2025Activity report​​​‌TeamCOATI

On the​​​‌ $k$-clique graph and‌ the $k$-clique operator‌​‌

Given a graph $\mathrm{G}$, a clique is​​​‌ a maximal complete subgraph‌ of $G$ and a‌​‌ biclique is a maximal​​ induced complete bipartite subgraph​​​‌ of $G$. The‌ intersection graphs of cliques‌​‌ and bicliques, denoted by​​ $K(G)‌$ and $KB(‌G)$ respectively, have‌​‌ been studied largely in​​ the last years. Several​​​‌ articles have been made‌ from a structural point‌​‌ of view, about characterizations,​​ recognition problem, the behavior​​​‌ of their respective iterated‌ operator, etc. In 82‌​‌, we generalize the​​ clique and biclique graphs​​​‌ as $k$-clique graphs,‌ that is, the intersection‌​‌ graphs of the family​​ of all maximal induced​​​‌ complete $k$-partite subgraphs‌ of $G$. We‌​‌ consider first $k=3$, i.e., triclique​​​‌ graphs, $KT(‌G)$, and‌​‌ we then generalize our​​ results to $k>‌3$, $k$-clique‌ graphs, $KC(‌‌G)$. In​​ particular, we study the​​​‌ connectivity relation between $\mathrm{G‌}$ and $KT(‌‌G)$, its​​ generalization to $K\mathrm{C‌}\left(G\right)$,‌ and we propose several‌​‌ structural results. We then​​ investigate its characterization and​​​‌ recognition problem. Next, we‌ consider the iterated $\mathrm{k‌‌}$-clique operators $K\mathrm{T}$ and $KC$,​​​‌ giving sufficient conditions for‌ a graph to be‌​‌ convergent or divergent under​​ these operators. Finally, we​​​‌ compare all known results‌ so far between clique,‌​‌ biclique and $k$-clique​​ graphs, and we propose​​​‌ some general conjectures on‌ the subject.

This work‌​‌ has been done in​​ collaboration with Leandro Montero​​​‌ [LS2N and IMT Atlantique,‌ Nantes, France].

Weak coloring‌​‌ numbers of minor-closed graph​​ classes

In 55,​​​‌ we study the growth‌ rate of weak coloring‌​‌ numbers of graphs excluding​​ a fixed graph as​​​‌ a minor. Van den‌ Heuvel et al. (European‌​‌ J. of Combinatorics, 2017)​​ showed that for a​​​‌ fixed graph $X$,‌ the maximum $r$-th‌​‌ weak coloring number of​​​‌ $X$-minor-free graphs is​ polynomial in $r$.​‌ We determine this polynomial​​ up to a factor​​​‌ of $𝒪(\mathrm{r}logr)$.​‌ Moreover, we tie the​​ exponent of the polynomial​​​‌ to a structural property​ of $X$, namely,​‌ 2-treedepth. As a result,​​ for a fixed graph​​​‌ $X$ and an $\mathrm{X}$-minor-free graph $G$,​‌ we show that ${\mathrm{wcol}}_r(G)‌=𝒪\left({\mathrm{r}}^{\mathrm{td}(X)‌-1}\log \mathrm{r}\right)$, which improves​​​‌ on the bound ${\mathrm{wcol}}_r(G)‌=𝒪({\mathrm{r}}^{g(\mathrm{td}(‌X))})$ given by Dujmović et​‌ al. (SODA, 2024), where​​ $g$ is an exponential​​​‌ function. In the case​ of planar graphs of​‌ bounded treewidth, we show​​ that the maximum $\mathrm{r‌}$-th weak coloring number​ is in $𝒪(‌r^2\log \mathrm{r}$), which is best​​​‌ possible.

This work has​ been done in collaboration​‌ with Jędrzej Hodor [Jagiellonian​​ University, Krakow, Poland], Xuan​​​‌ Hoang [LISN, Université Paris-Saclay,​ France] and Piotr Micek​‌ [Jagiellonian University, Krakow, Poland].​​

The $\chi$-Binding Function​​​‌ of $d$-Directional Segment​ Graphs

Given a positive​‌ integer $d$, the​​ class $d$-DIR is​​​‌ defined as all those​ intersection graphs formed from​‌ a finite collection of​​ line segments in ${\mathrm{ℝ‌}}^2$ having at most​ $d$ slopes. Since each​‌ slope induces an interval​​ graph, it easily follows​​​‌ for every $G$ in​ $d$-DIR with clique​‌ number at most $\mathrm{\omega }$ that the chromatic number​​​‌ $\chi (G)$ of $G$ is at​‌ most $d\omega$.​​ In 38, we​​​‌ show for every even​ value of $\omega$ how​‌ to construct a graph​​ in $d$-DIR that​​​‌ meets this bound exactly.​ This partially confirms a​‌ conjecture of Bhattacharya, Dvořák​​ and Noorizadeh  87.​​​‌ Furthermore, we show that​ the $\chi$-binding function​‌ of $d$-DIR is​​ $\omega \mapsto d\mathrm{\omega ‌}$ for $\omega$ even and​ $\omega \mapsto d(‌\omega -1)+1$ for $\mathrm{\omega ‌}$ odd. This extends an​ earlier result by Kostochka​‌ and Nešetřil  98,​​ which treated the special​​​‌ case $d=\mathrm{2}$.

This work has​‌ been done in collaboration​​ with Lech Duraj [Jagiellonian​​​‌ University, Krakow, Poland], Ross​ Kang [University of Amsterdam,​‌ Netherlands], Xuan Hoang La​​ [Jagiellonian University, Krakow, Poland],​​​‌ Jonathan Narboni [Jagiellonian University,​ Krakow, Poland], Filip Pokrývka​‌ [Université Masaryk, Brno, Czech​​ Republic] and Amadeus Reinald​​​‌ [LIRMM, Montpellier, France].

Directed​​ colouring

A directed colouring​​​‌ or dicolouring of a​ digraph is a colouring​‌ such that every colour​​ class induces an acyclic​​​‌ digraph. The dichromatic number​ of a digraph $\mathrm{D‌}$, denoted $\stackrel{\to }{\chi }\left(D\right)$,​​​‌ is the minimum number​ of colours needed to​‌ partition $D$ into acyclic​​ induced subdigraphs. This is​​​‌ a natural generalization of​ (unidirected) graph colouring. Indeed,​‌ if $G$ is an​​ undirected graph, and $\mathrm{D}$ is the symmetric digraph​​​‌ obtained from $G$ by‌ replacing each edge with‌​‌ the pair of oppositely​​ directed arcs joining the​​​‌ same pair of vertices,‌ then $\chi \left(\mathrm{G‌‌}\right)=\stackrel{\to }{\chi }\left(D\right)$,​​​‌ where $\chi \left(\mathrm{G‌}\right)$ denotes the chromatic‌​‌ number of $G$,​​ that is the minimum​​​‌ $k$ such that there‌ exists a proper $\mathrm{k‌‌}$-colouring of $G$.​​ One of the important​​​‌ research directions of our‌ team involves trying to‌​‌ generalize results from graph​​ coloring to dicoloring.

Brooks'​​​‌ Theorem is a fundamental‌ result on graph colouring,‌​‌ stating that the chromatic​​ number of a graph​​​‌ $G$ is almost always‌ upper bounded by its‌​‌ maximal degree $\Delta (G)$. Lovász​​​‌ showed that such a‌ colouring may then be‌​‌ computed in linear time​​ when it exists. Many​​​‌ analogues are known for‌ variants of (di)graph colouring,‌​‌ notably for list-colouring and​​ partitions into subgraphs with​​​‌ prescribed degeneracy. One of‌ the most general results‌​‌ of this kind is​​ due to Borodin, Kostochka,​​​‌ and Toft, when asking‌ for classes of colours‌​‌ to satisfy “variable degeneracy”​​ constraints. An extension of​​​‌ this result to digraphs‌ has recently been proposed‌​‌ by Bang-Jensen, Schweser, and​​ Stiebitz, by considering colourings​​​‌ as partitions into “variable‌ weakly degenerate” subdigraphs. Unlike‌​‌ earlier variants, there exists​​ no linear-time algorithm to​​​‌ produce colourings for these‌ generalizations. In 41,‌​‌ we introduce the notion​​ of (variable) bidegeneracy for​​​‌ digraphs, capturing multiple (di)graph‌ degeneracy variants. We define‌​‌ the corresponding concept of​​ $F$-dicolouring, where $\mathrm{F‌}=(f_{\mathrm{1‌}},...,{\mathrm{f‌‌}}_s)$ is a​​ vector of functions, and​​​‌ an $F$-dicolouring requires‌ vertices coloured $i$ to‌​‌ induce a “strictly-${\mathrm{f}}_i$-bidegenerate” subdigraph. We​​​‌ prove an analogue of‌ Brooks' theorem for $\mathrm{F‌‌}$-dicolouring, generalizing the result​​ of Bang-Jensen et al.,​​​‌ and earlier analogues in‌ turn. Our new approach‌​‌ provides a linear-time algorithm​​ that, given a digraph​​​‌ $D$, either produces‌ an $F$-dicolouring of‌​‌ $D$, or correctly​​ certifies that none exist.​​​‌ This yields the first‌ linear-time algorithms to compute‌​‌ (di)colourings corresponding to the​​ aforementioned generalizations of Brooks'​​​‌ theorem. In turn, it‌ gives an unified framework‌​‌ to compute such colourings​​ for various intermediate generalizations​​​‌ of Brooks' theorem such‌ as list-(di)colouring and partitioning‌​‌ into (variable) degenerate sub(di)graphs.​​

Reed in 1998 conjectured​​​‌ the following strengthening of‌ Brooks Theorem : $\mathrm{\chi ‌‌}\left(G\right)\le \lceil (\Delta (‌G)+\mathrm{1‌}+\omega \left(\mathrm{G‌‌}\right))/\mathrm{2}\rceil$ for every graph​​​‌ $G$. As a‌ partial result, he proved‌​‌ the existence of $\mathrm{\epsilon }>0$ for which​​​‌ every graph $G$ satisfies‌ $\chi (G)‌‌\le \lceil (\mathrm{1}-\epsilon )(‌\Delta (G)‌+1)+‌‌\epsilon \omega (\mathrm{G})\rceil$. In​​​‌ 57, we propose‌ an analogue conjecture for‌​‌ digraphs. Given a digraph​​ $D$, we let​​​‌ $\overleftrightarrow{\omega }\left(\mathrm{D‌}\right)$ denote the size‌​‌ of the largest biclique​​​‌ (a set of vertices​ inducing a complete digraph)​‌ of $D$ and $\widetilde{\mathrm{\Delta }}(D)‌={max}_{v\in V(D)‌}\sqrt{d^{+}\left(\mathrm{v}\right)·d^{-‌}\left(v\right)}$.​ We conjecture that every​‌ digraph $D$ satisfies $\overrightarrow{\mathrm{\chi }}(D)‌\le \lceil (\widetilde{\mathrm{\Delta }}(D)‌+1+\overleftrightarrow{\mathrm{\omega }}(D)‌)/2\rceil $, which if true​‌ implies Reed's conjecture. As​​ a partial result, we​​​‌ prove the existence of​ $\epsilon >0$ for​‌ which every digraph $\mathrm{D}$ satisfies $\overrightarrow{\chi }(‌D)\le \lceil (1-\mathrm{\epsilon ‌})(\stackrel{˜}{\Delta }\left(D\right)+‌1)+\mathrm{\epsilon }\overleftrightarrow{\omega }\left(\mathrm{D‌}\right)\rceil$. This​​ implies both Reed's result​​​‌ and an independent result​ of Harutyunyan and Mohar​‌ for oriented graphs. To​​ obtain this upper bound​​​‌ on $\overrightarrow{\chi }$,​ we prove that every​‌ digraph $D$ with $\overleftrightarrow{\mathrm{\omega }}(D)‌>\frac{2}{3}({\Delta }_{max}\left(\mathrm{D‌}\right)+1)$, where ${\Delta }_{max‌}\left(D\right)={max}_{v\in \mathrm{V‌}\left(D\right)}max(d^{+}(‌v),{\mathrm{d}}^{-}(v)‌)$, admits an​​ acyclic set of vertices​​​‌ intersecting each biclique of​ $D$, which generalizes​‌ a result of King.​​

