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
itself 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 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 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/).

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. 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 attack 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.

The emerging 5G induces a great diversity of use cases, a multiplication of the number of connections, an increase in throughput as well as stronger constraints in terms of quality of service such as low latency and isolation of requests. To support these new constraints, Network Function Virtualization (NFV) and Software Defined Network (SDN) technologies have been coupled to introduce the network slicing paradigm. Due to the high dynamicity of the demands, it is crucial to regularly reconfigure the network slices in order to maintain an efficient provisioning of the network. A major concern is to find the best frequency to carry out these reconfigurations, as there is a trade-off between a reduced network congestion and the additional costs induced by the reconfiguration.

In 59, we tackle the problem of deciding the best moment to reconfigure by taking into account this trade-off. By coupling Deep Reinforcement Learning for decision and a Column Generation algorithm to compute the reconfiguration, we propose Deep-REC and show that choosing the best time during the day to reconfigure allows to maximize the profit of the network operator while minimizing the use of network resources and the congestion of the network. Moreover, by selecting the best moment to reconfigure, our approach allows to decrease the number of needed reconfigurations compared to an algorithm doing periodic reconfigurations during the day.

In 48, we present PRISMA (Packet Routing Simulator for Multi-Agent Reinforcement Learning), an open source simulation ns-3-based module. To the best of our knowledge, this is the first tool specifically conceived to develop and test Reinforcement Learning (RL) algorithms for the Distributed Packet Routing (DPR) problem. In this problem, where a communication node selects the outgoing port to forward a packet using local information, distance-vector routing protocol (e.g., RIP) are traditionally applied. However, when network status changes very dynamically, is uncertain, or is partially hidden (e.g., wireless ad-hoc networks or wired multi-domain networks), RL is an alternate solution to discover routing policies better fitted to these cases. Unfortunately, no RL tools have been developed to tackle the DPR problem, forcing the researchers to implement their own simplified RL simulation environments, complicating reproducibility and reducing realism. To overcome these issues, we present PRISMA, which offers to the community a standardized framework where: (i) communication process is realistically modelled (thanks to ns3); (ii) distributed nature is explicitly considered (nodes are implemented as separated threads); (iii) and, RL proposals can be easily developed (thanks to a modular code design and real-time training visualization interfaces) and fairly compared them. This work has been done in collaboration with Lucile Sassatelli from SIS team of I3S laboratory.

In recent years, several works have studied Multi-Agent Deep Reinforcement Learning for the Distributed Packet Routing problem, with promising results in various scenarios where network status changes dynamically, is uncertain, or is partially hidden (e.g., wireless ad hoc networks or wired multidomain networks). Unfortunately, these previous works focus on an ideal scenario where the impact of control signalling is neglected, and network simulation is tailored to simplistic assumptions. In 47, we present the first experimental investigation of control signalling mechanisms for distributed learning-based packet routing. We rely on PRISMA. We formulate two signalling mechanisms between agents (value sharing and model sharing). We investigate the net gains considering in-band signalling and show that routing policies close to those provided by an oracle with full knowledge of traffic and network topology can be discovered with a control overhead of 150% with respect to injected data packets, if neighboring agents share their Deep Neural Network models. We discuss the generality of our results to underline the importance of assessing net gains of Multi-Agent Deep Reinforcement Learning (MA-DRL)-based routing. This work has been done in collaboration with Lucile Sassatelli from SIS team of I3S laboratory.

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 63, 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.

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 Martin Heusse and Andrzej Duda (Drakkar, LIG, Grenoble), we propose a model fitting the unslotted ALOHA protocol for channel access in a LoRaWAN cell 60, 68. It combines the effects of collisions with channel fading, by which reception may get buried in noise. Unlike the existing models of LoRaWAN, our model takes into account the frame arrival timing: it distinguishes, on the one hand, the interference created by earlier transmissions with respect to the frame of interest, and on the other hand, the interference by the frames arriving later on. From the results of the model, we draw three observations regarding the improvement of Packet Delivery Ratio (PDR). First, it puts back under the spotlight the often overlooked fact that repeating frames is always beneficial when the desired PDR is above 60%, even though the extra packet transmissions create more collisions. Second, as soon as the node density becomes notable and collisions have a similar impact on losses as attenuation, adding a smaller spreading factor SF6 modulation into the cell list of transmission parameters allows increasing the coverage range. Third, the model shows that cell capacity sometimes grows with the distance to the gateway or with decreased node transmission power, a trend seldom observed in wireless networks.

