Section: New Results
Graph Algorithms
Participants : Julio Araújo, Jean-Claude Bermond, David Coudert, Frédéric Havet, Frédéric Giroire, Bi Li, Fatima Zahra Moataz, Christelle Molle-Caillouet, Nicolas Nisse, Ronan Pardo Soares, Stéphane Pérennes.
Coati is also interested in the algorithmic aspects of Graph Theory. In general we try to find the most efficient algorithms to solve various problems of Graph Theory and telecommunication networks. More information on several results presented in this section may be found in R. Soares's thesis [14] .
Complexity and Computation of Graph Parameters
We use graph theory to model various network problems. In general we study their complexity and then we investigate the structural properties of graphs that make these problems hard or easy. In particular, we try to find the most efficient algorithms to solve the problems, sometimes focusing on specific graph classes from which the problems are polynomial-time solvable.
Parameterized Complexity
Parameterized complexity is a way to deal with intractable computational problems having some parameters that can be relatively small with respect to the input size. This area has been developed extensively during the last decade.
More precisely, we consider problems that consist in deciding whether a graph
We study the parameterized complexity of the edge-modification problems. Given a graph
We also design a unified parameterized algorithm for computing various widths of graphs (such as branched tree-width, branch-width, cut-width, etc.) [60] .
Convexity in Graphs
The geodesic convexity of graphs naturally extends the notion of convexity in euclidean metric spaces. A set
Hyperbolicity
The Gromov hyperbolicity is an important parameter for analyzing complex networks since it expresses how the metric structure of a network looks like a tree. In other words, it provides bounds on the stretch resulting from the embedding of a network topology into a weighted tree. It is therefore used to provide bounds on the expected stretch of greedy-routing algorithms in Internet-like graphs. However, the best known algorithm for computing this parameter has time complexity in
Graph searching and applications
Pursuit-evasion encompasses a wide variety of combinatorial problems related to the capture of a fugitive residing in a network by a team of searchers. The goal consists in minimizing the number of searchers required to capture the fugitive in a network and in computing the corresponding capture strategy. We investigated several variants of these games.
Variants of graph searching.
We study non-deterministic graph searching where the searchers have to capture an invisible fugitive but can see him a bounded number of times. This variant generalizes the notion of pathwidth and treewidth of graphs. In this setting, we provide a polynomial-time algorithm that approximates the minimum number of searchers needed in trees, up to a factor of two [56] .
In [34] , [61] , we define another variant of graph searching, where searchers have to capture an invisible fugitive with the constraint that no two searchers can occupy the same node simultaneously. This variant seems promising for designing approximation algorithms for computing the pathwidth of graphs. The main contribution in [34] , [61] is the characterization of trees where
We also study graph searching in directed graphs. We prove that the graph processing variant is monotone which allows us to show its equivalence with a particular digraph decomposition [47] .
Surveillance Game and Fractional Game.
A surprising application of some variant of pursuit-evasion games is the problem for a web-browser to download documents in advance while an internaut is surfing on the Web. In a previous work, we model this problem as a Pursuit-evasion game called Surveillance game. In [40] , [67] , we continue our study of the Surveillance game. We provide some bounds on the connected and online variants of this game. In particular, we show that, in the online variant (when the searchers discover the graph during the game), the best strategy is the trivial one that consists in downloading the document in the neighborhood of the position of the internaut.
In [69] , [48] , [52] , we define a framework generalizing and relaxing many games (including the Surveillance game) where Players use fractions of their token at each turn. We design an algorithm for solving the fractional games. In particular, our algorithm runs in polynomial-time when the length of the game is bounded by 2 (in contrast, computing the surveillance game is NP-hard even when the game is limited to two turns). For some games, we also prove that the fractional variant provides some good approximation. This direction of research seems promising for solving many open problems related to Pursuit-evasion games.
Robots in anonymous networks.
Motivated by the understanding of the limits of distributed computing, we consider a recent model of robot-based computing which makes use of identical, memoryless mobile robots placed on nodes of anonymous graphs. The robots operate in Look-Compute-Move cycles that are performed asynchronously for each robot. In particular, we consider various problems such as graph exploration, graph searching and gathering in various graph classes. We provide a new distributed approach which turns out to be very interesting as it neither completely falls into symmetry-breaking nor into symmetry-preserving techniques. We proposed a general approach [38] , [66] to solve the three problems in rings even in case of symmetric initial configurations.
Algorithm design in biology
In Coati , we have recently started a collaboration with EPI ABS (Algorithms Biology Structure) from Sophia Antipolis on “minimal connectivity complexes in mass spectrometry based macro-molecular complex reconstruction” [28] , [55] . This problem turns out to be a minimum color covering problem (minimum number of colors to cover colored edges with connectivity constraints on the subgraphs induced by the colors) of the edges of a graph, and is surprizingly similar to a capacity maximization problem in a multi-interfaces radio network we were studying.