Coati is a joint team between the Inria Centre at Université Côte d'Azur and the I3S laboratory (Informatique Signaux et Systèmes de Sophia Antipolis), which
belongs to CNRS (Centre National de la Recherche Scientifique) and Université 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 8.5) such as structural biology, transportation networks, economy, sociology, etc. For instance, we collaborate with Inria project-team CRONOS (Computational modelling of brain dynamical networks) from Sophia Antipolis in the field of computational neuro-sciences. 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.

These collaborations benefit to the above mentioned domains via the dissemination of our tools. Also, 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 besides our research activity, we are deeply involved in Terra Numerica and contribute to disseminating our domain towards a general audience.

Coati is mostly interested in telecommunications networks but also in the network structures 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 and with project-team CRONOS (formerly ATHENA) on problems arising in computational neurosciences. 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 curricula (studies and career paths of persons). Last, we collaborate 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.

Joanna Moulierac is member of the I3S CO2 group since 2019. The objective of this working group is to evaluate the environmental impact of our research activities and to propose ways to make them evolve by 2030, in order to fulfill the Paris agreements (www.i3s.unice.fr/co2/).

SageMath is a free mathematics software system written in Python, 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 contributes the improvement and maintenance of the graph module and its 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.

Integrated Assessment Modeling is a research area that focuses on developing and applying integrated models of human and earth systems to help understand how key aspects of society may evolve in the future and how they might interact with the natural environment and with a changing climate.

Despite the importance of the field, there has been no general software framework which allows scientists to investigate integrated assessment models of sustainable development, by using and adapting the most famous world models, from Meadows et al.'s World3 to recent proposals.

WorldDynamics.jl is a new software library written by members of the COATI Team, that aims to fill such crucial gap, by providing a modern framework in the Julia programming language based on current software engineering and scientific machine learning techniques.

Idawi is a middleware for the development and experimentation of distributed applications for multi-hop dynamic networks, such as the IoT, the Edge, Mobile Ad hoc Networks, etc. Such networks can be represented by mixed (featuring both directed and undirected edges) dynamic multi-graphs.

The development of Idawi was initially motivated by the need of the COATI Research group to deploy scientific applications in clusters of computers, in order to run large experimentation campaigns of graph algorithms. Idawi is an innovative arrangement of many features found in existing tools into a fresh Open Source Java reference implementation, but it our Research context we were led to introduce new features and ideas not found in other middleware solutions for distributed computing - such as a fully decentralized network model, a by-default collective communication/computation model, etc.

These naturally turned Idawi into a Research software platform for the experimentation of middleware-level techniques.

Idawi defines applications elements as components organized into a multi-hop overlay network on top of agnostic transport layers such as TCP, UDP and SSH (to enable component deployment and communication even in the presence of NATs and firewalls). In the usual use case, there will be only one component per device. But, in order to enable the simulation/emulation of large systems, components can deploy other components in their Java Virtual Machine (JVM) or in another JVM(s) in the same device.

Idawi proposes a structuring model of distributed applications, which then must conform to a specific Object-Oriented model in the style of SOA: it defines that components expose their functionality via services. Services hold data and implement functionality about the specific concern they are about. Functionality is then exposed via operations, which can be triggered remotely from anywhere in the component overlay.

The decentralized communication model of Idawi matches the very nature of mobile multi-hop networks. It defines that components communicate with each other via messages of bounded size. Messaging can be both synchronous (imperative) and asynchronous (reactive/event-driven). It is powered by a default routing scheme and APIs that are tailored to collective communication, so as to offer native support of parallel processing.

Idawi comes with a set of built-in fully decentralized services for automatized quick deployment/bootstrapping of components through SSH, interoperability through a REST-based web interface, service provisioning and discovery, overlay management, and many other system-level functionality.

In 2023, Idawi was completely refactored so as to introduce the notion of "digital twin" as the core of its network management model. This idea is that the environment of a component is represented by a digital twin — each remote component being represented by a local twin counterpart. Messages can then be configured to be delivered to a twin or to the real component. Doing this smoothens the API and enables any component to simulate distributed processes.

OnlineGraph is a decentralized mixed graph library. OnlineGraph is designed and developed as a specific application-level service of the Idawi middleware. It implements a decentralized application distributed over a multi-hop overlay network of components. In this application dedicated to the storage, edition and graphical rendering of graphs, graphs are partitioned over the set of components. The application imposes no allocation strategy: it is left to the application. A graph may be entirely allocated on one specific component, or a subset of the components forming the application, or all of them.