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 56, we present a mixed integer linear program to find the drones’ optimal trajectories to form and maintain this network through time while minimizing their movements and energy consumption. Furthermore we analyze the trade-off between distance and energy, where increasing the drones’ mobility can reduce their energy consumption, and derive a fair trade-off optimal solution to balance the two opposite objectives.

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 Atlantique) 55, we focus on the self-organised deployment of a fleet of drones to solve this problem, taking into account the multiple constraints involved. We propose a two-step optimization framework based on an optimal orchestration solution to reduce the recharging time of a complete sensor system, by optimizing the number of drones, the overall flight time and their energy consumption. We illustrate the performance of our framework that ensures the drones avoid conflicts to guarantee a higher energy harvesting efficiency (establishment of optimal drone positions and planning of the global flight plan).

In the context of studies on decentralized algorithms for mobile dynamic networks, we investigated the state of the art of the experimentation tools. We discovered that existing solutions, either coming from labs or companies, do not match the requirements of experimentation as it is usually done by researchers. Indeed commercial products focus on reliability and interoperability at the expense of versatility, while lab tools most often serve as proof of concepts. The experimental study of algorithms requires the availability, in a single solution, of the following features: support for both synchronous and asynchronous communication, simulation/emulation of large systems, fast deployment, Web interoperability, and because less is more, full decentralization, just to name a few. Idawi was designed and implemented to the very purpose of providing the research community with a tool tailored to its needs. The resulting middleware, called Idawi is released via the MAVEN global distribution platform for Java software. MAVEN Central statistics indicate that Idawi binaries have been downloaded (for integration) 350 times in the year 2022. In addition to this, Idawi and its satellite tools are all Open Source. They are released under the Apache V2 license. For the sake of Open Science, their source code are fully available on GitHub. See 61, 62, 85

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.

A path-decomposition of a graph bags, that satisfy some connectivity properties.
The length of a path-decomposition of a graph pathlength, denoted by etc. However, deciding if the pathlength of a graph

If a graph locally irregular. In 67, we introduce and study the problem of identifying a largest induced subgraph of a given graph

Then, looking for more positive results, we turn our attention towards computing

This is a joint work with with Nikolaos Melissinos (LAMSADE, Université Paris-Dauphine) and Theofilos Triomatis (School of Electrical Engineering, University of Liverpool).

In 23, we introduce the largest connected subgraph game played on an undirected graph reflection graphs) in which the game is a draw. We then show 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 also prove 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 give an algorithm to determine the outcome of the game in cographs in linear time. This is a collaboration with Fionn Mc Inerney (postdoc at CISPA Helmholtz Center for Information Security, Saarbrücken, Germany).

Then, in 49, we study the Maker-Breaker variant of this game, where Alice wins if there is a connected red component of order at least A-perfect graphs, which are the graphs i.e., those in which Alice can ensure that the red subgraph is connected. We give sufficient conditions, in terms of the minimum and maximum degrees or the number of edges, for a graph to be A-perfect. Also, we show that, for any

Inspired by the board game Kahuna, we have introduced and studied in 34 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.

In 79, 51 we revisit the problem entitled Sharing a Pizza stated by P. Winkler by considering a new puzzle called Sharing a Pissaladiere. The game is played by two polite coatis Alice and Bob who share a pissaladière (a Sharing a linear pizza. In that case each player can take only an end vertex of the remaining path. These problems are particular cases of a new class of games called

Our main results are the following. We give optimal strategies for paths (linear pizzas) with no two adjacent weighty vertices. We also give a recurrence formula to compute the gains which depend only on the parity of n and of the respective parities of weighty vertices with a complexity in
Sharing pissaladière with

The optimal stopping theory is the art of choosing a proper time to take a particular action in order to maximize gain or minimize cost. In the series of articles we undertake several optimal stopping problems and come up with some tools that help to estimate the winning probabilities.

