on planar digraphs by answering some of the questions left open in 100. We have exhibited planar digraphs needing 8 colours to be NP-hard for every

In the recent studies of crossing numbers, drawings of graphs that can be extended to an arrangement of pseudolines (pseudolinear drawings) have played an important role as they are a natural combinatorial extension of rectilinear (or straight-line) drawings. A characterization of the pseudolinear drawings of the complete graph

was found recently. With Alan Arroyo (IST Austria) and Bruce Richter (University of Waterloo, Canada), we have extended this characterization to all graphs in

16, by describing the set of minimal forbidden subdrawings for pseudolinear drawings. Our characterization also leads to a polynomial-time algorithm to recognize pseudolinear drawings and construct the pseudolines when it is possible.

Motivated by the proper orientations and the 1-2-3 Conjecture, we investigate the semi-proper orientations.
A weighted orientation of graph semi-proper orientation is a weighted orientation such that for every two adjacent vertices semi-proper orientation number of a graph

A semi-proper orientation optimal if

A tournament is an orientation of a complete graph.
A digraph is $n$-unavoidable if it is contained (as a subdigraph) in every tournament of order

With François Dross (LaBRI, Université de Bordeaux), we improve in 33 on the results towards those conjectures.
We first prove that every arborescence (tree in which all arcs are directed away from the root) of order

With
David Munhá Correia, Nemanja Draganić and Benny Sudakov (ETH Zürich, Switzerland),
François Dross (LaBRI, Université de Bordeaux),
Jacob Fox (Stanford University, CA, USA),
António Girão (Heidelberg University, Germany),
Dániel Korándi and Alex Scott (University of Oxford, England) and
William Lochet (University of Bergen, Norway),
we consider in 32 powers of directed paths. We show that every tournament contains the

All these results were obtained using the notion of median order.

A digraph is eulerian if it is connected and every vertex has its in-degree equal to its outdegree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle.
In 86, together with Jørgen Bang-Jensen and Anders Yeo (University of Southern Denmark), we first characterize the pairs

The number of embeddings of a partially ordered set

We collaborate with experts in various areas (transport, bio-informatics, e-health, etc.). In this section, we present the results we have obtained in the context of these collaborations.

One important objective of Coati is to use its expertise on graph algorithms and Operations Research to address problems in other scientific domains (transport, bio-informatics, e-health, ed-tech, etc.). During the last years, we have initiated several collaborations with academic and industrial partners in this direction.
In this section, we present the last results we have obtained in the context of these collaborations.

A major problem in structural biology is the characterization of low resolution structures of macro-molecular assemblies. One subproblem of this very difficult question is to determine the plausible contacts between the subunits (e.g. proteins) of an assembly, given the lists of subunits involved in all the complexes. This problem can be conveniently modelled by graphs and hypergraphs, and we collaborate with Dorian Mazauric (ABS) in order to better understand its computational complexity.

Let

In collaboration with Rachid Deriche, Samuel Deslauriers-Gauthier and Matteo Frigo (ATHENA), we investigated in 35 the problem of aligning brain atlases. More precisely, the interactions between different brain regions can be modeled as a graph, called connectome, whose nodes correspond to parcels from a predefined brain atlas, and whose edges encode the strength of the axonal connectivity between regions of the atlas that can be estimated via diffusion magnetic resonance imaging (MRI) tractography. In 35, we have provided a novel perspective on the problem of choosing a suitable atlas for structural connectivity studies by assessing how robustly an atlas captures the network topology across different subjects in a homogeneous cohort. We have measured this robustness by assessing the alignability of the connectomes, namely the possibility to retrieve graph matchings that provide highly similar graphs. We have introduced two novel concepts. First, the graph Jaccard index (GJI), a graph similarity measure based on the well-established Jaccard index between sets; the GJI exhibits natural mathematical properties that are not satisfied by previous approaches. Second, we have devised WL-align, a new technique for aligning connectomes obtained by adapting the Weisfeiler-Leman (WL) graph-isomorphism test. We have validated the GJI and WL-align on data from the Human Connectome Project database, inferring a strategy for choosing a suitable parcellation for structural connectivity studies.

