## Section: New Results

### Understanding graph representations

#### Distributed algorithms without knowledge of global parameters

Participants : Amos Korman, Jean-Sébastien Sereni, Laurent Viennot.

Many fundamental local distributed algorithms are non-uniform, that is, they assume that all nodes know good estimations of one or more global parameters of the network, e.g., the maximum degree $\Delta $ or the number of nodes $n$. In [28] , we introduce a rather general technique for transforming a non-uniform algorithm into a uniform one with same asymptotic complexity.

#### Asymptotic modularity

Participants : Fabien de Montgolfier, Mauricio Soto, Laurent Viennot.

Modularity has been introduced as a quality measure for graph partitioning by Newman and Girvan. It has received considerable attention in several disciplines, especially complex systems. In order to better understand this measure from a graph theoretical point of view, we study in [32] , [31] the asymptotic modularity of a variety of graph classes.

#### Internet Structure

Participants : Fabien de Montgolfier, Mauricio Soto, Laurent Viennot.

In [33] , [1] , we study the measurement of the Internet according to two graph parameters: treewidth and hyperbolicity.

#### Multipath Spanners

Participants : Cyril Gavoille, Quentin Godfroy, Laurent Viennot.

Motivated by multipath routing, we introduce in [23] , [39] a multi-connected variant of spanners.

#### $\delta -$hyperbolicity

Participants : Victor Chepoi [CNRS LIF, University of Marseille, France] , Feodor Dragan [University of Ohio, USA] , Bernard Estrellon [CNRS LIF, University of Marseille, France] , Michel Habib [CNRS LIAFA, University of Paris Diderot, France] , Yann Vaxes [University of Florence, Italy] , Yang Xiang [University of Ohio, USA] .

$\delta -$Hyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4-point condition: for any four points $u,v,w,x$, the two larger of the distance sums $d(u,v)+d(w,x),d(u,w)+d(v,x),d(u,x)+d(v,w)$ differ by at most $2\delta $. They play an important role in geometric group theory, geometry of negatively curved spaces, and have recently become of interest in several domains of computer science, including algorithms and networking. In [5] paper, we study un- weighted $\delta -$hyperbolic graphs. Using the Layering Partition technique, we show that every $n-$vertex $\delta $-hyperbolic graph with $\delta \ge 1/2$ has an additive $O(\delta logn)-$spanner with at most $O\left(\delta n\right)$ edges and provide a simpler, in our opinion, and faster con- struction of distance approximating trees of $\delta $-hyperbolic graphs with an additive error $O(\delta logn)$. The construction of our tree takes only linear time in the size of the input graph. As a consequence, we show that the family of $n-$vertex $\delta -$hyperbolic graphs with $\delta \ge 1/2$ admits a routing labeling scheme with $O\left(\delta {log}^{2}n\right)$ bit labels, $O(\delta logn)$ additive stretch and $O\left({log}_{2}\left(4\delta \right)\right)$ time routing protocol, and a distance labeling scheme with $O\left({log}^{2}n\right)$ bit labels, $O(\delta logn)$ additive error and constant time distance decoder.

#### Perfect Phylogeny

##### Perfect Phylogeny Is $NP-$ Hard

Participants : Michel Habib [CNRS LIAFA, University of Paris Diderot, France] , Juraj Stacho [University of Haifa, Israel] .

We answer in the affirmative [24] , to the question pro- posed by Mike Steel as a $100 challenge: “Is the following problem $NP-$ hard? Given a ternary 1 phylogenetic $X$-tree $T$ and a collection $Q$ of quartet subtrees on $X$, is $T$ the only tree that displays $Q$?” As a particular consequence of this, we show that the unique chordal sandwich problem is also $NP-$hard.

##### Compatibility of Multi-states Characters

Participants : Michel Habib [CNRS LIAFA, University of Paris Diderot, France] , Thu-Hien To [CNRS LIAFA, University of Paris Diderot, France] .

Perfect phylogeny consisting of determining the compatibil- ity of a set of characters is known to be $NP-$complete. We propose in [25] , a conjecture on the necessary and sufficient conditions of compatibility: Given a set $C$ of $r-$states full characters, there exists a function $f\left(r\right)$ such that $C$ is compatible $iff$ every set of $f\left(r\right)$ characters of $C$ is compatible. According to numerous references, $f\left(2\right)=2$, $f\left(3\right)=3$ and $f\left(r\right)\ge r$. Some conjectured that $f\left(r\right)=r$ for any $r\ge 2$. In this paper, we present an example showing that $f\left(4\right)\ge 5$. Therefore it could be the case that for $r\ge 4$ characters, the problem behavior drastically changes. In a second part, we propose a closure operation for chordal sandwich graphs. The later problem is a common approach of perfect phylogeny.

