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 $G$ satisfies some property (i.e., if $G$ belongs to some given family of graphs). For decision problems with input size $n$ and parameter $k$, the goal is to design an algorithm with running time $f\left(k\right).n$, where $f$ depends only on $k$. Problems for which we can find an optimal algorithm with such time complexity are said to be fixed-parameter tractable (FPT). Equivalently, the goal is to design a polynomial-time algorithm (in $k$ and $n$) that computes a pair $(H,k^{\text{'}})$ where $H$ is a graph (the kernel) with size polynomial in $k$ and $P\left(G\right)\le k$ if and only if $P\left(H\right)\le k^{\text{'}}$.

We study the parameterized complexity of the edge-modification problems. Given a graph $G=(V,E)$ and a positive integer $k$, an edge modification problem for a graph property $\Pi$ consists in deciding whether there exists a set $F$ of pairs of $V$ of size at most $k$ such that the graph $H=(V,E\Delta F)$ satisfies the property $\Pi$. In [25] , it is proved that parameterized cograph edge-modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for the $P_l$-free edge-deletion and the $C_l$-free edge-deletion problems for $l\ge 7$ and $l\ge 4$ respectively.

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 $S$ of vertices of a graph $G=(V,E)$ is convex if any vertex on a shortest path between two vertices of $S$ also belongs to $S$. The convex hull of $S\subset V$ is the smallest convex set containing $S$. Finally, a hull set of a graph is a set of vertices whose convex hull is $V$. The hull number of a graph $G$ is the minimum size of a hull set in $G$. In [16] , we prove that computing the hull number is NP-complete in bipartite graphs. We also provide bounds and design various polynomial-time algorithms for this problem in different graph classes such as co-bipartite graphs, $P_4$-sparse graphs, etc. In [30] , we first show a polynomial-time algorithm to compute the hull number of any $P_5$-free triangle-free graph. Then, we present four reduction rules based on vertices with the same neighborhood. We use these reduction rules to propose a fixed-parameter tractable algorithm to compute the hull number of any graph $G$, where the parameter is the size of a vertex cover of $G$ or, more generally, its neighborhood diversity. We also use these reductions to characterize the hull number of the lexicographic product of any two graphs.

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 $O\left(n^{3.69}\right)$, which is prohibitive for large-scale graphs. In [36] , we proposed a novel algorithm for determining the hyperbolicity of a graph that is scalable for large graphs. The time complexity of this algorithm is output-sensitive and depends on the shortest-path distances distribution in the graph and on the computed value of the hyperbolicity. Although its worst case time complexity is in $O\left(n^4\right)$, it is in practice much faster than previous proposals as it uses bounds to cut the search space. This algorithm allowed us for computing the hyperbolicity of all maps of the Internet provided by CAIDA and DIMES.

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 $k$ searchers are necessary and sufficient to win. Our characterization leads to a polynomial-time algorithm to compute the minimum number of searchers needed in trees.

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.