In 41 together with Fabrício Siqueira Benevides (UFC, Ceará, Brazil) we obtain a surprising result about a certain family of optimal stopping problems defined on graphs. The vertices of a graph

Next, we turn to a particular problem defined classically for uniform random variables. A decision maker observes a sequence of

In cooperation with Alexander Gnedin (QMUL, London, UK) and Patryk Kozieł (WUST, Wrocław, Poland) we investigate much more general, full-information, models in 42. The full-information best choice problem asks one to find a strategy maximizing the probability of stopping at the minimum (or maximum) of a sequence

Finally, we come up with some tools that may be helpful by estimating winning probabilities in optimal stopping problems defined for tree-like graphs or partially ordered sets (posets). When deriving the optimal stopping rule, one needs to compare the probability of success by stopping the process at the current moment with the probability of success if one decided to play further. Then one is often forced to derive the number of embeddings of a currently generated structure (e.g. graph or poset) in the whole underlying structure.
More precisely, the number of embeddings of a partially ordered set

Modularity is a well-established parameter measuring the presence of community structure in the network. It was introduced by Newman and Girvan in 2004. Nowadays it is widely used as a quality function for community detection algorithms. The popular heuristic clustering algorithms (e.g., Louvain algorithm or Leiden algorithm) find a partition using modularity-based approach.

In 44 (in collaboration with Michał Lasoń from the Polish Academy of Sciences, Poland) we prove 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, for every

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 also just a few models allow to both preserve a power-law degree distribution and a high modularity indicating the presence of communities. In 83, 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. This is a collaboration with Thibaud Trolliet (MillionRoads).

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 37. 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.

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

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 36, 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 53, 52 a new approach that uses a hierarchy of distance-

The C++ code of all our algorithms is available at 92.

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 70, in collaboration with Isabella Ziccardi (Università degli Studi dell'Aquila), we 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 we introduce a simple feature of uniform communication noise, following 95. We prove that in the fully connected communication network of 3-Majority dynamics exhibits a phase transition. For a noise probability 3-Majority dynamics surprisingly turns out to be less resilient to noise than the Undecided-State dynamics 95, whose noise threshold value is

In several real Multi-Agent Systems (MAS), it has been observed that only weaker forms of
metastable consensus are achieved, in which a large majority of agents agree on some opinion while other opinions continue to be supported by a (small) minority of agents.
In 45, we take a step towards the investigation of metastable consensus for complex (non-linear) opinion dynamics by considering the popular Undecided dynamics in the binary setting, which is known to reach consensus exponentially faster than the Voter dynamics.
We propose a simple form of uniform noise in which each message can change to another one with probability

We study opinion dynamics in multi-agent networks where agents hold binary opinions and are influenced by their neighbors while being biased towards one of the two opinions, called the superior opinion.
The dynamics is modeled by the following process: at each round, a randomly selected agent chooses the superior opinion with some probability

In collaboration with George Giakkoupis (WIDE Team, IRISA, Rennes) and Andrea Clementi (Univ. of Rome 2 "Tor Vergata", Rome, Italy), we investigated in 91 a parallel version of the famous Lévy walk stochastic process, the most famous general model of animal movement. This year, we presented our work at the Highlights of Algorithms (HALG) flagship international conference 86.

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 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 H. Hocquard, D. Lajou and É. 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 28, we have gotten 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 27, 65, 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 29, 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 lists 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 also 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 32, with F. Mc Inerney (CISPA Helmholtz Center for Information Security, Saarbrücken, Germany)
and K. Lyngsie (Technical University of Denmark), we have investigated the generalisation of existing results with labels

Also, in 50, 77, with H. Hocquard and P.-M. Marcille (LaBRI, Université de Bordeaux),
we have investigated the so-called Weak

In some other works, we have also introduced new variants of distinguishing labelling problems,
and have provided a few results on them.
For instance, in 30, with H. Hocquard and P.-M. Marcille,
we have introduced a variant of the 1-2-3 Conjecture where the vertex sums are fetched within a larger radius

In 78, 31, 19, with B. Li (Northwestern Polytechnical University, China), 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 results on their order, their minimum degree, their maximum degree, and their clique number.

In 20, 21, 22, 75 with T. Das, S. Das, S. Nandi and S. Sen (from various institutes in India), N. Oijid and T. Pierron (LIRIS, Université de Lyon), and D. Lajou and É. 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. We have also conducted a study of various types of grids, summarising the state of research of the field to date, and raising new results and questions.

Finally, we also came up with attempts to generalise classical types of colourings to signed graphs and oriented graphs, such as complete colourings and total colourings. For each such attempt, we investigated how results for the undirected case adapt to our case (if they did), and proposed new directions and questions to motivate further research on the topic.