Any component in the application can expose graphs to the Web—not just its component-local part, but all the parts it has access to in the overlay networks. Graphs are then accessible to clients through a set of Web servers exposing services.

A component-local part of a partitioned graph consists of three correlated sets: the vertex set, the arc set, and the edge set. These sets can store the elements they contain in RAM or on disk, thereby enabling persistence. In these sets, graph elements are represented by a 64-bit numerical ID, and they can be associated to any data. This graph data structures can be used out of the distributed infrastructure, just any graph library.

OnlineGraph is developed within the COATI Research group at Inria Sophia Antipolis and I3S Computer Science Laboratory of Université Côte d'Azur.

The Temporal Brain Networks dataset 77 contains networks representing the brain activity of 1047 subjects. These networks are derived from resting-state functional Magnetic Resonance Imaging (fMRI) data from the Human Connectome Project (HCP). Network nodes represent brain regions, and edges represent the correlation of a window of two brain region activity values. The networks are undirected and weighted. The diagonal values are always 1 because they represent the correlation of a brain region with itself. The number of nodes remains constant over time, while the weights of the edges change. The dataset can be used to investigate brain activity as a temporal graph, it can be found in the Inria space on the Recherche Data Gouv Platform ( https://entrepot.recherche.data.gouv.fr/dataset.xhtml?persistentId=doi%3A10.57745%2FPR8VUV).

This dataset has been assembled in collaboration with Samuel Deslauriers-Gauthier (CRONOS).

The Labeled Temporal Brain Networks dataset contains a collection of temporal brain networks from 100 subjects. Each subject is labeled with their biological sex (M for male and F for female) and age range (22-25, 26-30, 31-35, and 36+). The networks are obtained from resting-state fMRI data from the Human Connectome Project (HCP) and are undirected and weighted. The number of nodes is fixed at 202, while the edge weights vary over time. The dataset is labeled and intended for use in machine learning tasks, specifically related to temporal graph neural network architectures. It can be found in the Inria space on the Recherche Data Gouv Platform ( https://entrepot.recherche.data.gouv.fr/dataset.xhtml?persistentId=doi:10.57745/HHNT10).

Network design is a very wide subject, which concerns all kinds of networks. In telecommunications, networks can be either physical (backbone, access, wireless, etc.) or virtual (logical). The objective is to design a network able to route a (given, estimated, dynamic, etc.) 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 or LoRa networks, and of new usages like Internet of Things (IoT), high quality video streaming. Furthermore, the development of machine-learning based methods brings new tools that can help solving optimization problems. 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 tackle these problems with tools from operations research and discrete mathematics (some of them developed in our teams, see Sections 8.2 and 8.3). 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.

Video streaming traffic dominates the Internet traffic. In such a context, assessing the carbon footprint of video streaming has received recently a significant attention with a number of models proposed to associate a CO2 cost to one hour of streaming. The topic is even becoming sensitive, as highlighted by the recent debate between the SHIFT project in France and the International Energy Agency (IEA) experts. Our objective in this work is to compare the modeling assumptions and computation methods used by a number of recent works to inform the debate. Indeed, initial results can be at odds, with up to one order of magnitude difference in the results.

In 74, we focus on five different models. Our contributions are: (i) we relate the difference in the results primarily to the perimeter of the study, e.g. including grey energy (production cost) or not; (ii) we demonstrate that some of the assumptions, while necessary, are difficult to estimate due to a lack of public data; (iii) we question some of the modeling assumptions made using a real deployment of a streaming server in a controlled environment with up to 2000 clients; (iv) we propose a technique to reconcile the models and obtain a CO2 estimate in between 60 and 400 grams when considering the average worldwide electricity efficiency.

HTTP-based streaming has become the dominant technology for streaming due to the widespread adoption of the HTTP protocol. Many streaming providers use a combination of Content Delivery Network (CDN) and Viewer-to-Viewer (V2V) technology, known as Hybrid CDN/V2V live streaming, for both efficiency and cost-effectiveness. V2V technology allows for offloading streaming traffic from the CDN and reducing operational costs, and WebRTC technology facilitates direct V2V transfer, as it is natively supported by all browsers. In a WebRTC-based V2V network, some viewers cache the video chunks on their devices, while others wait and fetch chunks from their neighbors. A common strategy used to determine when a viewer should stop waiting for chunk delivery and revert to the CDN is called Random Waiting Time Control (RWC). However, due to the complex dynamics in the V2V system, RWC is far from optimal. In 50, we have formulated the Waiting Time Control determination problem as a reinforcement learning problem and proposed a Q-learning-based Waiting Time Control (QWC) solution. We conducted offline experiments in the Grid5000 84 testbed and validated our results through a 14-day A/B testing in the wild. Our findings showed that QWC improves overall streaming Quality-of-Experience (QoE) in rebuffering (-29% fewer events), video quality (+17% higher), and buffer length (+5% longer), with a slightly improved V2V ratio (+5% more).