In 34, we​​​‌ give both lower and​ upper bounds on the​‌ dichromatic number of super-orientations​​ of chordal graphs. In​​​‌ general, the dichromatic number​ of such digraphs is​‌ bounded above by the​​ clique number of the​​​‌ underlying graph (because chordal​ graphs are perfect). However,​‌ this bound can be​​ improved when we restrict​​​‌ the symmetric part of​ such a digraph. Let​‌ $D=(\mathrm{V},A)$ be​​​‌ a super-orientation of a​ chordal graph $G$.​‌ Let $B\left(\mathrm{D}\right)$ be the undirected​​​‌ graph with vertex set​ $V$ in which $\mathrm{u‌}v$ is an edge​​ if and only if​​​‌ both $uv$ and​ $vu$ belongs to​‌ $A$. An easy​​ greedy procedure shows $\overrightarrow{\mathrm{\chi ‌}}(D)\le ⌈(\omega (‌G)+\mathrm{\Delta }\left(B\right(\mathrm{D‌}\left)\right))/2⌉$. In 34​‌, we show that​​ this bound is the​​​‌ best possible by constructing,​ for every fixed $\mathrm{k‌},\ell$ with $\mathrm{k}\ge \ell +\mathrm{1‌}$, a super-orientation ${\mathrm{D}}_{k,\ell }$ of​‌ a chordal graph ${\mathrm{G}}_{k,\ell }$ such​​​‌ that $\omega ({\mathrm{G}}_{k,\ell })‌=k$, $\mathrm{\Delta }\left(B({\mathrm{D‌}}_{k,\ell })\right)=\ell$ and​‌ $\overrightarrow{\chi }({\mathrm{D}}_{k,\ell })‌=⌈(k+\ell )/\mathrm{2‌}⌉$. When $\Delta (B(D)‌)=0$ (​i.e.$D$ is an​‌ orientation of $G$),​​ we give another construction​​​‌ showing that this is​ tight even for orientations​‌ of interval graphs. Next,​​ we show that $\overrightarrow{\mathrm{\chi }}(D)‌\le \frac{1}{2}\mathrm{\omega ‌}\left(G\right)+‌‌O\left(\sqrt{d·\omega (G)‌}\right)$ with $d$ the‌ maximum average degree of‌​‌ $B(D)$. Finally, we show​​​‌ that if $B(‌D)$ contains no‌​‌ cycle of order 4,​​ $C_4$, as​​​‌ a subgraph, then $\overrightarrow{\mathrm{\chi ‌}}(D)‌‌\le ⌈(\omega (G)+\mathrm{3‌})/2⌉$.‌ We justify that this‌​‌ is almost best possible​​ by constructing, for every​​​‌ fixed $k$, a‌ super-orientation $D_k$ of‌​‌ a chordal graph ${\mathrm{G}}_k$ with clique number​​​‌ $k$ such that $\mathrm{B‌}(D_k)‌‌$ is a disjoint union​​ of paths and $\overrightarrow{\mathrm{\chi ‌}}\left(D_{\mathrm{k‌}}\right)=⌊(\mathrm{k‌‌}+3)/2⌋$.We also exhibit​​​‌ a family of orientations‌ of cographs for which‌​‌ the dichromatic number is​​ equal to the clique​​​‌ number of the underlying‌ graph.

The acyclic number‌​‌ $\overrightarrow{\alpha }\left(\mathrm{D}\right)$ is the maximum​​​‌ order of an acyclic‌ induced subdigraph. In 21‌​‌, we study $\overrightarrow{\mathrm{a}}(n)‌$ and $\overrightarrow{t}(‌n)$ which are‌​‌ the minimum of $\overrightarrow{\mathrm{\alpha }}(D)‌$ and the maximum of‌ $\overrightarrow{\chi }\left(\mathrm{D‌‌}\right)$, respectively, over​​ all oriented triangle-free graphs​​​‌ of order $n$.‌ For every $ϵ>‌‌0$ and $n$ large​​ enough, we show that​​​‌ $(1/\sqrt{\mathrm{2‌}}-ϵ)\sqrt{\mathrm{n‌‌}logn}\le \overrightarrow{\mathrm{a}}(n)‌\le \frac{107}{8}\sqrt{\mathrm{n‌}}logn$ and $\frac{\mathrm{8‌‌}}{107}\sqrt{n}/logn\le \stackrel{\to ‌}{t}\left(n\right)\le ‌(\sqrt{2}+\mathrm{ϵ‌‌})\sqrt{n/logn}$. We also​​​‌ construct an oriented triangle-free‌ graph on 25 vertices‌​‌ with dichromatic number 3,​​ and show that every​​​‌ oriented triangle-free graph of‌ order at most 17‌​‌ has dichromatic number at​​ most 2.

This work​​​‌ has been done in‌ collaboration with Pierre Aboulker‌​‌ [ENS, Paris, France], Stéphane​​ Bessy [LIRMM, Université de​​​‌ Montpellier, France], Daniel Gonçalves‌ [ LIRMM, Montpellier, France],‌​‌ Ken-Ichi Kawarabayashi [National Institute​​ of Informatics and University​​​‌ of Tokyo, Japan], François‌ Pirot [LISN, Université Paris-Saclay,‌​‌ Fance], Amadeus Reinald [​​ LIRMM, Montpellier, France], and​​​‌ Juliette Schabanel [LaBRI, Université‌ de Bordeaux, France].

Semi-proper‌​‌ orientations of dense graphs​​

An orientation$D$ of​​​‌ a graph $G$ is‌ a digraph obtained from‌​‌ $G$ by replacing each​​ edge by exactly one​​​‌ of the two possible‌ arcs with the same‌​‌ ends. An orientation $\mathrm{D}$ of a graph $\mathrm{G‌}$ is a $k$-orientation‌ if the in-degree of‌​‌ each vertex in $\mathrm{D}$ is at most $\mathrm{k‌}$. An orientation $\mathrm{D‌}$ of $G$ is proper‌​‌ if any two adjacent​​ vertices have different in-degrees​​​‌ in $D$. The‌ proper orientation number of‌​‌ a graph $G$,​​ denoted by $\stackrel{\to ‌}{\chi }\left(G\right)$,‌ is the minimum $\mathrm{k‌‌}$ such that $G$ has​​ a proper $k$-orientation.​​​‌ A weighted orientation of‌ a graph $G$ is‌​‌ a pair $(\mathrm{D‌},w)$,​ where $D$ is an​‌ orientation of $G$ and​​ $w$ is an arc-weighting​​​‌ $A(D)\to \mathbb{N}\setminus \{‌0\}$. A​​ semi-proper orientation of $\mathrm{G‌}$ is a weighted orientation​ $(D,\mathrm{w‌})$ of $G$ such​​ that for every two​​​‌ adjacent vertices $u$ and​ $v$ in $G$,​‌ we have that ${\mathrm{S}}_{(D,\mathrm{w‌})}(v)\ne S_{(\mathrm{D‌},w)}(u)$, where​​​‌ $S_{(D,w)}\left(\mathrm{v‌}\right)$ is the sum​​ of the weights of​​​‌ the arcs in $(D,w)‌$ with head $v$.​​ For a positive integer​​​‌ $k$, a semi-proper​ $k$-orientation$(\mathrm{D‌},w)$ of​​ a graph $G$ is​​​‌ a semi-proper orientation of​ $G$ such that ${max‌}_{v\in V(G)}{S}_{(‌D,w)}\left(v\right)\le ‌k$. The semi-proper​​ orientation number of a​​​‌ graph $G$, denoted​ by $\stackrel{\to ‌}{{\chi }_s}\left(G\right)$,​​ is the least $\mathrm{k‌}$ such that $G$ has​ a semi-proper $k$-orientation.​‌ In 22, we​​ first prove that $\overrightarrow{{\mathrm{\chi ‌}}_s}\left(\mathrm{G}\right)\in \{\mathrm{\omega ‌}\left(G\right)-1,\omega (‌G)\}$ for​ every split graph $\mathrm{G‌}$, and that, given​​ a split graph $\mathrm{G‌}$, deciding whether $\overrightarrow{{\mathrm{\chi }}_s}\left(\mathrm{G‌}\right)=\omega (G)-\mathrm{1‌}$ is an NP-complete problem.​ We also show that,​‌ for every $k$,​​ there exists a (chordal)​​​‌ graph $G$ and a​ split subgraph $H$ of​‌ $G$ such that $\overrightarrow{\mathrm{\chi }}(G)‌\le k$ and $\overrightarrow{\mathrm{\chi }}(H)‌=2k-2$. In the​​​‌ sequel, we show that,​ for every $n\ge ‌p(p+1)$, $\overrightarrow{{\mathrm{\chi ‌}}_s}\left({\mathrm{P}}_n^p\right)=‌⌈\frac{3}{2}p⌉$,​​ where $P_n^{\mathrm{p‌}}$ is the $p^{\mathrm{t}h}$ power of the​‌ path on $n$ vertices.​​ We investigate further unit​​​‌ interval graphs with no​ big clique: we show​‌ that $\overrightarrow{\chi }(G)\le \mathrm{3‌}$ for any unit interval​ graph $G$ with $\mathrm{\omega ‌}\left(G\right)=3$, and present​​​‌ a complete characterization of​ unit interval graphs with​‌ $\overrightarrow{\chi }\left(\mathrm{G}\right)=\omega (‌G)=\mathrm{3}$. Then, we show​‌ that deciding whether $\overrightarrow{{\mathrm{\chi }}_s}\left(\mathrm{G‌}\right)=\omega (G)-\mathrm{1‌}$ can be solved in​​ polynomial time in the​​​‌ class of cobipartite graphs.​ Finally, we prove that​‌ computing $\stackrel{\to }{{\chi }_s}\left(G\right)$ is​​​‌ fixed-parameter tractable (FPT) when​ parameterized by the minimum​‌ size of a vertex​​ cover in $G$ or​​​‌ by the treewidth of​ $G$. We also​‌ prove that not only​​ computing $\stackrel{\to ‌}{{\chi }_s}\left(G\right)$,​ but also $\stackrel{\to ‌}{\chi }\left(G\right)$,​​ admits a polynomial kernel​​ when parameterized by the​​​‌ neighborhood diversity plus the‌ value of the solution.‌​‌ These results imply kernels​​ of size $4^{\mathrm{𝒪‌}(k^2)‌}$ and $𝒪({\mathrm{2‌‌}}^k{k}^2)$, in chordal graphs​​​‌ and split graphs, respectively,‌ for the problem of‌​‌ deciding whether $\overrightarrow{{\chi }_{\mathrm{s}}}(G)‌\le k$ parameterized by‌ $k$. We also‌​‌ present exponential kernels for​​ computing both $\stackrel{\to ‌}{\chi }\left(G\right)$ and‌ $\overrightarrow{{\chi }_s}(‌‌G)$ parameterized by​​ the value of the​​​‌ solution when $G$ is‌ a cograph. On the‌​‌ other hand, we show​​ that computing $\overrightarrow{{\chi }_{\mathrm{s‌}}}(G)‌$ does not admit a‌​‌ polynomial kernel parameterized by​​ the value of the​​​‌ solution when $G$ is‌ a chordal graph, unless‌​‌ NP$\subseteq$ coNP/poly.

This​​ work has been done​​​‌ in collaboration with Júlio‌ Araújo [Universidade Federal do‌​‌ Ceará, Fortaleza, Brazil], Claudia​​ Linhares Sales [Universidade Federal​​​‌ do Ceará, Fortaleza, Brazil]‌ and Karol Suchan [Universidad‌​‌ Diego Portales, Santiago, Chile]​​ in the context of​​​‌ the SticAm-Sud project GALOP‌ and of the EA‌​‌ Inria CANOE.

Backbone colouring​​ of chordal graphs

A​​​‌ proper $k$-colouring of‌ a graph $G=‌‌(V,\mathrm{E})$ is a function​​​‌ $c:V(‌G)\to \{‌‌1,...,k\}$ such that​​​‌ $c(u)‌\ne c(\mathrm{v‌‌})$ for every edge​​ $uv\in \mathrm{E‌}\left(G\right)$.‌ Given a spanning subgraph‌​‌ $H$ of $G$,​​ a $q$-backbone $\mathrm{k‌}$-colouring of $(\mathrm{G‌},H)$ is‌​‌ a proper $k$-colouring​​ $c$ of $G$ such​​​‌ that $|c(‌u)-\mathrm{c‌‌}\left(v\right)|\ge q$ for every​​​‌ edge $uv\in ‌E(H)‌‌$. The $q$-backbone​​ chromatic number ${\mathrm{BBC}}_{\mathrm{q‌}}(G,\mathrm{H‌})$ is the smallest‌​‌ $k$ for which there​​ exists a $q$-backbone​​​‌ $k$-colouring of $(‌G,H)‌‌$. In their seminal​​ paper, Broersma et al.​​​‌ ask whether, for any‌ chordal graph $G$ and‌​‌ any spanning forest $\mathrm{H}$ of $G$, we​​​‌ have that ${\mathrm{BBC}}_{\mathrm{2‌}}(G,\mathrm{H‌‌})\le \chi (G)+\mathrm{O‌}\left(1\right)$.‌

In 45, 70‌​‌, we first show​​ that this is true​​​‌ as long as $\mathrm{H‌}$ is bipartite and $\mathrm{G‌‌}$ is an interval graph​​ in which each vertex​​​‌ belongs to at most‌ two maximal cliques. We‌​‌ then show that this​​ does not extend to​​​‌ bipartite graphs as backbone‌ by exhibiting a family‌​‌ of chordal graphs $\mathrm{G}$ with spanning bipartite subgraphs​​​‌ $H$ satisfying ${\mathrm{BBC}}_{\mathrm{2‌}}(G,\mathrm{H‌‌})\le 5\mathrm{\chi }\left(G\right)/‌3$. Then, we‌ show that if $\mathrm{G‌‌}$ is chordal and $\mathrm{H}$ has bounded maximum average​​​‌ degree (in particular, if‌ $H$ is a forest),‌​‌ then ${\mathrm{BBC}}_2(G,H)‌\le \chi \left(\mathrm{G‌}\right)+O(‌‌\chi (G)‌)$. We finally​ show that ${\mathrm{BBC}}_{\mathrm{2‌}}(G,\mathrm{H})\le \frac{3}{\mathrm{2‌}}\chi (G)+O\left(\mathrm{1‌}\right)$ holds whenever $\mathrm{G}$ is chordal and $\mathrm{H‌}$ is $C_4$-free.​

This work has been​‌ done in collaboration with​​ Júlio Araújo [Universidade Federal​​​‌ do Ceará, Fortaleza, Brazil]​ in the context of​‌ the EA Inria CANOE.​​

NSD Edge-Colourings with Strong​​​‌ Assignment Constraints

In 74​, we introduce and​‌ study a combination of​​ two types of graph​​​‌ edge-colourings, namely strong edge-colourings​ (in which every two​‌ edges at distance at​​ most 2 must be​​​‌ assigned distinct colours) and​ neighbor-sum-distinguishing edge-colourings (in which​‌ adjacent edges must be​​ assigned distinct colours, and​​​‌ every two adjacent vertices​ must be incident to​‌ distinct sums of colours).​​ In particular, we investigate​​​‌ how the smallest number​ of colours in such​‌ combined edge-colourings behaves. For​​ several classes of graphs,​​​‌ we prove that such​ edge-colourings are very close​‌ from strong edge-colourings, in​​ the sense that no​​​‌ additional colours are required.​ For others classes of​‌ graphs, we prove that​​ introducing more colours is​​​‌ sometimes necessary. We conjecture​ that, in general, designing​‌ this new type of​​ edge-colourings should always be​​​‌ possible provided we are​ allowed to introduce and​‌ assign a constant number​​ of additional colours. We​​​‌ prove this conjecture for​ a few classes of​‌ graphs, including trees and​​ graphs of bounded maximum​​​‌ degree.