In collaboration with Laurent Viennot (GANG, Inria Paris), we investigate in 82 the problem of making a neural network perform some hidden computation whose result can be easily retrieved from the network output. In particular, we consider the following scenario. A user is provided a neural network for a classification task by a company. We further assume that the company has limited access to the user's computation, and can only observe the output of the network when the user evaluates it. The user's input to the network contains some sensible information. We provide a simple and efficient training procedure, called Hidden Learning, that produces two networks such that i) One of the networks solves the original classification task with comparable performance to state of the art solutions of the task; ii) The other network takes as input the output of the first and solves another classification task that retrieves the sensible information with considerable accuracy. Our result might expose important issues from an information security point of view, as for the use of artificial neural networks in sensible applications.

Federated learning (FL) offers naturally a certain level of privacy, as clients’ data is not collected at a third party. However, maintaining the data locally does not provide itself formal privacy guarantees. An (honest-but-curious) adversary can still infer some sensitive client information just by eavesdropping the exchanged messages (e.g., gradients).

We initiate, in collaboration with Giovanni Neglia (NEO), the study of local model reconstruction attacks for federated learning, where a honest-but-curious adversary eavesdrops the messages exchanged between the client and the server and reconstructs the local model of the client 61. The success of this attack enables better performance of other known attacks, such as the membership attack, attribute inference attacks, etc. We provide analytical guarantees for the success of this attack when training a linear least squares problem with full batch size and arbitrary number of local steps. One heuristic is proposed to generalize the attack to other machine learning problems. Experiments are conducted on logistic regression tasks, showing high reconstruction quality, especially when clients’ datasets are highly heterogeneous (as it is common in federated learning).

In collaboration with Giovanni Neglia (NEO) and Oualid Zari (Intern, Université Côte d'Azur), we study in 62 the membership inference attack, where the adversary can infer whether the client owns a specific data instance. We propose a new passive inference attack that requires much less computation power and memory than existing methods. Our empirical results show that our attack achieves a higher accuracy on CIFAR100 dataset (more than 4 percentage points) with three orders of magnitude less memory space and five orders of magnitude less calculations.

Members of Coati are involved in the working group RESCOM (Réseaux de communications) of GDR RSD, CNRS
(gdr-rsd.fr/?page_id=159). In particular, David Coudert is co-chair of this working group since 2017.

We are also involved in the working group "Energy" of GDR RSD (gdr-rsd.fr/?page_id=166). In particular, Frédéric Giroire is co-chair of this working group.

Members of Coati are involved in the working group "Graphes" of GDR IM, CNRS.
(gtgraphes.labri.fr/). In particular, Frédéric Havet is member of the steering committee.

Members of Coati are involved in the working group GRAMINEES (GRaph data Mining in Natural, Ecological and Environnemental Sciences) of GDR MADICS (Masses de Données, Informations et Connaissances en Sciences).
(www.madics.fr/actions/actions-en-cours/graminees/).

Members of COATI have reviewed numerous manuscripts submitted to national and international conferences, including:

AAAI'22; AAMAS'21; ACM MMSys; ALGOCLOUD'21; ALGOSENSORS'21; AlgoTel'21; CoRes'21; Euro-Par'21; IEEE ICC'21; IEEE Globecom'21; LAGOS'21; MFCS'21; NeurIPS'21; ONDM'21; PODC'21; SEA'21; STACS'22; WADS'21.

Members of COATI have reviewed numerous manuscripts submitted to international journals, including:

ACM Journal of Experimental Algorithmics; Combinatorics, Probability and Computing; Computer Networks (COMNET); Computers & Operations Research (COR); Discrete Mathematics; Discrete Applied Mathematics (DAM); Discrete Mathematics and Theoretical Computer Sciences (DMTCS); European Journal of Operational Research (EJOR); IEEE Internet of Things Journal; IEEE Transactions on Mobile Computing (TOMC); IEEE Transactions on Network and Service Management; IEEE Transactions on Network Science and Engineering; INFORMS Journal on Computing; Journal of Combinatorial Theory Ser. B; MDPI Future Internet; MDPI Sensors; Networks (Wiley); SIAM Journal of Discrete Mathematics (SIDMA); Springer Photonic Network Communications; The Computer Journal; Theoretical Computer Science (TCS).

Members of Coati have taught for more that 1650 hours (ETD) this year:

Many members of Coati are importantly involved in Terra Numerica (see terra-numerica.org/) and participate to the creation of several popularization systems of science popularization : games, activities, on-line resources, ...

We have published a popularization article on optimization problems in telecommunications networks in a special issue on operations research of the science popularization journal Tangente 29.