This is a collaboration with Zhejiayu Ma and Sofiane Rouibia from EasyBroadcast and Guillaume Urvoy-Keller (I3S).

In 35, we present PRISMA-v2, a new release of PRISMA 82, 81, a Packet Routing Simulator for Multi-Agent Reinforcement Learning. PRISMA-v2 brings a new set of features. First, it allows simulating overlay network topologies, by integrating virtual links. Second, this release offers the possibility to simulate control packets, which allows to better evaluate the overhead of the network protocol. Last, we integrate the modules along with the core (ns-3) to a docker container, so that it can be run in any machine or platform. PRISMA-v2 is, to the best of our knowledge, the first realistic overlay network simulation playground that offers to the community the possibility to test and evaluate new network protocols.

This work has been done in collaboration with Lucile Sassatelli from SIS team of I3S laboratory.

LoRa is a low-power and long range radio communication technology designed for low-power Internet of Things (IoT) 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 Martin Heusse and Andrzej Duda (Drakkar, LIG, Grenoble), we present a model for frame delivery rates in LoRaWAN when the gateway is capable of receiving antenna diversity 48. Based on this model, we investigate the level of transmission redundancy required to guarantee reliable data reception without overloading the channel. We then show that receive antenna diversity combined with a well-chosen level of transmission redundancy guarantees maximum channel utilization while maintaining reliable reception of application data.

We also explore the capacity limits and the tradeoff between the antagonistic means of enabling reliable data delivery in a loaded LoRaWAN cell 33. In fact, channel attenuation and variability call for robust transmission settings but the associated load increase causes more collisions between frames. In addition to the physical layer parameters of the LoRa modulation, this work considers the benefits and tuning of inter-packet Error Correction Codes (ECC), which also trades transmission redundancy for reliability. We propose a refined Packet Delivery Ratio (PDR) model that improves the ones found in the literature in that, first, it takes into account the dependency between overcoming ambient noise and dominating colliding frames; and second, it considers the sum of interference powers when multiple colliding frames are present, even if the interference preexists. Moreover, the model extends to the case of a gateway with receiver diversity. In a second step, the model allows to set out the level of redundancy at which ECC is the most effective without hindering capacity: a coding rate of one third. When this is fixed, it allows to define the transmission parameters allocation within a cell and thus the size of the cell.

The problem of the lifetime of connected objects, in most use cases (Industrial Internet of Things (IIoT), disaster management, etc.) is an essential element of the proposed solutions. Radio frequency (RF) harvesting of sensor batteries is an attractive solution, however, it does not scale up if it has to be done by human operators, and becomes impossible if the objects are located in unreachable places. An innovative solution consists of using fleets of drones to take care of this regular recharge. In collaboration with Yann Busnel (IMT Nord Europe) 55, we propose a two-stage optimization framework for determining near-optimal trajectories for unmanned aerial vehicles (UAVs), or drones, in order to rapidly recharge a ground-based sensor network, thereby increasing its lifetime. The chosen wireless radio-frequency (RF) charging solution enables multiple sensors to be charged in parallel by a single UAV. However, when several drones charge the same sensor simultaneously, there is a high energy loss. We therefore optimize the flight plan of the drone fleet to minimize the time needed to charge all the sensors, while avoiding this wastage.

In collaboration with Takako Kodate (Tokyo Woman's Christian University, Japan) and Joseph Yu (University of the Fraser Valley, B.C., Canada), we tackled in 27 the problem of gossiping with interference constraint in radio ring networks. Gossiping (or total exchange information) is a protocol where each node in the network has a message and is expected to distribute its own message to every other node in the network. The gossiping problem consists in finding the minimum running time (makespan) of a gossiping protocol and algorithms that attain this makespan. We focus on the case in which the transmission network is a ring network. We consider synchronous protocols where it takes one unit of time (step) to transmit a unit-length message. During one step, a node receives at most one message only through one of its two neighbors. We also suppose that, during one step, a node cannot be both a sender and a receiver (half duplex model). Moreover communication is subject to interference constraints. We use a primary node interference model where, if a node receives a message from one of its neighbors, its other neighbor cannot send at the same time. With these assumptions we completely solve the problem for ring networks. We first show lower bounds and then present gossiping algorithms that meet these lower bounds and so are optimal. The number of rounds depends on the congruences of

Idawi (see Section 7.1.4) is a research software platform for the experimentation of Edge Computing algorithms, whose the infrastructure network can be represented by a mixed (featuring both directed and undirected edges) dynamic multi-graph.