This work has​ been done in collaboration​‌ with Leandro Montero [LS2N​​ and IMT Atlantique, Nantes,​​​‌ France].

Arbitrarily Partitionable Graphs​

In 24, 25​‌, we pursue our​​ investigations on so-called arbitrarily​​​‌ partitionable graphs, being those​ graphs, answering a practical​‌ network sharing problem, that​​ can be partitioned into​​​‌ arbitrarily many connected subgraphs​ with arbitrary orders. In​‌ particular, we wonder about​​ generalizations to partitioning other/more​​​‌ elements into connected subgraphs,​ namely edges and both​‌ vertices and edges, respectively.​​ In each case, we​​​‌ wonder about similarities and​ discrepancies with the vertex-only​‌ case, leading to interesting​​ questions and problems.

This​​​‌ work has been done​ in collaboration with Olivier​‌ Baudon [LaBRI, Université de​​ Bordeaux, France] and Lyn​​​‌ Vayssieres [IREM Aquitaine, Université​ de Bordeaux, France].

Problems, Proofs,‌ and Disproofs on the‌​‌ Inversion Number

The inversion​​ of a set $\mathrm{X‌}$ of vertices in a‌ digraph $D$ consists in‌​‌ reversing the direction of​​ all arcs of $\mathrm{D‌}\langle X\rangle$.‌ The inversion number of‌​‌ an oriented graph $\mathrm{D}$, denoted by $\mathrm{i‌}nv\left(\mathrm{D‌}\right)$, is the‌​‌ minimum number of inversions​​​‌ needed to transform $\mathrm{D}$ into an acyclic oriented​‌ graph. We study a​​ number of problems involving​​​‌ the inversion number of​ oriented graphs. Firstly, in​‌ 23, we give​​ bounds on $i\mathrm{n‌}v(n)$, the maximum of​‌ the inversion numbers of​​ the oriented graphs of​​​‌ order $n$. We​ show $n-\sqrt{(‌nlogn}\le inv(‌n)\le \mathrm{n}-\lceil log(‌n+1)\rceil$. Secondly, we​​​‌ disprove a conjecture of​ Bang-Jensen et al. asserting​‌ that, for every pair​​ of oriented graphs $\mathrm{L‌}$ and $R$, we​ have $in\mathrm{v‌}(L\to \mathrm{R})=i\mathrm{n‌}v(L)+in\mathrm{v‌}(R)$,​​ where $L\to \mathrm{R‌}$ is the oriented graph​ obtained from the disjoint​‌ union of $L$ and​​ $R$ by adding all​​​‌ arcs from $L$ to​ $R$. Finally, we​‌ investigate whether, for all​​ pairs of positive integers​​​‌ $k_1,{\mathrm{k}}_2$, there exists​‌ an integer $f(k_1,{\mathrm{k‌}}_2)$ such that​ if $D$ is an​‌ oriented graph with $\mathrm{i}nv\left(\mathrm{D‌}\right)\ge f(k_1,{\mathrm{k‌}}_2)$ then there​​ is a partition $(‌V_1,{\mathrm{V}}_2)$ of $\mathrm{V‌}\left(D\right)$ such​​ that $in\mathrm{v‌}\left(D\langle {\mathrm{V}}_i\rangle \right)\ge ‌k_i$ for $\mathrm{i}=1,\mathrm{2‌}$. We show that​ $f(1,‌k)$ exists and​​ $f(1,‌k)\le \mathrm{k}+10$ for all​‌ positive integers $k$.​​ Further, we show that​​​‌ $f(k_{\mathrm{1}},k_2)‌$ exists for all pairs​​ of positive integers ${\mathrm{k‌}}_1,k_{\mathrm{2}}$ when the oriented graphs​‌ in consideration are restricted​​ to be tournaments.

In​​​‌ 39, we study​ ${\mathrm{sinv}}_k^{\text{'}}(‌D)$ (resp. ${\mathrm{sinv}}_k(D)‌$) which is (for​ some positive integer $\mathrm{k‌}$) the minimum number​​ of inversions needed to​​​‌ transform $D$ into a​ $k$-arc-strong (resp. $\mathrm{k‌}$-strong) digraph or $+\infty$ if no such​​​‌ transformation exists. Note that​ ${\mathrm{sinv}}_k^{\text{'}}(‌D)\le {\mathrm{sinv}}_k(D)‌$. We set ${\mathrm{sinv}}_k^{\text{'}}\left(\mathrm{n‌}\right)=max\{{\mathrm{sinv}}_k^{\text{'}}(‌D)\mid \mathrm{D}\text{is}\text{a}2\mathrm{k‌}\text{-edge-connected}\text{digraph}\text{of}\text{order}n\}$. We​​​‌ show the following results​ where $k$ is a​‌ fixed integer for $(i)-(‌vi)$:​

• (i)
  $\frac{1}{2}log‌(n-\mathrm{k}+1)\le ‌{\mathrm{sinv}}_k^{\text{'}}(n)\le log‌n+4\mathrm{k}-3$ for every​​​‌ $n\ge k$;​
• (ii)
  for any fixed​‌ positive integer $t$,​​ deciding whether a given​​​‌ oriented graph $D$ with​ ${\mathrm{sinv}}_k^{\text{'}}(‌D)<+\infty$ satisfies ${\mathrm{sinv}}_{\mathrm{k}}^{\text{'}}(D)‌\le t$ is NP-complete;‌
• (iii)
  for any fixed‌​‌ positive integer $t$,​​ deciding whether a given​​​‌ oriented graph $D$ with‌ ${\mathrm{sinv}}_k\left(\mathrm{D‌‌}\right)<+\mathrm{\infty }$ satisfies ${\mathrm{sinv}}_k(‌D)\le \mathrm{t‌}$ is NP-complete;
• (iv)
  if‌​‌ $T$ is a tournament​​ of order at least​​​‌ $2k+\mathrm{1‌}$, then ${\mathrm{sinv}}_{\mathrm{k‌‌}}\left(T\right)\le 2k$, and​​​‌ ${\mathrm{sinv}}_k^{\text{'}}(‌T)\le \frac{\mathrm{4‌‌}}{3}k+\mathrm{o}\left(k\right)$;​​​‌
• (v)
  $\frac{1}{2}log‌(2k+‌‌1)\le {\mathrm{sinv}}_k^{\text{'}}\left(\mathrm{T‌}\right)$ for some tournament‌ $T$ of order $\mathrm{2‌‌}k+1$;​​
• (vi)
  if $T$ is​​​‌ a tournament of order‌ at least $19\mathrm{k‌‌}-2$ (resp. $\mathrm{11}k-2$),​​​‌ then ${\mathrm{sinv}}_k(‌T)\le \mathrm{1‌‌}$ (resp. ${\mathrm{sinv}}_k(T)\le \mathrm{3‌}$);
• (vii)
  for every‌ $ϵ>0$,‌​‌ there exists $C$ such​​ that for every positive​​​‌ integer $k$ and every‌ tournament $T$ on at‌​‌ least $2k+1+ϵ\mathrm{k‌}$ vertices, we have ${\mathrm{sinv‌}}_k(T)‌‌\le C$.

This​​ work has been done​​​‌ in collaboration with Guillaume‌ Aubian [IRIF, Université Paris-Cité,‌​‌ Paris, France], Julien Duron​​ [LIP, ENS Lyon, France],​​​‌ Florian Hörsch [CISPA, Helmholtz‌ Center for Information Security,‌​‌ Saarbrücken, Germany], Felix Kingelhoefer​​ [G-SCOP, Université Grenoble Alpes,​​​‌ France] and Quentin Vermande‌ [STAMP] in the context‌​‌ of the ANR Digraphs.​​

$k$-shortest simple paths​​​‌ in bounded treewidth graphs​

The $k$-shortest simple​‌ paths problem asks to​​ compute a set of​​​‌ top-$k$ shortest simple​ paths from a source​‌ to a sink in​​ a graph $G=‌(V,\mathrm{E})$ with $\left|\mathrm{V‌}\right|=n$ vertices​​ and $|E|‌=m$ edges. The​ most well-known algorithm for​‌ solving this problem is​​ due to Yen (1971)​​​‌ with time complexity in​ $O\left(k\mathrm{n‌}\right(m+\mathrm{n}logn))‌$ and the fastest algorithm​ is due to Gotthilf​‌ and Lewenstein (2009) with​​ time complexity in $\mathrm{O‌}(kn(m+nlog‌logn))$. For bounded treewidth​​​‌ graphs, Eppstein and Kurz​ (2017) lowered the computational​‌ complexity to $O(kn)$ by​​​‌ retrieving paths from the​ $k$ smallest solutions of​‌ a monadic second-order formula,​​ and to $O(‌n+klog\left(n\right))‌$ to retrieve the $\mathrm{k}$ shortest simple distances only.​​​‌ In 36, we​ provide an algorithm that​‌ answers $k$-shortest simple​​ distances in $O(‌k+n)$ time on graphs with​‌ treewidth at most 2,​​ and a constructive algorithm,​​​‌ simpler than that of​ Eppstein and Kurz, that​‌ solves the $k$-shortest​​ simple paths problem in​​​‌ $O\left(k\mathrm{n}\right)$ time on bounded​‌ treewidth graphs.

This is​​ a joint work with​​​‌ Andrea d'Ascenzo [Luiss University,​ Rome, Italy].

New lower​‌ bounds on the cutwidth​​ of graphs

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 33‌, we give new‌​‌ lower bounds based on​​ different graph density parameters​​​‌ such as the minimum,‌ the average and the‌​‌ maximum average degree. Our​​ main result is a​​​‌ new bound using the‌ notion of traffic grooming‌​‌ on a path network,​​ which appears to be​​​‌ in many cases better‌ than bounds in the‌​‌ literature. Furthermore, the bound​​ based on grooming can​​​‌ be computed quickly, in‌ $O(log\mathrm{n‌‌})$ time, and so​​ is of interest to​​​‌ design faster branch-and-bound algorithms.‌ Through extensive experiments, we‌​‌ show that this bound​​ behaves very well compared​​​‌ to other bounds. Furthermore,‌ we show how to‌​‌ obtain even better results​​ when combining it with​​​‌ heuristics for finding dense‌ subgraphs.