Panagopoulou and Spirakis (A game theoretic approach for efficient graph coloring. Algorithms and Computation
pages 183–195, 2008) proposed a colouring algorithm based on a game on a graph c is the combination of the actions of each vertex.
Given a configuration, a player's payoff is 0 if he selects the same colour as one of his neighbours, otherwise it is the number of vertices that selected the same colour.
An elementary improvement occurs when a player unilaterally deviates to other colour and increases its payoff.
A pure Nash equilibrium (PNE) is a configuration in which no vertex can do elementary improvements.
It can be seen as a state of the game that is sustainable.
Note that pure Nash equilibria are colourings of

In 39, we introduce and study Nash colourings, that correspond to pure Nash equilibria in Panagopoulou and Spirakis's game: a Nash $k$-colouring is a

We then study the Nash number and the diminishing Grundy number of trees and forests, and prove that

Finally we study the complexity of related problems. We show that computing the
Nash number or the diminishing Grundy number is NP-hard even when the input graph is bipartite or chordal.
We also show that deciding whether a graph satisfies

In 16, with Pierre Aboulker (DI ENS Paris) and Kolja Knauer and Clément Rambaud (Aix Marseille Université), we give bounds on the dichromatic number

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

A folklore result in Graph Theory, attributed to Erdös, is that every graph Max-Cut problem, which is well-known to be NP-complete, but finding a subgraph

For directed graphs, we showed few years ago that there is no

A conjecture due to Kreutzer, Oum, Seymour, van der Zypen, and Wood (Electr. J. Comb., 24(2):P2.25, 2017) implies that every digraph of minimum out-degree

A digraph is eulerian if it is connected and every vertex has its in-degree equal to its out-degree.
Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle.
In 18, we first characterize the pairs

A digraph eulerian-connected if for any two distinct vertices

All our results may be seen as arc analogues of well-known results on hamiltonian paths and cycles in semicomplete digraphs.

The metric dimension

In 33, we study, for particular graph families, the maximum metric dimension over all strongly-connected orientations, by exhibiting lower and upper bounds on this value. We first exhibit general bounds for graphs with bounded maximum degree. In particular, we prove that, in the case of subcubic

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 average properties of the well-known Subset Sum Problem can be studied by the means of its randomised version, where we are given a target value

The lottery ticket hypothesis states that a randomly-initialized neural network contains a small subnetwork which, when trained in isolation, can compete with the performance of the original network. Recent theoretical works proved an even stronger version: every sufficiently overparameterized (dense) neural network contains a subnetwork that, even without training, achieves accuracy comparable to that of the trained large network. These works left as an open problem to extend the result to convolutional neural networks (CNNs). In 54 we provide such generalization by showing that, with high probability, it is possible to approximate any CNN by pruning a random CNN whose size is larger by a logarithmic factor.

The high energy demands of modern Artificial Intelligence not only imply huge costs to run it at scale, but also constrains its deployment on edge devices. While analog computing offers a way to run those algorithms with orders of magnitude more efficiency, most proposals for its actual implementation require very high precision components and would depend on non-standard manufacturing. We propose a new method that uses a few inaccurate components to build accurate and programmable resistors, allowing analog neuromorphic devices to be manufactured with standard processes. It leverages the possibility of approximating any target value by summing a subset of given random values.

In federated learning, clients such as mobile devices or data silos (e.g. hospitals and banks) collaboratively improve a shared model, while maintaining their data locally. Multiple recent works show that client's private information can still be disclosed to an adversary who just eavesdrops the messages exchanged between the targeted client and the server. In 66, we propose a novel model-based attribute inference attack in federated learning which overcomes the limits of gradient-based ones. Furthermore, we provide an analytical lower-bound for the success of this attack. Empirical results using real world datasets confirm that our attribute inference attack works well for both regression and classification tasks. Moreover, we benchmark our novel attribute inference attack against the state-ofthe-art attacks in federated learning. Our attack results in higher reconstruction accuracy especially when the clients' datasets are heterogeneous (as it is common in federated learning). Most importantly, our model-based fashion of designing powerful and explainable attacks enables an effective quantification of the privacy risk in FL.

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 8.2.5.