The development of Idawi was initially motivated by the need of Coati to deploy scientific applications in clusters of computers, in order to run large experimentation campaigns of graph algorithms, and it progressively turned to a research platform for decentralized algorithms/systems.

In order to enable the Idawi middleware interoperability with Web appplications, and in particular to build a user Web interface for the control of running distributed computations, we introduced into the middleware a specific service. The role of the service is bridge Web (HTTP/JSON) queries to the internal messaging system of Idawi. The very decentralized nature of Idawi allowed us to design and implement a novel architecture for a Web backend that consists of a set of application-specific services accessible via bridge services available in one or several component of the Idawi system. This enables the resulting decentralized Web backend to be accessible from anywhere, regardless of the presence of network obstacles (NATs, firewall), and to natively support redundancy 49.

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 approximations, 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.

Let leanness of geodetic graphs, also have received special attention in graph theory. The practical computation of leanness in real-life complex networks has been studied recently 88. In collaboration with Samuel Coulomb (ENS Paris) and Guillaume Ducoffe (IRO, Bucharest, Romania), we give in 69 a finer-grained complexity analysis of two related problems, namely: deciding whether the leanness of a graph

Cutwidth is a parameter used in many layout problems. Determining the cutwidth of a graph is an NP-complete problem, but it is possible to design efficient branch-and-bound algorithms if good lower bounds are available for cutting branches during exploration. Knowing how to quickly evaluate good bounds in each node of the search tree is therefore crucial. In 54, we present new lower bounds based on different graph density parameters. We also propose a lower bound based on the grooming of traffic requests on the path.

The support of a flow

The length of a tree-decomposition of a graph is the maximum distance (in the graph) 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

We first fully describe 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 Hunters and Rabbit game is played on a graph

Let infected. In 18, we consider the process in which, at every step, each non-infected vertex with at least

We consider the following 2-player combinatorial game taking a graph

We first show that computing PSPACE-complete1 in bipartite graphs with diameter 4, in split graphs and in planar graphs. Then, we focus on A-perfect graphs, namely, graphs for which i.e., the maximum possible value. While there exist arbitrary large A-perfect

Finally, graph classes where

In 21, we initiate the study of the algorithmic complexity of Maker-Breaker games played on the edge sets of general graphs. We mainly consider the perfect matching game and the

We then give several positive results for the

Another natural direction to take is to consider the

The degree distributions of complex networks are usually considered to follow a power law distribution. However, it is not the case for a large number of them. We thus propose a new model able to build random growing networks with (almost) any wanted degree distribution 32. 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, the broken power-law, and the geometric distributions. We also use the model on an undirected version of the Twitter network, for which the degree distribution has an unusual shape. We finally show that the divergence of chosen attachment functions is directly linked to the heavy-tailed property of the obtained degree distributions. This is a collaboration with Thibaud Trolliet (TSE).

In the field of complex networks, hypergraph models have so far received significantly less attention than graphs. However, many real-life networks feature multiary relations (coauthorship, protein reactions) may therefore be better modeled by hypergraphs. Also, a recent study by Broido and Clauset suggests that a power-law degree distribution is not as ubiquitous in the natural systems as it was thought so far. They experimentally confirm that a majority of networks (56% of around 1000 networks that undergone the test) favor a powerlaw with an exponential cutoff over other distributions. We address the two above observations by introducing a preferential attachment hypergraph model that allows for vertex deactivations. The phenomenon of vertex deactivations is rare in existing theoretical models and omnipresent in real-life scenarios (social network accounts that are not maintained forever, collaboration networks in which people retire, technological networks in which devices break down). We prove that the degree distribution of the proposed model follows a power-law with an exponential cutoff. We also check experimentally that a Scopus collaboration network has the same characteristic. We believe that our model will predict well the behavior of systems from a variety of domains 46. In collaboration with Małgorzata Sulkowska (Wrocław University of Science and Technology).