#### Graph sandwich

Participants : Arnaud Durand [CNRS LIAFA, University of Paris Diderot, France] , Michel Habib [CNRS LIAFA, University of Paris Diderot, France] .

Graph sandwich problems were introduced by Golumbic et al. (1994) in [12] for DNA physical mapping problems and can be described as follows. Given a property $\Pi $ of graphs and two disjoint sets of edges ${E}_{1}$ , ${E}_{2}$ with ${E}_{1}\subseteq {E}_{2}$ on a vertex set $V$ , the problem is to find a graph $G$ on $V$ with edge set ${E}_{s}$ having property $\Pi $ and such that ${E}_{1}\subseteq {E}_{s}\subseteq {E}_{2}$ . In [8] paper, we exhibit a quasi-linear reduction between the problem of finding an independent set of size $k\ge 2$ in a graph and the problem of finding a sandwich homogeneous set of the same size $k$. Using this reduction, we prove that a number of natural (decision and counting) problems related to sandwich homogeneous sets are hard in general. We then exploit a little further the reduction and show that finding efficient algorithms to compute small sandwich homogeneous sets would imply substantial improvement for computing triangles in graphs.

#### Diameter of Real-World Undirected Graphs

Participants : Pierluigi Crescenzi [University of Florence, Italy] , Roberto Grossi [University of Pisa, Italy] , Michel Habib [CNRS LIAFA, University of Paris Diderot, France] , Lorenzo Lanzi [University of Florence, Italy] , Andrea Marino [University of Florence, Italy] .

In [16] , we propose a new algorithm for computing the diameter of undirected unweighted graphs. Even though, in the worst case, this algorithm has complexity $O\left(nm\right)$, where $n$ is the number of nodes and $m$ is the number of edges of the graph, we experimentally show (on almost 200 real-world graphs) that in practice our method works in linear time. Moreover, we show how to extend our algorithm to the case of undirected weighted graphs and, even in this case, we present some preliminary very positive experimental results.

#### Parsimonious flooding in dynamic graphs

Participants : Hervé Baumann [CNRS LIAFA, University of Paris Diderot, France] , Pierluigi Crescenzi [University of Florence, Italy] , Pierre Fraigniaud [CNRS LIAFA, University of Paris Diderot, France] .

An edge-Markovian process with birth-rate $p$ and death-rate $q$ generates infinite sequences of graphs $({G}_{0},{G}_{1},{G}_{2},\dots )$ with the same node set $\left[n\right]$ such that ${G}_{t}$ is obtained from ${G}_{t-1}$ as follows: if $e\notin E\left({G}_{t-1}\right)$ then $e\in E\left({G}_{t}\right)$ with probability $p$, and if $e\in E\left({G}_{t-1}\right)$ then $e\notin E\left({G}_{t}\right)$ with probability $q$. In [2] , we establish tight bounds on the complexity of flooding in edge-Markovian graphs, where flooding is the basic mechanism in which every node becoming aware of an information at step t forwards this information to all its neighbors at all forthcoming steps $t\prime >t$. These bounds complete previous results obtained by Clementi et al. Moreover, we also show that flooding in dynamic graphs can be implemented in a parsimonious manner, so that to save bandwidth, yet preserving efficiency in term of simplicity and completion time. For a positive integer $k$, we say that the flooding protocol is $k-$active if each node forwards an information only during the $k$ time steps immediately following the step at which the node receives that information for the first time. We define the reachability threshold for the flooding protocol as the smallest integer $k$ such that, for any source s[n] , the $k-$active flooding protocol from s completes (i.e., reaches all nodes), and we establish tight bounds for this parameter. We show that, for a large spectrum of parameters p and q, the reachability threshold is by several orders of magnitude smaller than the flooding time. In particular, we show that it is even constant whenever the ratio $p/(p+q)$ exceeds $logn/n$. Moreover, we also show that being active for a number of steps equal to the reachability threshold (up to a multiplicative constant) allows the flooding protocol to complete in optimal time, i.e., in asymptotically the same number of steps as when being perpetually active. These results demonstrate that flooding can be implemented in a practical and efficient manner in dynamic graphs. The main ingredient in the proofs of our results is a reduction lemma enabling to overcome the time dependencies in edge-Markovian dynamic graphs.