On the rank‌​‌ and the general position​​ number in cycle convexity​​​‌

Graph convexities have been‌ widely investigated through various‌​‌ combinatorial parameters that capture​​ different aspects of convexity-related​​​‌ structures. The cycle convexity‌ has been introduced due‌​‌ to its applications in​​ the field of Knot​​​‌ Theory. In the cycle‌ convexity, the interval $\mathrm{I‌‌}\left(S\right)$ of​​ a subset $S$ of​​​‌ vertices of a graph‌ $G=(\mathrm{V‌‌},E)$ is​​ the set of all​​​‌ vertices in $S$ union‌ the vertices with two‌​‌ neighbors in a same​​ connected component of $\mathrm{G‌}\left[S\right]$.‌ The hull of $\mathrm{S‌‌}$ is the closure $\mathrm{H}\left(S\right)=‌I^{*}\left(\mathrm{S‌}\right)$, i.e., the‌​‌ set of vertices that​​ can be obtained from​​​‌ $S$ by recursively adding‌ to $S$ any vertex‌​‌ that is the neighbor​​ of two vertices in​​​‌ a same connected component‌ of $S$. In‌​‌ any graph convexity, it​​ is classical to study​​​‌ the minimum size of‌ a hull set $\mathrm{S‌‌}$ of $G$, i.e.,​​ a minimum set such​​​‌ that $H\left(\mathrm{S‌}\right)=V$.‌​‌ Previous works on other​​ convexities also study the​​​‌ general position number of‌ $G$, i.e., the‌​‌ maximum size of a​​ set $S$ such that​​​‌ $v\notin I(‌S\setminus \left\{\mathrm{v‌‌}\right\})$ for every​​ $v\in S\subseteq ‌V(G)‌$, and the rank‌​‌ of $G$, i.e.,​​ the maximum size of​​​‌ a set $S$ such‌ that $v\notin \mathrm{H‌‌}(S\setminus \{v\})$ for​​​‌ every $v\in \mathrm{S‌}\subseteq V\left(\mathrm{G‌‌}\right)$. In 68​​, we first show​​​‌ that the general position‌ number of $G$ in‌​‌ the cycle convexity equals​​ the maximum order of​​​‌ an induced forest in‌ $G$. Then, we‌​‌ focus on the computational​​ complexity of computing the​​​‌ rank of a given‌ $G$ in the cycle‌​‌ convexity. We show that​​ it is NP-hard, even​​​‌ if the input graph‌ is known to be‌​‌ bipartite, and W[1]-hard parameterized​​​‌ by the size of​ the solution. Then, we​‌ prove that it is​​ polynomial-time computable in various​​​‌ graph classes such as​ forests, cycles, complete graphs,​‌ complete bipartite graphs, cographs,​​ Cartesian grids, chordal graphs,​​​‌ starlike graphs, and (​$q,q-‌4)$-graphs. Surprisingly,​​ the classes of Cartesian​​​‌ grids and of not​ 2-connected chordal graphs are​‌ quite involved. We conclude​​ by showing fixed-parameter tractable​​​‌ algorithms for computing the​ rank of a given​‌ graph when parameterized by​​ the neighborhood diversity, by​​​‌ vertex cover, or by​ the treewidth of the​‌ input graph.

This work​​ has been done in​​​‌ collaboration with Júlio Aráujo​ [UFC Fortaleza, Brazil] in​‌ the context of the​​ EA CANOE.

The General​​​‌ Expiration Streaming Model: Diameter,​ $k$-Center, Counting, Sampling,​‌ and Friends

An important​​ thread in the study​​​‌ of data-stream algorithms focuses​ on settings where stream​‌ items are active only​​ for a limited time.​​​‌ In 78, we​ introduce a new expiration​‌ model, where each item​​ arrives with its own​​​‌ expiration time. The special​ case where items expire​‌ in the order that​​ they arrive, which we​​​‌ call consistent expirations, contains​ the classical sliding-window model​‌ of Datar, Gionis, Indyk,​​ and Motwani [SICOMP 2002]​​​‌ and its timestamp-based variant​ of Braverman and Ostrovsky​‌ [FOCS 2007]. Our first​​ set of results presents​​​‌ algorithms (in the expiration​ streaming model) for several​‌ fundamental problems, including approximate​​ counting, uniform sampling, and​​​‌ weighted sampling by efficiently​ tracking active items without​‌ explicitly storing them all.​​ Naturally, these algorithms have​​​‌ many immediate applications to​ other problems. Our second​‌ and main set of​​ results designs algorithms (in​​​‌ the expiration streaming model)​ for the diameter and​‌ $k$-center problems, where​​ items are points in​​​‌ a metric space. Our​ results significantly extend those​‌ known for the special​​ case of sliding-window streams​​​‌ by Cohen-Addad, Schwiegelshohn, and​ Sohler [ICALP 2016], including​‌ also a strictly better​​ approximation factor for the​​​‌ diameter in the important​ special case of high-dimensional​‌ Euclidean space. We develop​​ new decomposition and coordination​​​‌ techniques along with a​ geometric dominance framework, to​‌ filter out redundant points​​ based on both temporal​​​‌ and spatial proximity.

This​ work has been done​‌ in collaboration with Lotte​​ Blank [University of Bonn,​​​‌ Germany], Sergio Cabello [University​ of Ljubljana, Slovenia], Mohammadtaghi​‌ Hajiaghayi [University of Maryland,​​ USA], Robert Krauthgamer [Weizmann​​​‌ Institute of Science, Rehovot,​ Israel], Sepideh Mahabadi [Microsoft​‌ Research, Redmond, USA], Jeff​​ Phillips [University of Utah,​​​‌ Salt Lake City, USA]​ and Jonas Sauer [University​‌ of Bonn, Germany].

Teoria dos Jogos​ Combinatórios em Grafos

The​‌ book 63 has been​​ written in the context​​​‌ of the 35th Brazilian​ Mathematics Colloquium (Rio de​‌ Janeiro, July 27-August 1st,​​ 2025). It is intended​​​‌ to be a textbook​ on Combinatorial games in​‌ graphs for a public​​ from highschool to University.​​​‌ It has been done​ in collaboration with Nicolas​‌ Martins [UNILAB - Universidade​​ da Integração Internacional da​​ Lusofonia Afro-Brasileira, Brazil] and​​​‌ Rudini Sampaio [UFC Fortaleza,‌ Brazil] in the context‌​‌ of the EA CANOE.​​

The Convex Set Forming​​​‌ Game

In 1984, Frank‌ Harary introduced the first‌​‌ graph convexity game, focused​​ on the geodesic convexity.​​​‌ A set $S\subseteq ‌V$ of vertices of‌​‌ a graph $G=(V,\mathrm{E‌})$ is convex if‌ every shortest path between‌​‌ two vertices of $\mathrm{S}$ is also included in​​​‌ $S$. In 35‌, we introduce the‌​‌ Convex Set Forming Game​​ CFG: two players alternately​​​‌ select vertices in such‌ a way that the‌​‌ set of selected vertices​​ is always a convex​​​‌ set. In the normal‌ (resp., misère) variant, the‌​‌ last player to be​​ able to select a​​​‌ vertex wins (resp., loses).‌ We also define a‌​‌ new graph invariant $\mathrm{gc}\left(G\right)$ as​​​‌ the largest integer $\mathrm{k‌}$ such that the first‌​‌ player has a strategy​​ ensuring that, at the​​​‌ end of the game,‌ at least $k$ vertices‌​‌ of the graph $\mathrm{G}$ have been selected. We​​​‌ first show that the‌ problems of deciding the‌​‌ outcome (does the first​​ player win?) of the​​​‌ game in both variants‌ (normal and misère), as‌​‌ well as the problem​​ of deciding whether $\mathrm{gc‌}\left(G\right)\ge ‌k$, are PSPACE-complete.‌​‌ As a by-product, we​​ prove that the optimization​​​‌ variant of the classical‌ Kayles game is PSPACE-complete.‌​‌ Then, we focus on​​ convexable graphs, i.e., $\mathrm{n‌}$-node graphs $G$ for‌ which $\mathrm{gc}\left(\mathrm{G‌‌}\right)=n$.​​ For this purpose, we​​​‌ say that a set‌ $S=\{{\mathrm{v‌‌}}_1,\cdots ,v_{|S|‌}\}\subseteq V$ in‌ a graph $G$ admits‌​‌ a Convex Elimination Ordering​​ (CEO) if $\{{\mathrm{v‌}}_1,\cdots ,‌v_i\}$ is‌​‌ convex for every $\mathrm{1}\le i\le |‌S|$. We‌ show that the class‌​‌ of graphs whose vertex-set​​ admits a CEO coincides​​​‌ with the chordal graphs‌ and that this class‌​‌ strictly contains the convexable​​ graphs. Moreover, every graph​​​‌ which is Ptolemaic (distance-hereditary‌ chordal) or unit interval‌​‌ is convexable. Finally, we​​ give a polynomial-time algorithm​​​‌ for computing a largest‌ set admitting a CEO‌​‌ in outerplanar graphs, which​​ gives upper bounds on​​​‌ $\mathrm{gc}(G)‌$ in outerplanar graphs $\mathrm{G‌‌}$.

This is a​​ joint work with Caroline​​​‌ Brosse [LIFO, Université d'Orléans,‌ France], Nicolas Martins [UNILAB‌​‌ - Universidade da Integração​​ Internacional da Lusofonia Afro-Brasileira,​​​‌ Brazil] and Rudini Sampaio‌ [UFC Fortaleza, Brazil] in‌​‌ the context of the​​ CANOE associated team.

The​​​‌ Graph colouring Game in‌ $4\times n$-Grids‌​‌

The graph colouring game​​ is a famous two-player​​​‌ game (re)introduced by Bodlaender‌ in 1991. Given a‌​‌ graph $G$ and $\mathrm{k}\in \mathbb{N}$, Alice​​​‌ and Bob alternately (starting‌ with Alice) colour an‌​‌ uncoloured vertex with some​​ colour in $\{\mathrm{1‌},\cdots ,\mathrm{k‌}\}$ such that no‌​‌ two adjacent vertices receive​​ a same colour. If​​​‌ eventually all vertices are‌ coloured, then Alice wins‌​‌ and Bob wins otherwise.​​​‌ The game chromatic number​ ${\chi }_g\left(\mathrm{G‌}\right)$ is the smallest​​ integer $k$ such that​​​‌ Alice has a winning​ strategy with $k$ colours​‌ in $G$. It​​ has been recently (2020)​​​‌ shown that, given a​ graph $G$ and $\mathrm{k‌}\in \mathbb{N}$, deciding​​ whether ${\chi }_g(‌G)\le \mathrm{k}$ is PSPACE-complete. Surprisingly, this​‌ parameter is not well​​ understood even in “simple”​​​‌ graph classes. Let ${\mathrm{P}}_n$ denote the path​‌ with $n\ge \mathrm{1}$ vertices. For instance, in​​​‌ the case of Cartesian​ grids, it is easy​‌ to show that ${\mathrm{\chi }}_g(P_{\mathrm{m‌}}\square P_n)\le 5$ since ${\mathrm{\chi ‌}}_g(G)\le \Delta +\mathrm{1‌}$ for any graph $\mathrm{G}$ with maximum degree $\mathrm{\Delta ‌}$ (here $\square$ denotes the​​ Cartesian product of two​​​‌ graphs). However, the exact​ value is only known​‌ for small values of​​ $m$, namely ${\mathrm{\chi ‌}}_g(P_{\mathrm{1}}\square P_n)‌=3$, ${\mathrm{\chi }}_g(P_{\mathrm{2‌}}\square P_n)=4$ and ${\mathrm{\chi ‌}}_g(P_{\mathrm{3}}\square P_n)‌=4$ for $\mathrm{n}\ge 4$ 99.​‌ In 48, we​​ prove that, for every​​​‌ $n\ge 18$,​ ${\chi }_g({\mathrm{P‌}}_4\square P_{\mathrm{n}})=4$.​​​‌

This is a joint​ work with Caroline Brosse​‌ [LIFO, Université d'Orléans, France],​​ Nicolas Martins [UNILAB -​​​‌ Universidade da Integração Internacional​ da Lusofonia Afro-Brasileira, Brazil]​‌ and Rudini Sampaio [UFC​​ Fortaleza, Brazil] in the​​​‌ context of the CANOE​ associated team.

The Closed​‌ Hull Game and the​​ Closed Interval Game

Given​​​‌ a set S of​ vertices in a graph​‌ $G$, its geodesic​​ interval is the set​​​‌ $I(S)$ containing $S$ and all​‌ vertices on a shortest​​ path between vertices of​​​‌ $S$. A set​ $S$ is convex if​‌ $I(S)=S$. Moreover,​​​‌ the convex hull $\mathrm{H}\left(S\right)$ of​‌ $S$ is the smallest​​ convex set containing $\mathrm{S‌}$. In 1984, Harary​ introduced convexity games where​‌ two players, Alice and​​ Bob, alternately select vertices​​​‌ of a graph $\mathrm{G}=(V,‌E)$ such that,​​ if the set of​​​‌ already selected vertices is​ $S$, the next​‌ player can only select​​ a vertex in $\mathrm{V‌}\setminus I\left(\mathrm{S}\right)$ (closed interval game)​‌ or in $V\setminus H(S)‌$ (closed hull game). Normal​ and misère version of​‌ these games have been​​ studied and we introduce​​​‌ in 69 the optimization​ variants of them. Formally,​‌ given a graph $\mathrm{G}$ and $k\in \mathrm{N‌}$, Alice wins if​ the game ends after​‌ at most $k$ vertices​​ have been selected and​​​‌ Bob wins otherwise. The​ corresponding problem consists of​‌ determining which player has​​ a winning strategy. In​​​‌ 69, we prove​ that the closed interval​‌ optimization game is PSPACE-complete​​ in graphs with diameter​​​‌ 4 and that the​ closed hull optimization game​‌ is NP-hard in bipartite​​ graphs and in split​​ graphs. On the positive​​​‌ side, we prove that‌ both games can be‌​‌ solved in polynomial time​​ in trees and that​​​‌ the closed hull optimization‌ game can be solved‌​‌ in polynomial time in​​ cobipartite graphs. We conjecture​​​‌ that the closed interval‌ optimization game is NP-hard‌​‌ in cobipartite graphs and​​ that the closed hull​​​‌ optimization game is PSPACE-complete‌ in general graphs.

This‌​‌ is a joint work​​ with Fabrício Benevides [UFC​​​‌ Fortaleza, Brazil], Nicolas Martins‌ [UFC Fortaleza, Brazil] and‌​‌ Rudini Sampaio [UFC Fortaleza,​​ Brazil] in the context​​​‌ of the CANOE associated‌ team.