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

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

We are putting an effort on better understanding directed graphs and partitioning problems, and in particular coloring problems. We also try to better understand the many relations between orientations and colorings. We study various substructures and partitions in (di)graphs. For each of them, we aim at giving sufficient conditions that guarantee their 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 60, 26, 65, with O. Baudon and M. Boivin (LaBRI, Université de Bordeaux), we have investigated several properties of arbitrarily partitionable graphs (AP graphs for short), which are those graphs that can be vertex-partitioned into arbitrarily many connected subgraphs with arbitrary order. While AP graphs form a superclass of Hamiltonian graphs, we proved results showing both the similarities and the discrepancies between AP graphs and Hamiltonian graphs. In particular, we proved that a few sufficient conditions for Hamiltonicity can be weakened to APness, while we proved some do not, sometimes in a strong sense. We also investigated AP graphs that are minimal w.r.t. the AP property, giving, among others, results on their order, their minimum degree, their maximum degree, and their clique number.

Let

In 20 with T. Das, S. Nandi and S. Sen (from various institutes in India), and D. Lajou (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), in particular in the context of various types of grids. The bounds we established apart, we also summarized the state of research of the field to date, and raised new results and questions, to motivate researchers to undertake further studies on the topic.

The dichromatic number

In 67, we give both lower and upper bounds on the dichromatic number of super-orientations of chordal graphs (i.e. digraphs for which the underlying graph is chordal).
When restricted to oriented graphs, we show that the easy bound on the dichromatic number of such digraphs is best possible even for orientations of interval graphs.
We then show, for every digraph

We define

Let

We then extend a theorem of Feghali, Johnson and Paulusma to digraphs. We prove that, for every digraph

A digraph

In 76, we prove the first conjecture when

In this work, we generalize several results on graph recolouring to digraphs.

Given two

A digraph is said to be

We pose as a conjecture that, for every digraph

Restricted to the special case of oriented graphs, we prove that the dicolouring graph of any subcubic oriented graph on

One of our goals is to establish structural results on digraphs that can be used to design efficient algorithms. In particular, we are looking for substructures with certain properties or ways to represent or approximate efficiently the digraphs.

An orientation $k$-orientation if the in-degree of each vertex in

A weighted orientation of a graph semi-proper orientation of semi-proper $k$-orientation

We first prove that

A (di)graph H has the Erdős-Pósa (EP) property for (butterfly) minors if there exists a function

The inversion of a set

An s-branching flow f in a network

Given a digraph cycle-degree of

In 75, using this new definition of cycle-degeneracy, we extend several evidences for Cereceda's conjecture to digraphs. The $k$-dicolouring graph of

We show that

We then show that two proofs of Bonamy and Bousquet for undirected graphs can be extended to digraphs. The first one uses the digrundy number of a digraph $D$-width of a digraph, denoted by

Finally, we give a general theorem which makes a connection between the recolourability of a digraph

The (directed) metric dimension of a digraph

In 17, we investigate the graph parameters

The most significant results deal with oriented forests.
We provide a linear-time algorithm to compute the metric dimension of an oriented forest and a linear-time algorithm that, given a forest

Collaboration within the Inria EA CANOE and the CAPES-Cofecub project with the PARGO team from UFC, Fortaleza, Brazil.

In the last years, Coati has started investigating machine-learning-based methods to enhance algorithms or solve optimization problems in networks (see e.g., Sections 8.1.2 and 8.1.3). It also investigates how to use tools from graph theory, algorithmic and combinatorics to improve machine-learning tools. We here present our last results in this direction.

The Strong Lottery Ticket Hypothesis (SLTH) states that randomly-initialized neural networks likely contain subnetworks that perform well without any training. Although unstructured pruning has been extensively studied in this context, its structured counterpart, which can deliver significant computational and memory efficiency gains, has been largely unexplored. One of the main reasons for this gap is the limitations of the underlying mathematical tools used in formal analyses of the SLTH. In 40, we overcome these limitations: we leverage recent advances in the multidimensional generalization of the Random Subset-Sum Problem and obtain a variant that admits the stochastic dependencies that arise when addressing structured pruning in the SLTH. We apply this result to prove, for a wide class of random Convolutional Neural Networks, the existence of structured subnetworks that can approximate any sufficiently smaller network. This result provides the first sub-exponential bound around the SLTH for structured pruning, opening up new avenues for further research on the hypothesis and contributing to the understanding of the role of over-parameterization in deep learning. Furthermore, in 42, we present an alternative proof for the classical result on the Random Subset Sum problem, which has been leveraged in previous proofs of the SLTH. Our new result offer a more direct approach and make use of more elementary tools.