A macromolecular assembly is composed of subunits (e.g. proteins or nucleic acids). We assume that the composition, in terms of individual subunits, of selected complexes of the assembly is known. Indeed, a given assembly can be chemically split into complexes by manipulating chemical conditions, and the composition of these complexes can then be inferred using native mass spectrometry.

We then get a data set that can be represented by a hypergaph:
a node represents a subunit, and the hyperedges represent the different complexes. The problem is then to find a graph $\Delta \le k)$-$F$-Overlay problem, consists in deciding whether there is a graph with maximum degree at most

Once the contact graph is obtained by solving the abiove problem, it serves as input graph for a second problem, called the domino problem, that consists in determining the high resolution structure of a given assembly. A configuration of a node is so a conformation of the corresponding protein (that is a position of each of its atoms in Conflict colouring consists in deciding whether there exists a conflict colouring, that is a colouring in which

In collaboration with Samuel Deslauriers-Gauthier (ATHENA), we investigated in

87properties of temporal brain networks extracted from functional Magnetic Resonance Imaging data. In particular, we focus on the temporal small worldness and, in order to test certain hypotheses of the observed functional connectivity, we propose three temporal null models: the geometric euclidean model on a square and on a torus, and the hyperbolic geometric graph model. The latter became famous in the research community investigating real-world complex networks, since it is able to model both a high tailed degree distribution and small worldness. We compare these models to a dataset of 1050 subjects’ empirical data, taken from the WU-Minn Human Connectome Project, across different thresholds. Our analysis shows that the hyperbolic model is more consistent in reproducing the small worldness property of real data, thus providing evidence in its favor as a null model.

In collaboration with Daniel Mitropolsky and Christos Papadimitriou (Columbia University, New York) and Pierluigi Crescenzi (GSSI, L'Aquila), we revisit in 88, 64 the planning problem in the blocks world, and we implement a known heuristic for this task. The program is written in the Assembly Calculus, a recently proposed computational framework meant to model computation in the brain by bridging the gap between neural activity and cognitive function. Its elementary objects are assemblies of neurons (stable sets of neurons whose simultaneous firing signifies that the subject is thinking of an object, concept, word, etc.), its commands include project and merge, and its execution model is based on widely accepted tenets of neuroscience. A program in this framework essentially sets up a dynamical system of neurons and synapses that eventually, with high probability, accomplishes the task. The purpose of this work is to establish empirically that reasonably large programs in the Assembly Calculus can execute correctly and reliably; and that rather realistic — if idealized — higher cognitive functions, such as planning in the blocks world, can be implemented successfully by such programs.

Members of Coati are involved in the working group RESCOM (Réseaux de communications) of GDR RSD, CNRS
(gdr-rsd.fr/pole-rescom). In particular, David Coudert was co-chair of this working group since 2017 until July 2022, and 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" 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 taught for more that 1400 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 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 comité ; 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.

Coati members have participated in the development of numerous mediation devices for Terra Numerica. We only list below the most important ones.

Frédéric Havet and Luc Hogie co-supervised with Dorian Mazauric (ABS team) five TER of EUR DS4H to develop various popularization softwares (Lucas Lyon, Quentin Scordo, David Prigodin, Milena Kostov, and Chahan Movsessian).

Nicolas Nisse participated to the WECAM (WeekEnd Création d'Activité de Médiation) Türing, Lyon, February 5-6th, 2022.

Members of Coati participated to the receptions of 3ème trainees:

Members of Coati participated to the training of teachers:

Many members of Coati (J.-C. Bermond, M. Cosnard, T. Dissaux, F. Havet, H. Lesfari, J. Moulierac, L. Picasarri-Arrieta) ran numerous popularization activities in front of schoolgirls and schoolboys, either at TerraNumerica@Sophia, at MIA (Maison de l'Intelligence Artificielle), or in their schools. Some were part of some national programmes such as “Chiche !”, “Cordées de la réussite”, or “Maths en Jeans”.

Many members of Coati (T. Dissaux, F. Giroire, F. Havet, H. Lesfari, J. Moulierac, L. Picasarri-Arrieta, S. Pérennes, C. Rambaud, M. Syska) participated some general audience science fairs like the inauguration of TerraNumerica@Sophia on June 11th 2022, and Fête de la science in October 2022 (we were present on the “Village des Sciences” in Antibes-Juan-les-Pins, Valbonne, Villeneuve-Loubet, Vinon-sur Verdon).

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