Complexity of Maker-Breaker‌​‌ Games on Edge Sets​​ of Graphs

In 37​​​‌, we initiate the‌ study of the algorithmic‌​‌ complexity of Maker-Breaker games​​ played on edge sets​​​‌ of graphs for general‌ graphs. We mainly consider‌​‌ three of the big​​ four such games: the​​​‌ connectivity game, perfect matching‌ game, and $H$-game.‌​‌ Maker wins if she​​ claims the edges of​​​‌ a spanning tree in‌ the first, a perfect‌​‌ matching in the second,​​ and a copy of​​​‌ a fixed graph $\mathrm{H‌}$ in the third. We‌​‌ prove that deciding who​​ wins the perfect matching​​​‌ game and the $\mathrm{H‌}$-game is PSPACE-complete, even‌​‌ for the latter in​​ graphs of small diameter​​​‌ if $H$ is a‌ tree. Seeking to find‌​‌ the smallest graph $\mathrm{H}$ such that the $\mathrm{H‌}$-game is PSPACE-complete, we‌ also prove that there‌​‌ exists such an $\mathrm{H}$ of order 51 and​​​‌ size 57. On the‌ positive side, we show‌​‌ that the connectivity game​​ and arboricity-$k$ game​​​‌ are polynomial-time solvable. We‌ then give several positive‌​‌ results for the $\mathrm{H}$-game, first giving a​​​‌ structural characterization for Breaker‌ to win the P‌​‌ 4-game, which gives a​​ linear-time algorithm for the​​​‌ $P_4$-game. We‌ provide a structural characterization‌​‌ for Maker to win​​ the $K_{1,‌\ell }$-game in trees,‌ which implies a linear-time‌​‌ algorithm for the ${\mathrm{K}}_{1,\ell }$-game​​​‌ in trees. Lastly, we‌ prove that the ${\mathrm{K‌‌}}_{1,\ell }$-game​​ in any graph, and​​​‌ the $H$-game in‌ trees are both FPT‌​‌ parameterized by the length​​ of the game. We​​​‌ leave the complexity of‌ the last of the‌​‌ big four games, the​​ Hamiltonicity game, as an​​​‌ open question.

This is‌ a joint work with‌​‌ Eric Duchêne [LIRIS, Lyon],​​ Valentin Gledel [LAMA, Chambery],​​​‌ Fionn Mc Inerney [Telefonica,‌ Barcelona, Spain], Nacim Oijid‌​‌ [Umea University, Sweden], Aline​​ Parreau [CNRS, LIRIS, Lyon]​​​‌ and Miloš Stojaković [University‌ of Novi Sad, Serbia]‌​‌ in the context of​​ the ANR P-Gase.

Algorithm Engineering​ of SSSP with Negative​‌ Edge Weights

Computing shortest​​ paths is one of​​​‌ the most fundamental algorithmic​ graph problems. It is​‌ known since decades that​​ this problem can be​​​‌ solved in near-linear time​ if all weights are​‌ nonnegative. A recent break-through​​ by Aaron Bernstein et​​​‌ al.  93 presented a​ randomized near-linear time algorithm​‌ for this problem. A​​ subsequent improvement in  94​​​‌ significantly reduced the number​ of logarithmic factors and​‌ thereby also simplified the​​ algorithm. It is surprising​​​‌ and exciting that both​ of these algorithms are​‌ combinatorial and do not​​ contain any fundamental obstacles​​​‌ for being practical. In​ 50, we launch​‌ the, to the best​​ of our knowledge, first​​​‌ extensive investigation towards a​ practical implementation of  94​‌. To this end,​​ we give an accessible​​​‌ overview of the algorithm​ and discuss what adaptions​‌ are necessary to obtain​​ a fast algorithm in​​​‌ practice. We manifest these​ adaptions in an efficient​‌ implementation. We test our​​ implementation on a benchmark​​​‌ data set that is​ adapted to be more​‌ difficult for our implementation​​ in order to allow​​​‌ for a fair comparison.​ As in  94 as​‌ well as in our​​ implementation there are multiple​​​‌ parameters to tune, we​ empirically evaluate their effect​‌ and thereby determine the​​ best choices. Our implementation​​​‌ is then extensively compared​ to one of the​‌ state-of-the-art algorithms for this​​ problem  96. On​​​‌ the hardest instance type,​ we are faster by​‌ up to almost two​​ orders of magnitude.

This​​​‌ work has been done​ in collaboration with Alejandro​‌ Cassis [MPII and Saarland​​ University Saarbrücken, Germany], Andreas​​​‌ Karrenbauer [MPII and Saarland​ University Saarbrücken, Germany] and​‌ Paolo Luigi Rinaldi [MPII​​ and Saarland University Saarbrücken,​​​‌ Germany].

Threshold-Driven​​ Streaming Graph: Expansion and​​​‌ Rumor Spreading

A randomized​ distributed algorithm called RAES​‌ was introduced in  86​​ to extract a bounded-degree​​​‌ expander from a dense​ $n$-vertex expander graph​‌ $G=(\mathrm{V},E)$.​​​‌ The algorithm relies on​ a simple threshold-based procedure.​‌ A key assumption in​​  86 is that the​​​‌ input graph $G$ is​ static – i.e., both​‌ its vertex set $\mathrm{V}$ and edge set $\mathrm{E‌}$ remain unchanged throughout the​ process – while the​‌ analysis of RAES in​​ dynamic models is left​​​‌ as a major open​ question. In 67,​‌ we investigate the behavior​​ of RAES under a​​​‌ dynamic graph model induced​ by a streaming node-churn​‌ process (also known as​​ the sliding window model),​​​‌ where, at each discrete​ round, a new node​‌ joins the graph and​​ the oldest node departs.​​​‌ This process yields a​ bounded-degree dynamic graph $\mathrm{𝒢‌}=\{G_{\mathrm{t}}=(V_{\mathrm{t‌}},E_t):t\in \mathrm{\mathbb{N}‌}\}$ that captures essential​​ characteristics of peer-to-peer networks​​​‌ – specifically, node churn​ and threshold on the​‌ number of connections each​​ node can manage. We​​​‌ prove that every snapshot​ $G_t$ in the​‌ dynamic graph sequence has​​ good expansion properties with​​ high probability. Furthermore, we​​​‌ leverage this property to‌ establish a logarithmic upper‌​‌ bound on the completion​​ time of the well-known​​​‌ PUSH and PULL rumor‌ spreading protocols over the‌​‌ dynamic graph $𝒢$.​​

This work has been​​​‌ done in collaboration with‌ Flora Angileri [University of‌​‌ Rome Tor Vergata, Rome,​​ Italy], Andrea Clementi [University​​​‌ of Rome Tor Vergata,‌ Rome, Italy], Michele Salvi‌​‌ [University of Rome Tor​​ Vergata, Rome, Italy] and​​​‌ Isabella Ziccardi [IRIF, Université‌ Paris-Cité, France].

On the‌​‌ h-majority dynamics with many​​ opinions

In 62,​​​‌ we present the first‌ upper bound on the‌​‌ convergence time to consensus​​ of the well-known $\mathrm{h‌}$-majority dynamics with $\mathrm{k‌}$ opinions, in the synchronous‌​‌ setting, for $h$ and​​ $k$ that are both​​​‌ non-constant values. We suppose‌ that, at the beginning‌​‌ of the process, there​​ is some initial additive​​​‌ bias towards some plurality‌ opinion, that is, there‌​‌ is an opinion that​​ is supported by $\mathrm{x‌}$ nodes while any other‌ opinion is supported by‌​‌ strictly fewer nodes. We​​ prove that, with high​​​‌ probability, if the bias‌ is $\omega \left(\sqrt{\mathrm{x‌‌}}\right)$ and the initial​​ plurality opinion is supported​​​‌ by at least $\mathrm{x‌}=\omega (log‌‌n)$ nodes, then​​ the process converges to​​​‌ plurality consensus in $\mathrm{O‌}(logn)‌‌$ rounds whenever $h=\omega (nlog‌n/x)‌$. A main corollary‌​‌ is the following: if​​ $k=o(‌n/log\mathrm{n‌})$ and the process‌​‌ starts from an almost-balanced​​ configuration with an initial​​​‌ bias of magnitude $\mathrm{\omega ‌}\left(\sqrt{n/\mathrm{k‌‌}}\right)$ towards the initial​​ plurality opinion, then any​​​‌ function $h=\mathrm{\omega ‌}(klog\mathrm{n‌‌})$ suffices to guarantee​​ convergence to consensus in​​​‌ $O(log\mathrm{n‌})$ rounds, with high‌​‌ probability. Our upper bound​​ shows that the lower​​​‌ bound of $\Omega (‌k/h^{\mathrm{2‌‌}})$ rounds to reach​​ consensus given by Becchetti​​​‌ et al.  92 cannot‌ be pushed further than‌​‌ $\tilde{\Omega }(\mathrm{k}/h)$.​​​‌ Moreover, the bias we‌ require is asymptotically smaller‌​‌ than the $\Omega (\sqrt{nlogn})‌$ bias that guarantees plurality‌ consensus in the 3-majority‌​‌ dynamics: in our case,​​ the required bias is​​​‌ at most any (arbitrarily‌ small) function in $\mathrm{\omega ‌‌}\left(\sqrt{x}\right)$ for​​ any value of $\mathrm{k‌}\ge 2$.

This‌ work has been done‌​‌ in collaboration with Francesco​​ d'Amore [Gran Sasso Science​​​‌ Institute, L'Aquila, Italy] and‌ George Giakkoupis [WIDE, Rennes,‌​‌ France].

Quantization​ vs Pruning: Insights from​‌ the Strong Lottery Ticket​​ Hypothesis

Quantization is an​​​‌ essential technique for making​ neural networks more efficient,​‌ yet our theoretical understanding​​ of it remains limited.​​​‌ Previous works demonstrated that​ extremely low-precision networks, such​‌ as binary networks, can​​ be constructed by pruning​​​‌ large, randomly-initialized networks, and​ showed that the ratio​‌ between the size of​​ the original and the​​​‌ pruned networks is at​ most polylogarithmic. The specific​‌ pruning method they employed​​ inspired a line of​​​‌ theoretical work known as​ the Strong Lottery Ticket​‌ Hypothesis (SLTH), which leverages​​ insights from the Random​​​‌ Subset Sum Problem. However,​ these results primarily address​‌ the continuous setting and​​ cannot be applied to​​​‌ extend SLTH results to​ the quantized setting. In​‌ 81, we build​​ on foundational results by​​​‌ Borgs et al.  88​ on the Number Partitioning​‌ Problem to derive new​​ theoretical results for the​​​‌ Random Subset Sum Problem​ in a quantized setting.​‌ Using these results, we​​ then extend the SLTH​​​‌ framework to finite-precision networks.​ While prior work on​‌ SLTH showed that pruning​​ allows approximation of a​​​‌ certain class of neural​ networks, we demonstrate that,​‌ in the quantized setting,​​ the analogous class of​​​‌ target discrete neural networks​ can be represented exactly,​‌ and we prove optimal​​ bounds on the necessary​​​‌ over parameterization of the​ initial network as a​‌ function of the precision​​ of the target network.​​​‌

Improved Learning via $\mathrm{k}$-DTW: A Novel Dissimilarity​‌ Measure for Curves

In​​ 58, we introduce​​​‌ $k$-Dynamic Time Warping​ ($k$-DTW), a​‌ novel dissimilarity measure for​​ polygonal curves. $k$-DTW​​​‌ has stronger metric properties​ than Dynamic Time Warping​‌ (DTW) and is more​​ robust to outliers than​​​‌ the Fréchet distance, which​ are the two gold​‌ standards of dissimilarity measures​​ for polygonal curves. We​​​‌ show interesting properties of​ $k$-DTW and give​‌ an exact algorithm as​​ well as a $(‌1+ϵ)$-approximation algorithm for $\mathrm{k‌}$-DTW by a parametric​​ search for the $\mathrm{k‌}$-th largest matched distance.​ We prove the first​‌ dimension-free learning bounds for​​ curves and further learning​​​‌ theoretic results. $k$-DTW​ not only admits smaller​‌ sample size than DTW​​ for the problem of​​​‌ learning the median of​ curves, where some factors​‌ depending on the curves'​​ complexity $m$ are replaced​​​‌ by $k$, but​ we also show a​‌ surprising separation on the​​ associated Rademacher and Gaussian​​​‌ complexities: $k$-DTW admits​ strictly smaller bounds than​‌ DTW, by a factor​​ $\tilde{\Omega }\left(\sqrt{\mathrm{m‌}}\right)$ when $k\ll m$. We complement​‌ our theoretical findings with​​ an experimental illustration of​​​‌ the benefits of using​ $k$-DTW for clustering​‌ and nearest neighbor classification.​​

This work has been​​​‌ done in collaboration with​ Amer Krivošija [Technische Universität​‌ Dortmund, Germany], Alexander Munteanu​​ [University of Cologne, Germany]​​​‌ and Chris Schwiegelshohn [​ Aarhus University, Denmark].