FlyHash is a locality-sensitive hashing algorithm inspired by the nervous system of the Drosophila fly. It has demonstrated to be particularly effective for similarity search, especially in the federated context where multiple players collaborate to solve a statistical learning task. FlyHash mainly relies on a process called winner-take-all, which is used to binarize information. However, the implementation of this process is a major challenge and limits the algorithm's usage in processing large data streams. In 41, 72, we propose a simple algorithm to make the winner-take-all operation efficient on GPUs. We create a FlyHash adaptation suitable for the CUDA architecture. We assess the speed of this version experimentally and present a comparison with the CPU version of FlyHash.

In 39, we investigate the problem of making an artificial neural network perform hidden computations whose result can be easily retrieved from the network's output. In particular, we consider the following scenario. A user is provided a neural network for a classification task by a third party. The user's input to the network contains sensitive information and the third party can only observe the output of the network. In this work, we provide a simple and efficient training procedure, which we call hidden learning, that produces two networks: (i) one that solves the original classification task with performance near to state of the art; (ii) a second one that takes as input the output of the first, retrieving sensitive information to solve a second classification task with good 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.

Trying to predict the evolution of human society in terms of its fundamental aspects is both a delicate and urgent issue for science. Over the years, a number of models have been developed to help in this respect. In 56, 70, we present WorldDynamics.jl, an open-source framework that includes the software presented in Section 7.1.3, with the aim of enabling scientists to easily use and adapt different integrated assessment models for sustainable development. The library has been developed using the Julia programming language and incorporates different famous integrated assessment models. The implementation provided can directly benefit from a wide range of tools already available in the Julia language ecosystem, including the JuMP modeling language for performing mathematical optimization on aspects of the models.

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.
In addition, some results motivated by transportation networks are presented in Section 8.2.4.

In collaboration with Samuel Deslauriers-Gauthier (CRONOS) we investigated the properties of temporal brain networks extracted from functional Magnetic Resonance Imaging (fMRI) data from the WU-Minn Human Connectome Project (HCP). We first created and published 78 the dataset that collects the networks (see Section 7.2.1). We presented it with a poster at the annual NeuroMod meeting. We analyzed the dataset focusing on temporal small-worldness. To test certain hypotheses of the observed functional connectivity, we proposed three temporal null models: the geometric euclidean model on a square and on a torus, and the hyperbolic geometric graph model. The hyperbolic model is popular in the research community for studying real-world complex networks. It can model both a high-tailed degree distribution and small-worldness. Our analysis indicates that the hyperbolic model is more effective in reproducing the small-worldness property of real data, making it a favorable null model. We presented our initial findings at the Complex Network Conference 52. A complete version of the work will soon be appearing in the journal Network Neuroscience.

We collaborated with Carlo Lucibello (Bocconi University, Italy) to contribute to the Julia open source libraries GraphNeuralNetworks.jl and MLDatasets.jl. Our project's objective was to add support for temporal graph neural networks (TGNNs) to GraphNeuralNetworks.jl by creating a temporal graph type. We added new layers to the package and included new datasets of brain connectomes (see Section 7.2.2) and traffic networks in MLDatasets.jl. Tutorials have been created to introduce the new temporal graph type and demonstrate how to perform temporal graph classification using the added data, model, and features. Examples were presented in a poster at the Julia Days in Paris 2023 79.

Communicating finite-state machines (CFMs) are a Turing powerful model of asynchronous message-passing distributed systems. In weakly synchronous systems, processes communicate through phases in which messages are first sent and then received, for each process. Such systems enjoy a limited form of synchronization, and for some communication models this restriction is enough to make the reachability problem decidable. In particular, we explore the intriguing case of p2p (FIFO) communication, for which the reachability problem is known to be undecidable for four processes, but decidable for two. We show in 43 that the configuration reachability problem for weakly synchronous systems of three processes is undecidable. This result is heavily inspired by our study on the treewidth of the Message Sequence Charts (MSCs) that might be generated by such systems. In this sense, the main contribution of this work is a weakly synchronous system with three processes that generates MSCs of arbitrarily large treewidth. In collaboration with Cinzia di Giusto and Étienne Lozes (I3S, UniCA).

The dAIEDGE Network of Excellence (NoE) seeks to strengthen and support the development of a dynamic European cutting-edge AI ecosystem under the umbrella of the European Lighthouse for AI and to sustain the development of advanced AI.

dAIEDGE fosters the exchange of ideas, concepts, and trends on cutting-edge next generation AI, creating links between ecosystem actors to help both the European Commission (EC) and the European Union (EU) and the peripheral AI constituency identify strategies for future developments in Europe.

Our main objective is to advance Europe's innovation and technology base by developing a comprehensive policy and governance approach to AI in order for the EU to become a world leader in innovation in the data economy and its applications.

The 5G network and the networks of the future represent a key issue for French and European industry, society and digital sovereignty. This is why the French government has decided to launch a dedicated national strategy. One of this strategy's priority ambitions is to produce significant public research efforts so the national scientific community contributes fully to making progress that clearly responds to the challenges of 5G and the networks of the future. In this context, the CNRS, the CEA and the Institut Mines-Télécom (ITM) are co-leading the '5G' acceleration PEPR to support upstream research into the development of advanced technologies for 5G and the networks of the future.

Inria is involved into 8 research projects over the 10 supported by the program, with the participation of 11 project-teams of the theme "Networks and Telecommunications" and the coordination of the PC9-Founds.

Coati is involed in PC1 NF-MUST (End-to-end multi domain services management), coordinated by Djamal Zeghlache (IMT), which involves Inria project-teams Coati, DIANA and ERMINE.

The 5G and 6G end-to-end Multi-Domain Services Management Architecture (NF-MUST) aims at automating production of inter-domain (business and application level) services for 5G, 5G Beyond and 6G networks. This is a challenging and still unrealized evolution of the networks compared with single domain services or pre-established static multi-domain services that are gradually emerging in 5G and Beyond.

Project NF-MUST of the PEPR 5G and Future Networks, focuses mainly on transforming client requests into end-to-end service orderings and on mapping them to resources and network level services (to be) provisioned by the multiple underlying networks. There is a clear evolution of 5 and 6G networks towards the provisioning of services involving multiple players and multiple technologies. Project NF-MUST addresses the related roles and interactions between customers and multiple domains in connection to the other “PEPR 5G and Future Networks” projects, to ensure automated production and operation of multi-domain services across multiple providers. Besides ordering services, NF-MUST will drive the management of the life cycle of the infrastructures” provisioned services and partake in their dynamic and automated adaptation and operation.

NF-MUST operates at the business subsystem (BSS) level and at the service side of the operation subsystem (OSS) level. NF-MUST interacts directly with network services treated by project 2 of the overall program.

PC CARECLoud - Understanding, improving, reducing the environmental impacts of Cloud Computing.

Cloud computing and its many variations offer users considerable computing and storage capacities. The maturity of virtualization techniques has enabled the emergence of complex virtualized infrastructures, capable of rapidly deploying and reconfiguring virtual and elastic resources in increasingly distributed infrastructures. This resource management, transparent to users, gives the illusion of access to flexible, unlimited and almost immaterial resources. However, the power consumption of these clouds is very real and worrying, as are their overall greenhouse gas (GHG) emissions and the consumption of critical raw materials used in their manufacture. In a context where climate change is becoming more visible and impressive every year, with serious consequences for people and the planet, all sectors (transport, building, agriculture, industry, etc.) must contribute to the effort to reduce GHG emissions. Despite their ability to optimize processes in other sectors (transport, energy, agriculture), clouds are not immune to this observation: the increasing slope of their greenhouse gas emissions must be reversed, otherwise their potential benefits in other sectors will be erased. This is why the CARECloud project (understanding, improving, reducing the environmental impacts of Cloud Computing) aims to drastically reduce the environmental impacts of cloud infrastructures.

Cloud infrastructures are becoming more and more complex: both in width, with more and more distributed infrastructures, whose resources are scattered as close as possible to the user (edge, fog, continuum computing) and in depth, with an increasing software stacking between the hardware and the user's application (operating system, virtual machines, containers, orchestrators, micro- services, etc.) The first objective of the project is to understand how these infrastructures consume energy in order to identify sources of waste and to design new models and metrics to qualify energy efficiency. The second objective focuses on the energy efficiency of cloud infrastructures, i.e., optimizing their consumption during the usage phase. In particular, this involves designing resource allocation and energy lever orchestration strategies: mechanisms that optimize energy consumption (sleep modes, dynamic adjustment of the size of virtual resources, optimization of processor frequency, etc.). Finally, the third objective targets digital sobriety in order to sustainably reduce the environmental impact of clouds. Indeed, current clouds offer high availability and very high fault tolerance, at the cost of significant energy expenditure, particularly due to redundancy and oversizing. This third objective aims to design infrastructures that are more energy and IT resource efficient, resilient to electrical intermittency, adaptable to the production of electricity from renewable energy sources and tolerant of the disconnection of a highly decentralized part of the infrastructure.