Solving​‌ the Traveling Salesman Problem​​ with Positional Encoding

In​​ 83, we propose​​​‌ transformer-based neural solvers for‌ the Euclidean Traveling Salesman‌​‌ Problem that rely on​​ positional encodings rather than​​​‌ coordinate projections. By adapting‌ ALiBi and RoPE, modern‌​‌ positional encodings originally developed​​ for large language models,​​​‌ to the Euclidean setting,‌ our Positional Encoding-based Neural‌​‌ Solvers (PENS) inherit useful​​ invariances and locality biases.​​​‌ To address the increased‌ density of large instances,‌​‌ we introduce a simple​​ yet effective rescaling of​​​‌ city coordinates that further‌ boosts performance. Trained only‌​‌ on TSP-100, PENS achieves​​ state-of-the-art results for instances​​​‌ with up to 10‌ 000 cities, a scale‌​‌ that was previously dominated​​ by methods requiring graph​​​‌ sparsification. These findings demonstrate‌ that positional encodings provide‌​‌ effective inductive biases for​​ neural combinatorial optimization.

Temporal​​​‌ graph neural networks

We‌ have contributed to the‌​‌ Julia open source libraries​​ GraphNeuralNetworks.jl and MLDatasets.jl. Our​​​‌ project’s objective was to‌ extend the support of‌​‌ temporal graph neural networks​​ in GraphNeuralNetworks.jl by creating​​​‌ several layers, known as‌ temporal graph convolutional layers.‌​‌ These layers were designed​​ specifically for a type​​​‌ of graph called TemporalSnapshotsGNNGraph‌. Particular emphasis was‌​‌ placed on layers combining​​ graph convolutions with recurrent​​​‌ cells, using the Flux.jl‌ machine learning framework as‌​‌ a reference for implementing​​ the latter. Some implemented​​​‌ layers were the DCRNN‌ layer for traffic prediction,‌​‌ the GConvGRU and GConvLSTM​​ layers, and the EGCN-O​​​‌ layer for tasks such‌ as node and edge‌​‌ classification as well as​​ link prediction. Additionally, we​​​‌ adapted all the implemented‌ temporal layers to work‌​‌ seamlessly with another Julia​​ machine learning framework, Lux.jl,​​​‌ ensuring compatibility across frameworks.‌ Beyond enhancing existing tools,‌​‌ we expanded the range​​ of datasets available in​​​‌ MLDatasets.jl, a Julia package‌ for machine learning datasets,‌​‌ by contributing new datasets.​​ We restructured the repository​​​‌ into a multi-repository setup,‌ created and deployed the‌​‌ multi-package documentation, as detailled​​ in 42. The​​​‌ package is available on‌ GitHub.

This work‌​‌ has been done in​​ collaboraiton with Carlo Lucibello​​​‌ [Bocconi University, Italy].

Characterizing‌ Dynamic Functional Connectivity Subnetwork‌​‌ Contributions in Narrative Classification​​ with Shapley Values

Functional​​​‌ connectivity derived from functional‌ Magnetic Resonance Imaging (fMRI)‌​‌ data has been increasingly​​ used to study brain​​​‌ activity. In 43,‌ we model brain dynamic‌​‌ functional connectivity during narrative​​ tasks as a temporal​​​‌ brain network and employ‌ a machine learning model‌​‌ to classify in a​​ supervised setting the modality​​​‌ (audio, movie), the content‌ (airport, restaurant situations) of‌​‌ narratives, and both combined.​​ Leveraging Shapley values, we​​​‌ analyze subnetwork contributions within‌ Yeo parcellations (7- and‌​‌ 17-subnetworks) to explore their​​ involvement in narrative modality​​​‌ and comprehension. This work‌ represents the first application‌​‌ of this approach to​​ functional aspects of the​​​‌ brain, validated by existing‌ literature, and provides novel‌​‌ insights at the whole-brain​​ level. Our findings suggest​​​‌ that schematic representations in‌ narratives may not depend‌​‌ solely on pre-existing knowledge​​ of the top-down process​​​‌ to guide perception and‌ understanding, but may also‌​‌ emerge from a bottom-up​​ process driven by the​​​‌ temporal parietal subnetwork.

This‌ work has been done‌​‌ in collaboration with Yanis​​​‌ Aeschlimann [CRONOS], Samuel Deslauriers-Gauthier​ [CRONOS] and Peter Ford​‌ Dominey [Inserm, Université de​​ Bourgogne, Dijon, France].

Supervised Classification​ in Federated Learning via​‌ Locality-Sensitive Filters

Federated Learning​​ (FL) can struggle with​​​‌ high communication costs, a​ problem that is only​‌ exacerbated by the fact​​ that, in FL, it​​​‌ is common to have​ heterogeneous data distributions among​‌ parties. Recently, the study​​ of the brain of​​​‌ fruit flies inspired two​ novel ML ideas: a​‌ Locality-Sensitive Hashing (LSH) scheme,​​ called FlyHash, and a​​​‌ data structure for novelty​ detection, called FlyBloomFilter. A​‌ recent study combined these​​ two tools to provide​​​‌ a simple and efficient​ method for performing FL​‌ in a single shot,​​ which is also agnostic​​​‌ to the heterogeneity of​ the data: FlyNN. Yet,​‌ despite their empirical success,​​ theoretical understanding of both​​​‌ Fly-Hash and FlyBloomFilter is​ still limited. In 79​‌, we distil the​​ ideas underlying FlyHash into​​​‌ a variant of SimHash,​ one of the most​‌ famous LSH schemes. We​​ provide a theoretical basis​​​‌ for the proposed algorithm​ and leverage the insights​‌ obtained to connect the​​ novelty detection structure to​​​‌ classical Bayesian theory, yielding​ improvements also for FlyBloom-Filter.​‌ Ultimately, we propose a​​ simplification of FlyNN, for​​​‌ which we provide both​ a theoretical motivation and​‌ extensive experiments demonstrating its​​ competitive performance.

This work​​​‌ has been done in​ collaboration with Arthur Carvalho​‌ Walraven da Cuhna [Aarhus​​ University, Denmark] and Paulo​​​‌ Bruno Serafim [Gran Sasso​ Science Institute, L'Aquila, Italy].​‌

Attribute Inference Attacks for​​ Federated Regression Tasks

Federated​​​‌ Learning enables multiple clients,​ such as mobile phones​‌ and IoT devices, to​​ collaboratively train a global​​​‌ machine learning model while​ keeping their data localized.​‌ However, recent studies have​​ revealed that the training​​​‌ phase of FL is​ vulnerable to reconstruction attacks,​‌ such as attribute inference​​ attacks (AIA), where adversaries​​​‌ exploit exchanged messages and​ auxiliary public information to​‌ uncover sensitive attributes of​​ targeted clients. While these​​​‌ attacks have been extensively​ studied in the context​‌ of classification tasks, their​​ impact on regression tasks​​​‌ remains largely unexplored. In​ 51, we address​‌ this gap by proposing​​ novel model-based AIAs specifically​​​‌ designed for regression tasks​ in FL environments. Our​‌ approach considers scenarios where​​ adversaries can either eavesdrop​​​‌ on exchanged messages or​ directly interfere with the​‌ training process. We benchmark​​ our proposed attacks against​​​‌ state-of-the-art methods using real-world​ datasets. The results demonstrate​‌ a significant increase in​​ reconstruction accuracy, particularly in​​​‌ heterogeneous client datasets, a​ common scenario in FL.​‌ The efficacy of our​​ model-based AIAs makes them​​​‌ better candidates for empirically​ quantifying privacy leakage for​‌ federated regression tasks.

This​​ work has been done​​​‌ in collaboration with Othmane​ Marfoq [Meta, France], Giovanni​‌ Neglia [NEO] and Eoin​​ Thomas [Amadeus, France].

Trading-off​​​‌ Accuracy and Communication Cost​ in Federated Learning

Leveraging​‌ the training-by-pruning paradigm introduced​​ by Zhou et al.​​​‌  100, Isik et​ al.  89 introduced a​‌ federated learning protocol that​​ achieves a 34-fold reduction​​ in communication cost. In​​​‌ 85, 61,‌ we achieve a compression‌​‌ improvements of orders of​​ magnitude over the state-of-the-art.​​​‌ The central idea of‌ our framework is to‌​‌ encode the network weights​​ $\overrightarrow{w}$ by a​​​‌ the vector of trainable‌ parameters $\overrightarrow{p}$,‌​‌ such that $\stackrel{\to }{w}=Q·\overrightarrow{\mathrm{p‌}}$ where $Q$ is‌ a carefully-generate sparse random‌​‌ matrix (that remains fixed​​ throughout training). In such​​​‌ framework, the previous work‌  100 is retrieved when‌​‌ $Q$ is diagonal and​​ $\overrightarrow{p}$ has the​​​‌ same dimension of $\overrightarrow{\mathrm{w‌}}$. We instead‌​‌ show that $\stackrel{\to }{p}$ can effectively be chosen​​​‌ much smaller than $\overrightarrow{\mathrm{w‌}}$, while retaining‌​‌ the same accuracy at​​ the price of a​​​‌ decrease of the sparsity‌ of $Q$. Since‌​‌ server and clients only​​ need to share $\overrightarrow{\mathrm{p‌}}$, such a‌ trade-off leads to a‌​‌ substantial improvement in communication​​ cost. Moreover, we provide​​​‌ theoretical insight into our‌ framework and establish a‌​‌ novel link between training-by-sampling​​ and random convex geometry.​​​‌

This work has been‌ done in collaboration with‌​‌ Frederik Mallmann-Trenn [King‘s College​​ London, United Kingdom] and​​​‌ Mattia Jacopo Villani [JP‌ Morgan AI Research, USA].‌​‌

Cutting Through Privacy: A​​ Hyperplane-Based Data Reconstruction Attack​​​‌ in Federated Learning

Federated‌ Learning enables collaborative training‌​‌ of machine learning models​​ across distributed clients without​​​‌ sharing raw data, ostensibly‌ preserving data privacy. Nevertheless,‌​‌ recent studies have revealed​​ critical vulnerabilities in FL,​​​‌ showing that a malicious‌ central server can manipulate‌​‌ model updates to reconstruct​​ clients' private training data.​​​‌ Existing data reconstruction attacks‌ have important limitations: they‌​‌ often rely on assumptions​​ about the clients' data​​​‌ distribution or their efficiency‌ significantly degrades when batch‌​‌ sizes exceed just a​​ few tens of samples.​​​‌ In 52, we‌ introduce a novel data‌​‌ reconstruction attack that overcomes​​ these limitations. Our method​​​‌ leverages a new geometric‌ perspective on fully connected‌​‌ layers to craft malicious​​ model parameters, enabling the​​​‌ perfect recovery of arbitrarily‌ large data batches in‌​‌ classification tasks without any​​ prior knowledge of clients'​​​‌ data. Through extensive experiments‌ on both image and‌​‌ tabular datasets, we demonstrate​​ that our attack outperforms​​​‌ existing methods and achieves‌ perfect reconstruction of data‌​‌ batches two orders of​​ magnitude larger than the​​​‌ state of the art.‌

This work has been‌​‌ done in collaboration with​​ Giovanni Neglia [NEO].

Towards​​​‌ Estimating the Carbon Footprint‌ of Video Streaming

Video‌​‌ streaming dominates the Internet​​ traffic. Assessing the carbon​​​‌ footprint of video streaming‌ has received recently a‌​‌ significant attention with a​​ number of models proposed​​​‌ to associate a CO‌${}_2$ cost to one‌​‌ hour of streaming. In​​ 60, we compare​​​‌ the modeling assumptions and‌ computation methods used by‌​‌ five recent works to​​ inform the debate. Indeed,​​​‌ initial results can be‌ at odds, with up‌​‌ to one order of​​ magnitude difference in the​​​‌ estimates. Our contributions are:‌ (i) we relate the‌​‌ difference in the results​​ primarily to the perimeter​​​‌ of the study, e.g.‌ including production cost or‌​‌ not, (ii) 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 and‌ (iii) we propose a‌​‌ technique to reconcile the​​ models and obtain a​​​‌ CO2 estimate in between‌ 60 and 140 grams‌​‌ when considering the average​​ worldwide carbon intensity of​​​‌ electricity.

This work has‌ been done in collaboration‌​‌ with Guillaume Urvoy-Keller [I3S,​​ Université Côte d'Azur, France],​​​‌ Marco Dinuzzi [I3S, Université‌ Côte d'Azur, France] and‌​‌ Zhejiayu Ma [Easybroadcast, France].​​

Enhancing Energy Efficient Task​​​‌ Caching and Offloading in‌ Mobile Edge Computing

Mobile‌​‌ Edge Computing (MEC) enables​​ both to prolong the​​​‌ battery life of mobile‌ devices and support the‌​‌ execution of computationally intensive​​ applications at the edge.​​​‌ This can be achieved‌ by offloading these tasks‌​‌ to a server deployed​​ near the base station​​​‌ and/or by directly caching‌ them. Previous works focus‌​‌ on only one of​​ these two strategies or​​​‌ formulate optimization problems that‌ are hard to solve‌​‌ and propose a suboptimal​​​‌ solution.

In 59 we​ propose a linear model​‌ for the joint task​​ caching and offloading optimization​​​‌ problem. Moreover, we present​ two efficient heuristics which​‌ provide close-to-optimal results in​​ terms of energy efficiency​​​‌ with a low execution​ time. We further prove​‌ that the offloading subproblem​​ can be solved with​​​‌ an optimal algorithm. Finally,​ we demonstrate the performance​‌ and scalability of our​​ propositions by extensive simulations​​​‌ on a large number​ of ${10}^5$ mobile​‌ devices.