The PEPR Data Technology for Mobility in the Territories (MOBIDEC) is in line with France 2030's objective of developing sober, sovereign and resilient mobility, by placing the collection, analysis and processing of mobility data at the heart of its work. It aims to understand and anticipate the mobility behaviors of goods and people, to facilitate the interpretation and processing of data, and to offer decision-making tools to simulate the impact of public policies in advance, or to assess the relevance of a new transport offer.

Coati is involed in project “Mob Sci-Data Factory”, one of the three projects composing the PEPR MOBIDEC. It shares the PEPR's primary goal of contributing to developing more sustainable mobility strategies by providing decision-making support methodology and a digital toolbox fed by appropriately selecting and processing mobility data and by a deeper understanding of the involved transport uses and behaviors in mobility.
It aims at clarifying and extracting the elements determining and explaining the characteristics of mobility data, which also raise the following challenging questions:
(1) What data and what are their availability, accessibility, quality, and representativeness? (2) Which methods and digital tools are necessary for processing, calibrating, understanding, and enriching data while dealing with missing data and new acquisition ? (3) What are the specifications of the decision-support platform required for standard tools and data research sharing?

Mob Sci-Data Factory will make available in a secure and privacy-compliant cloud-based infrastructure different sources of mobility data together with open-source libraries and methods designed to be unified, modular, and interoperable from conception. Mob Sci-Dat Factory outcomes will facilitate data sovereignty and open-source development interoperability across multiple scientific actors in France, while accelerating research focused on mobility by offering privacy-compliant and secure data accessibility.

Members of Coati are involved in the working group RESCOM (Réseaux de communications) of GDR RSD, CNRS
(gdr-rsd.fr/pole-rescom). In particular, Christelle Caillouet is co-chair of this working group since July 2022.

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

Members of Coati are involved in the working group "Graphes" (gtgraphes.labri.fr/) and Complexité et algorithmes (CoA) www.gdr-im.fr/les-gt/gt-coa/ of GDR IM, CNRS.
(gtgraphes.labri.fr/). In particular, Frédéric Havet is member of the steering committee of the GT Graphes and Nicolas Nisse is member of the steering committee of the GT CoA.

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 taught for more that 1100 hours (ETD) this year:

Coati is deeply involved in Terra Numerica. Its members are very active in content creation, dissemination to the public, training of teaching or facilitating staff, and project governance.

Frédéric Havet and Nicolas Nisse are also involved in the ANR project ASMODEE (Analyse et conception de situations de médiation en informatique débranchée) headed by the LIRIS laboratory.

Frédéric Havet is co-head of Terra Numerica and one of the responsible of the “Comité Scientifique, Pédagogique et Technique” ; Nicolas Nisse is a member of this committee ; Joanna Moulierac is the referent of Terra Numerica for higher education ; Luc Hogie is in charge of hardware and software development.

Frédéric Havet is member of the editorial board of 1024, le bulletin de la SIF (Société Informatique de France, in which he draws cartoons to illustrate some articles.

Press articles related to Terra Numerica can be found at terra-numerica.org/presse/. Members of Coati have contributed to several of them.

Most of the members of Coati are involved in Terra Numerica. During the year 2023, more than 300 events held, Terra Numerica has been visited by 290 classes (about 8000 primary school/college/highschool student, for 2 hours in average). We have trained about 360 persons (including 250 teachers) and touched more than 15000 people during events such as Fête de la science, etc.

Many members of Coati (C. Brosse, M. Cosnard, F. Giroire, F. Havet, J. Moulierac, N. Nisse, L. Picasarri-Arrieta, S. Pérennes, C. Rambaud, M. Syska) participated some general audience science fairs, such as the Fête de la science in October 2024 (we were present on the “Village des Sciences” in Antibes-Juan-les-Pins, Valbonne, Villeneuve-Loubet, Vinon-sur Verdon) or some stages of the Science Tour Terra Numerica (20 days in several places : Digne-les-Bains, Forcalquier, Manosque, Aups, La Seyne sur Mer, Brignoles, Draguignan, Salernes, Roquebillière, Saint Martin de Vésubie, Breil sur Roya, Tende.
They also occasionally act as scientific facilitator at Terra Numerica.

Frédéric Havet also gave general audience conferences in several cities (Bonson, Brignoles, Draguigan, Falicon, Menton, Rians, Vinon-sur-Verdon) as well as in for Esope 21, Science pour Tous 06, and Terra Numerica.