This work has​​ been done in collaboration​​​‌ with Fabiano Lorusso [I3S,​ Université Côte d'Azur ]​‌ and Guillaume Urvoy-Keller [I3S,​​ Université Côte d'Azur ].​​​‌

CANOE

• Title:
  Combinatorial Algorithms​‌ for Networking prOblEms
• Duration:​​
  2023 - 2025
• Coordinator:​​​‌
  Julio Araujo (julio@mat.ufc.br)
• Partners:​
  Universidade Federal do Ceará​‌ Fortaleza (Brésil)
• Inria contact:​​
  Nicolas Nisse
• Summary:

  A​​​‌ graph is a mathematical​ structure that allows modeling​‌ networks in many contexts,​​ from route networks, telecommunication​​​‌ networks, biological networks, neural​ networks to social networks.​‌ There are graph problems​​ arising in each of​​​‌ these domains that are​ classified as computationally difficult,​‌ where the objective is​​ to obtain an efficient​​​‌ algorithm for any graph​ presented as input. However,​‌ studying algorithms for a​​ problem restricted to special​​ graphs can shed light​​​‌ on the problem. This‌ approach consists in assuming‌​‌ that the graph has​​ some special structural property​​​‌ and exploiting this property‌ in the algorithm. Such‌​‌ a structural property defines​​ a class of graphs,​​​‌ for example, trees or‌ planar graphs. The aim‌​‌ is to build an​​ efficient algorithm for a​​​‌ class of graphs, and‌ then explore the ideas‌​‌ used to solve larger​​ and larger classes of​​​‌ graphs or with fewer‌ structural constraints. While a‌​‌ lot of work has​​ been dedicated to the​​​‌ study of structural properties‌ of graphs, very few‌​‌ results are known concerning​​ directed graphs or hypergraphs​​​‌ which better model real‌ life networks. For instance,‌​‌ road networks are intrinsically​​ directed and so are​​​‌ many social networks (e.g.,‌ Twitter), co-authorship networks correspond‌​‌ to hypergraphs (where each​​ publication corresponds to an​​​‌ hyperedge gathering the co-authors),‌ etc. This project aims‌​‌ at tackling chalenging theoretical​​ open problems in digraphs​​​‌ and/or hypergraphs, with applications‌ in road, telecommunications and‌​‌ social networks.

  The purpose​​ of this project is​​​‌ also to pursue and‌ extend our fruitful collaboration‌​‌ with the ParGO team​​ from Universidade Federal do​​​‌ Ceara (Fortaleza), which is‌ one of the partner‌​‌ universities of LNCC (Laboratório​​ Nacional de Computação Científica).​​​‌

ELECTRON

• Title:
  Evolutionary LEarning‌ and Compressed TRaining Of‌​‌ Neural networks
• Webpage:
  team.inria.fr/electron/​​
• Duration:
  2025 - 2027​​​‌
• Coordinator:
  Emanuele Natale
• Partners:‌
  King's College London
• Inria‌​‌ contact:
  Emanuele Natale
• Summary:​​
  The ELECTRON team focuses​​​‌ on understanding the role‌ of topology in neural‌​‌ networks and the principles​​ behind their efficient design.​​​‌ Motivated and inspired by‌ insights from evolutionary neuroscience,‌​‌ on the one hand,​​ and by the goal​​​‌ of improving the efficiency‌ of deep learning techniques,‌​‌ on the other, the​​ team aims to combine​​​‌ theoretical insights on artificial‌ neural network sparsification and‌​‌ compression. To this end,​​ our initial focus is​​​‌ to (i) establish rigorous‌ guarantees for “training-by-pruning” heuristics‌​‌ in the context of​​ the Strong Lottery Ticket​​​‌ Hypothesis, and (ii) investigate‌ evolutionary frameworks for network‌​‌ topology learning. This integrated​​ approach seeks not only​​​‌ to advance the theory‌ of sparse neural architectures‌​‌ but also to catalyze​​ novel interdisciplinary collaborations between​​​‌ computer scientists and neuroscientists.‌

CAPES-Cofecub project​​ Ma 1004/23 : Graphs,​​​‌ Optimization, Combinatorics and Algorithms‌

Participants: Thomas Dissaux,‌​‌ Frédéric Giroire, Frédéric​​ Havet, Nicolas Nisse​​​‌, Lucas Picasarri-Arrieta,‌ Clément Rambaud.

• Title:‌​‌
  Graphs, Optimization, Combinatorics and​​ Algorithms
• Duration:
  2023 -​​​‌ 2026
• Coordinator:
  Nicolas Nisse‌
• Partners:
  ParGO Research Group,‌​‌ Department of Mathematics, Federal​​ University of Ceará, Fortaleza,​​​‌ Brazil
• Summary:
  Complementary project‌ of the Inria EA‌​‌ CANOE.

French-Indian Campus

Participants:​​ Niccolò D'Archivio, Emanuele​​​‌ Natale.

• Title:
  The‌ Franco-Indian Campus in Life‌​‌ Sciences of Université Côte​​ d'Azur
• Website:
  life.univ-cotedazur.eu/education/franco-indian-campus
• Partners:​​​‌
  IIIT-Delhi (Indraprastha Institute of‌ Information Technology Delhi), New‌​‌ Dehli, Dehli, India
• Summary:​​
  The Franco-Indian Campus of​​​‌ Université Côte d'Azur is‌ one of the 4‌​‌ projects selected in the​​ framework of the call​​​‌ for projects on the‌ creation of a Franco-Indian‌​‌ campus, by the Ministry​​​‌ of Europe and Foreign​ Affairs (MEAE), the Ministry​‌ of Higher Education, Research​​ and Innovation (MESRI), the​​​‌ Campus France agency and​ the Indian embassy in​‌ France. Emanuele Natale and​​ Niccolò D'Archivio have been​​​‌ part of a delegation​ of the Université Côte​‌ d'Azur which visited IIIT-Delhi​​ in November 2025 to​​​‌ discuss future collaborations.

Other international visits to​​​‌ the team

• Sarah houdaigou​ [National Institute of Informatics​‌ (NII), Japan], July 5-12,​​ 2025.
• Frederik Mallmann-Trenn [King's​​​‌ College London, UK], September​ 16-19, 2025.
• Frank Hirth​‌ [King's College London, UK],​​ September 16-19, 2025.
• Luciano​​​‌ Gualà [University of Rome​ Tor Vergata, Italy], November​‌ 10-16, 2025.
• Andrea Clementi​​ [University of Rome Tor​​​‌ Vergata, Italy], November 10-16,​ 2025.
• Alma Ademovic Tahirovic​‌ [Intelligent Systems Hub], June​​ 2025 - November 2025.​​​‌
• Caroline Aparecido De Paula​ Silva [UNICAMP, Brasil], until​‌ August 2025.
• Piotr Micek​​ [Jagiellonian University, Kraków, Poland],​​​‌ December 1-4, 2025.
• Michał​ Pilipczuk [Jagiellonian University, Kraków,​‌ Poland], December 1-4, 2025.​​
• Julio Cesar Silva Araujo​​​‌ [Associated Professor at UFC​ Fortaleza, Brazil], April 27​‌ - May 10, 2025.​​
• Malgorzata Sulkowska [Assistant Professor​​​‌ at Univ. Wroclawski, Poland],​ February 4 - March​‌ 1, 2025. 2025.

Research stays abroad

• Niccolò​ D'Archivio : visit to​‌ the team of Dr.​​ Frederik Mallmann Trenn [King's​​​‌ College London, United Kindom],​ July 21-25 and November​‌ 17-21, 2025.
• Niccolò D'Archivio​​ : visit IIT-Delhi under​​​‌ the Franco-India Health Campus​ Event, New Dehli, Dehli,​‌ India. October 11-16, 2025.​​
• Frédéric Giroire : visiting​​​‌ researchers in the group​ of Xavier Defago [Institute​‌ of Science Tokyo, Japan]​​ and in the group​​​‌ of Tsuyoshi Murata [Institute​ of Science Tokyo, Japan],​‌ Tokyo, Japan, June 3​​ - July 9, 2025.​​​‌
• Emanuele Natale : visit​ the team of Dr.​‌ Frederik Mallmann Trenn [King's​​ College London, United Kindom],​​​‌ November 22-28, 2025.
• Emanuele​ Natale : visit the​‌ IIIT-Delhi (Indraprastha Institute of​​ Information Technology Delhi), New​​​‌ Dehli, Dehli, India, October​ 11-16, 2025.
• Emanuele Natale​‌ : invited professor at​​ University of Rome Tor​​​‌ Vergata in Andrea Clementi​ [University of Rome Tor​‌ Vergata, Italy]'s Group, Rome,​​ Italy, March 1 -​​​‌ April 30, 2025.
• Emanuele​ Natale : visiting professor​‌ at University of Bonn​​ in Petra Mutzel [University​​​‌ of Bonn, Germany]'s Group,​ Bonn, Germany, September 1-15​‌ and 26-30, October 1-9​​ and 16-30, November 1-10​​​‌ and 15-22 and 28-30,​ December 1-8 and 12-23,​‌ 2025.
• Emanuele Natale :​​ visiting researcher at Institute​​​‌ of Science Tokyo in​ Tsuyoshi Murata [Institute of​‌ Science Tokyo, Japan]'s Group,​​ Tokyo, Japan, May 8​​​‌ - June 7, 2025.​
• Emanuele Natale : visit​‌ to the team of​​ Flavio Vella [University of​​​‌ Trento, Italy], January 29-31,​ 2025.
• Nicolas Nisse :​‌ ParGO team, Universidade Federal​​ do Ceára, October 19​​​‌ - November 17, 2025.​
• André Nusser : University​‌ of Bonn, Bonn, Germany.​​ April 2025 – April​​​‌ 2026.
• Clément Rambaud :​ Jagiellonian University, Kraów, Poland.​‌ March 31-April 26, 2025.​​

HORIZON-CL4-2022-HUMAN-02-02 dAIEDGE,​ 2023-2026

Participants: Chuan Xu​‌, Frédéric Giroire.​​

• Program:
  HORIZON-CL4-2022-HUMAN-02-02 European Network​​ of AI Excellence Centres:​​​‌ Expanding the European AI‌ lighthouse.
• Project acronym:
  dAIEDGE‌​‌
• Project title:
  A network​​ of excellence for distributed,​​​‌ trustworthy, efficient and scalable‌ AI at the Edge‌​‌ Granting Authority
• Duration:
  September​​ 2023 - August 2026​​​‌
• Coordinator:
  Alain Pagani [DFKI]‌
• Other partners:
  36 partners‌​‌ from 15 countries.
• Summary:​​

  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.‌

• Web:
  daiedge.eu

GDR RSD,​​ ongoing (since 2006)

Members​​​‌ of COATI are involved‌ in the working group‌​‌ RESCOM (Réseaux de​​ communications) of GDR​​​‌ RSD of CNRS (‌gdr-rsd.cnrs.fr/). In particular,‌​‌ Christelle Caillouet is in​​ the steering committee 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.

GDR​​​‌ IFM, ongoing (since 2006)​

Members of COATI are​‌ involved in the working​​ group "Graphes" (gtgraphes.labri.fr/​​​‌) and Complexité et​ algorithmes (CoA) www.irif.fr/gt-coa/ of​‌ GDR IM, CNRS. In​​ particular, Nicolas Nisse is​​​‌ member of the scientific​ committee of the GT​‌ Graphes and of the​​ steering committee of the​​​‌ GT CoA.

Member​‌ of the organizing committees​​

• André Nusser
  • Workshop Massive​​​‌ Data Models and Computational​ Geometry II, University of​‌ Bonn, Bonn, Germany, September​​ 22–-26, 2025

Chair of​ conference program committees

• Emanuele​‌ Natale :
  • UAI25: Area​​ Chair for the 41st​​​‌ Conference on Uncertainty in​ Artificial Intelligence, Rio de​‌ Janeiro, Brazil, 21-25 July,​​ 2025.
• Christelle Caillouet :​​​‌
  • Co-Chair of International Workshop​ on Networked Robotics and​‌ Communication Systems IEEE NetRobiCS​​ 2025 in conjunction with​​​‌ IEEE Infocom 2025, May​ 19, 2025, London, UK.​‌

Member of the conference​​ program committees

• Emanuele Natale​​​‌ :
  • AAAI: Program Committee​ member of the 39th​‌ Annual AAAI Conference on​​ Artificial Intelligence, Philadelphia, Pennsylvania,​​​‌ USA, February 25 –​ March 4, 2025;
• Christelle​‌ Caillouet :
  • Slices-FR School​​, July 7-11, 2025,​​​‌ ENS Lyon;
  • WIMOB: 21st​ International Conference on Wireless​‌ and Mobile Computing, Networking​​ and Communications, October 20-22,​​​‌ 2025, Marrakech Morocco;
  • ITC36:​ International Teletraffic Congress, June​‌ 2-5, 2025, Trondheim Norway;​​
  • ISCC: 30th IEEE Symposium​​​‌ on Computers and Communications,​ July 2-5, 2025, Bologna​‌ Italy;
  • ICCCN: 34th International​​ Conference on Computer Communications​​​‌ and Networks, August 4-7,​ 2025, Tokyo Japan.
• David​‌ Coudert :
  • ACDA: SIAM​​ Conference on Applied and​​​‌ Computational Discrete Algorithms, Montréal,​ Québec, Canada, July 30–August​‌ 1, 2025;
  • ATMOS: Symposium​​ on Algorithmic Approaches for​​​‌ Transportation Modelling, Optimization, and​ Systems, Warsaw, Poland on​‌ September 18-19, 2025;
  • ONDM:​​ Conference on Optical Network​​​‌ Design and Management, Pisa,​ Italy, May 6-9, 2025.​‌
• Frédéric Havet
  • LAGOS: XIII​​ Latin American Algorithms, Graphs,​​​‌ and Optimization Symposium, Buenos​ Aires, Argentina, 10–14 November,​‌ 2025.
• Nicolas Nisse
  • AlgoTel:​​ 27ème Rencontre Francophone sur​​​‌ les Aspects Algorithmiques des​ Télécommunications, Saint Valery-sur-Somme (France),​‌ June 2-6, 2025;
  • LAGOS:​​ XIII Latin American Algorithms,​​​‌ Graphs, and Optimization Symposium,​ Buenos Aires, Argentina, 10–14​‌ November, 2025;
  • COCOON: 31st​​ International Computing and Combinatorics​​​‌ Conference, Chengdu, China, August​ 15-17, 2025.
• Joanna Moulierac​‌
  • AlgoTel: 27ème Rencontre Francophone​​ sur les Aspects Algorithmiques​​​‌ des Télécommunications, Saint Valery-sur-Somme​ (France), June 2-6, 2025;​‌
• André Nusser
  • SEA: 23rd​​ Symposium on Experimental Algorithms,​​​‌ Venice, Italy, July 22–24,​ 2025;
  • WADS: 19th Algorithms​‌ and Data Structures Symposium,​​ York University, Toronto, Canada,​​​‌ August 11–13, 2025;
  • CCCG:​ 37th Canadian Conference on​‌ Computational Geometry, York University,​​ Toronto, Canada, August 11–15,​​​‌ 2025.

Reviewer

• Emanuele Natale​ :
  • ICML: Conference Reviewer​‌ for the Forty-Second International​​ Conference on Machine Learning,​​ Vancouver, Canada, July 13-19,​​​‌ 2025;
  • NeurIPS: Conference Reviewer‌ for the Thirty-Ninth Annual‌​‌ Conference on Neural Information​​ Processing Systems, San Diego,​​​‌ Dec 2 - 7,‌ 2025.

Member‌​‌ of the editorial boards​​

• Jean-Claude Bermond :
  • Computer​​​‌ Science Reviews (Elsevier);
  • Discrete‌ Applied Mathematics (Elsevier);
  • Discrete‌​‌ Mathematics (Elsevier);
  • Discrete Mathematics,​​ Algorithms and Applications (World​​​‌ Scientific);
  • Journal of Graph‌ Theory (Wiley);
  • Advisory board‌​‌ of Journal of Interconnection​​ Networks (World Scientific);
  • Networks​​​‌ (Wiley);
  • Parallel Processing Letters‌ (World Scientific);
  • the SIAM‌​‌ book series on Discrete​​ Mathematics (SIAM).
• Christelle Caillouet​​​‌ :
  • Computer Communications.
• Alexandre‌ Caminada :
  • IEEE Transactions‌​‌ on Mobile Computing (IEEE);​​
  • IEEE Transactions on Vehicular​​​‌ Technology (IEEE);
  • Journal of‌ Traffic and Transportation Engineering‌​‌ (Elsevier);
  • Soft Computing (Springer).​​
• David Coudert :
  • Discrete​​​‌ Applied Mathematics (Elsevier);
  • Networks‌ (Wiley).
• Frédéric Giroire :‌​‌
  • Journal of Interconnection Networks​​ (World Scientific).
• Frédéric Havet​​​‌ :
  • Discrete Mathematics and‌ Theoretical Computer Science (DMTCS);‌​‌
  • Innovations in Graph Theory.​​
• Emanuele Natale :
  • The​​​‌ WikiJournal of Science (Wikimedia‌ Foundation).
• Nicolas Nisse :‌​‌
  • Discrete Applied Mathematics (Elsevier).​​

PhD thesis​​​‌

• PhD in progress: Jamil​ Abou Ltaif , Energy-efficient​‌ QoE-aware Beyond 5G Future​​ Mobile Networks, since​​​‌ November 2024. Co-supervisors: Chadi​ Barakat [DIANA], Frédéric Giroire​‌ , Joanna Moulierac and​​ Thierry Turletti [DIANA];
• PhD​​​‌ in progress: Yanis Achaichia​ , Optimizing the orchestration​‌ of virtualized services in​​ a multi-domain environment,​​ since October 2024. Co-supervisors:​​​‌ Christelle Caillouet , Nicolas‌ Huin [IMT Atlantique], Géraldine‌​‌ Texier [IMT Atlantique];
• PhD​​ in progress: Niccolò D’Archivio​​​‌ , Bio-inspired algorithms for‌ collective search and decision-making‌​‌, since April 2024.​​ Co-supervisors: Emanuele Natale and​​​‌ Frédéric Giroire ;
• PhD‌ in progress: Carlo Castoldi‌​‌ , Unlearning across brains​​ and models: neuro-computational insights​​​‌, since November 2025.‌ Supervisors: Emanuele Natale and‌​‌ Bianca Silva [IPMC, Université​​ Côte d'Azur ];
• PhD​​​‌ in progress: Francesco Diana‌ , Privacy Attacks in‌​‌ Federated Learning, since​​ January 2024. Co-supervisors: Chuan​​​‌ Xu and Giovanni Neglia‌ [NEO];
• PhD in progress:‌​‌ Emi Dreckmeyr , Data​​ capture and collection by​​​‌ energy-free sensors and ultra-low‌ power transmission in hostile‌​‌ environments, since January​​ 2025. Co-supervisors: Christelle Caillouet​​​‌ and Nathalie Mitton [FUN];‌
• PhD in progress: Davide‌​‌ Ferré , Energy efficient​​ deployment of applications in​​​‌ the edge-network-cloud continuum,‌ since January 2024. Co-supervisors:‌​‌ Frédéric Giroire and Emanuele​​ Natale ;
• PhD in​​​‌ progress: Rémi Godet ,‌ Privacy on-demand and Security‌​‌ preserving Federated Generative Networks​​ or Models, since​​​‌ April 2025. Co-supervisors: Frédéric‌ Giroire and Chuan Xu‌​‌ ;
• PhD in progress:​​ Sayf Eddine Halmi ,​​​‌ Impact of multidisciplinarity on‌ research productivity, since‌​‌ October 2025. Co-supervisors: Frédéric​​ Giroire and Nicolas Nisse​​​‌ and Michele Pezzoni [Université‌ Côte d'Azur, Groupe de‌​‌ Recherche en Droit, Economie,​​ Gestion (GREDEG)];
• PhD in​​​‌ progress: Aakash Kumar ,‌ Phase Transitions in Artificial‌​‌ Neural Networks, since​​ September 2025. Supervisor: Emanuele​​​‌ Natale ;
• PhD in‌ progress: Henrique Lovisi Ennes‌​‌ , Calcul quantique en​​ topologie, since October​​​‌ 2023. Co-supervisors: Clément Maria‌ [DATASHAPE] and Nicolas Nisse‌​‌ ;
• PhD in progress:​​ Samuel Nascimento , Convexity​​​‌ Games on Graphs,‌ PhD student in the‌​‌ Postgraduate Program in Computer​​ Science at the Federal​​​‌ University of Ceará (UFC‌ Fortaleza, Brazil) since March‌​‌ 2023, one year in​​ France since November 2024.​​​‌ Supervisor : Rudini Menezes‌ Sampaio [UFC Fortaleza, Brazil]‌​‌ and Nicolas Nisse ;​​
• PhD in progress: Pierre​​​‌ Pereira , Problem Size‌ Generalization in Neural Combinatorial‌​‌ Optimization, since October​​ 2024. Co-supervisors: Emanuele Natale​​​‌ and Frédéric Giroire ;‌
• PhD in progress: Caroline‌​‌ Aparecida de Paula Silva​​ , Universality and madericity​​​‌ of digraphs, University‌ of Campinas, Campinas, São‌​‌ Paulo, Brazil. From September​​ 2024 till August 2025.​​​‌ Supervisor: Frédéric Havet ;‌
• PhD in progress: Adrien‌​‌ Sardi , Modèles d'intelligence​​ artificielle génératifs et gestion​​​‌ énergétique des ressources au‌ sein des réseaux distribués‌​‌ 6G, since January​​ 2025. Co-supervisors: Marie-Line Alborel​​​‌ [Nokia], Sara Alouf [NEO],‌ Frédéric Giroire and Joanna‌​‌ Moulierac ;
• PhD in​​ progress: Kyrylo Tymchenko ,​​​‌ Enhancing Large-Scale Distributed Caching‌ Systems with Erasure Coding:‌​‌ Performance, Reliability, and Design​​ Trade-offs, since October​​​‌ 2025. Co-supervisors: Sara Alouf‌ [NEO] and Frédéric Giroire‌​‌ ;
• PhD: Tiago da​​ Silva Barros , Optimizing​​​‌ Performance and Energy Consumption‌ for Machine Learning Inference‌​‌ 64, Defended: November​​ 3 2025. Co-supervisors: Ramon​​​‌ Aparicio Pardo [I3S, Université‌ Côte d'Azur ] and‌​‌ Frédéric Giroire ;
• PhD:​​ Clément Rambaud , Structures​​​‌ of graph classes and‌ of their excluded minors‌​‌ 65, Defended: December​​​‌ 3 2025. Supervisor: Frédéric​ Havet ;
• PhD: Aurora​‌ Rossi , Computational methods​​ and analysis of temporal​​​‌ networks : applications in​ neurosciences 66, Defended:​‌ September 25 2025. Supervisor:​​ David Coudert ;

Internships​​​‌

• Google Summer of Code:​ Janmenjaya Panda , addition​‌ of the class of​​ matching covered graphs in​​​‌ Sagemath, IIT Madras,​ India, from May till​‌ November 2025. Mentor: David​​ Coudert .
• Google Summer​​​‌ of Code: Yuta Inoue​ , implementation of faster​‌ algorithms for the enumeration​​ of (weighted) (directed) paths​​​‌ and cycles in Sagemath​, University of Tokyo,​‌ Japan, from May till​​ November 2025. Mentor: David​​​‌ Coudert .
• Licence 3:​ Pablo Bernard , réalisation​‌ d'une application d'un jeu​​ de labyrinthe, Université​​​‌ Côte d'Azur, from June​ until July 2025. Supervisor:​‌ Nicolas Nisse
• Licence 3:​​ Eloi Rathgeber Kivits ,​​​‌ Pushability of oriented graphs​, Université Côte d'Azur,​‌ from June until August​​ 2025. Supervisor: Frédéric Havet​​​‌
• Master 2: Sayf Eddine​ Halmi , étude de​‌ la recherche multidisciplinaire entre​​ domaines et au cours​​​‌ du temps, Université​ Côte d'Azur, from April​‌ until August 2025. Supervisors:​​ Frédéric Giroire and Nicolas​​​‌ Nisse
• Master 2: Skander​ Meziou , estimation de​‌ la proximité thématique des​​ chercheurs, Université Côte​​​‌ d'Azur, from April until​ August 2025. Supervisors: Frédéric​‌ Giroire and Nicolas Nisse​​
• Master 2: Matteo Stromieri​​​‌ , Mathematical Approaches to​ Comparative Evolutionary Neuroscience,​‌ Université Côte d'Azur, from​​ April until September 2025.​​​‌ Supervisor: Emanuele Natale
• Master​ 2: Davide Toniatti ,​‌ An Empirical Evaluation of​​ Expand-and-Sparsify Classifiers, Université​​​‌ Côte d'Azur, from March​ until August 2025. Supervisor:​‌ Emanuele Natale
• Master 2:​​ Kyrylo Tymchenko , Study​​​‌ of large distributed systems​, Université Côte d'Azur,​‌ from March until August​​ 2025. Supervisor: Frédéric Giroire​​​‌ , Stéphane Pérennes ,​ and Sara Alouf [NEO].​‌
• Relai-thèse (followed by PhD):​​ Rémi Godet , Privacy​​​‌ on-demand and Security preserving​ Federated Generative Networks or​‌ Models, August 2024​​ till March 2025. Co-supervisors:​​​‌ Chuan Xu , Frédéric​ Giroire and Marco Lorenzi​‌ [EPIONE].

Apprentices (for Terra​​ Numerica)

• Vincent Chayé​​​‌ [BUT Informatique, Université Côte​ d'Azur ], since September​‌ 2023. Supervisor: Frédéric Havet​​ .
• Hamadi Daghar [Master​​​‌ 2 Informatique, Université Côte​ d'Azur ], since September​‌ 2024. Supervisor: Nicolas Nisse​​ .
• Mael Rivière [Master​​​‌ 2 MIAGE, Université Côte​ d'Azur ], since September​‌ 2024. Supervisor: Joanna Moulierac​​ .