Section: New Results

Graph Algorithms

Participants : Julio Araújo, Jean-Claude Bermond, David Coudert, Guillaume Ducoffe, Frédéric Giroire, Aurélien Lancin, Bi Li, Fatima Zahra Moataz, Christelle Molle-Caillouet, Nicolas Nisse, 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 PhD thesis of B. Li [15] and A. Lancin [14] , and in the Habilitation thesis of N. Nisse [17] .

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.

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 [47] , we investigate some relations between the hyperbolicity of a graph and the hyperbolicity of its atoms, that are the subgraphs resulting from the decomposition of the graph according its clique minimal separators. More precisely, we prove that the maximum hyperbolicity taken over all the atoms is at least the hyperbolicity of $G$ minus one. We also give an algorithm to slightly modify the atoms, which is at no extra cost than computing the atoms themselves, and so that the maximum hyperbolicity taken over all the resulting graphs is exactly the hyperbolicity of $G$. An experimental evaluation of our methodology is provided for large collaboration networks. Finally, we deduce from our theoretical results the first linear-time algorithm to compute the hyperbolicity of an outerplanar graph.

The shortest-path metric $d$ of a connected graph $G$ is 1/2-hyperbolic if, and only if, it satisfies $\mathrm{d}(u,v)+\mathrm{d}(x,y)\le max\left\{\mathrm{d}\right(u,x)+\mathrm{d}(v,y),\mathrm{d}(u,y)+\mathrm{d}(v,x\left)\right\}+1$, for every 4-tuple $u,x,v,y$ of $G$. We show in [26] , [48] that the problem of deciding whether an unweighted graph is 1/2-hyperbolic is subcubic equivalent to the problem of determining whether there is a chordless cycle of length 4 in a graph. An improved algorithm is also given for both problems, taking advantage of fast rectangular matrix multiplication. In the worst case it runs in $O\left(n^{3.26}\right)$-time.

Branch and Bound Algorithm for computing Pathwidth

It is well known that many NP-hard problems are tractable in the class of bounded pathwidth graphs. In particular, path-decompositions of graphs are an important ingredient of dynamic programming algorithms for solving such problems. Therefore, computing the pathwidth and associated path-decomposition of graphs has both a theoretical and practical interest. In [36] , [51] , we design a Branch and Bound algorithm that computes the exact pathwidth of graphs and a corresponding path-decomposition. Our main contribution consists of several non-trivial techniques to reduce the size of the input graph (pre-processing) and to cut the exploration space during the search phase of the algorithm. We evaluate experimentally our algorithm by comparing it to existing algorithms of the literature. It appears from the simulations that our algorithm offers a significative gain with respect to previous work. In particular, it is able to compute the exact pathwidth of any graph with less than 60 nodes in a reasonable running-time (10 min.). Moreover, our algorithm also achieves good performance when used as a heuristic (i.e., when returning best result found within bounded time-limit). Our algorithm is not restricted to undirected graphs since it actually computes the vertex-separation of digraphs (which coincides with the pathwidth in case of undirected graphs).

To satisfy impatient Web surfers is hard

Prefetching is a basic mechanism for faster data access and efficient computing. An important issue in prefetching is the tradeoff between the amount of network's resources wasted by the prefetching and the gain of time. For instance, in the Web, browsers may download documents in advance while a Web surfer is surfing. Since the Web surfer follows the hyperlinks in an unpredictable way, the choice of the Web pages to be prefetched must be computed online. The question is then to determine the minimum amount of resources used by prefetching that ensures that all documents accessed by the Web surfer have previously been loaded in the cache. In [28] , we model this problem as a two-player game similar to Cops and Robber Games in graphs. Let $k\ge 1$ be any integer. The first player, a fugitive, starts on a marked vertex of a (di)graph $G$. The second player, an observer, marks at most $k$ vertices, then the fugitive moves along one edge/arc of $G$ to a new vertex, then the observer marks at most $k$ vertices, etc. The fugitive wins if it enters an unmarked vertex, and the observer wins otherwise. The surveillance number of a (di)graph is the minimum $k$ such that the observer marking at most $k$ vertices at each step can win against any strategy of the fugitive. We also consider the connected variant of this game, i.e., when a vertex can be marked only if it is adjacent to an already marked vertex. We study the computational complexity of the game. All our results hold for both variants, connected or unrestricted. We show that deciding whether the surveillance number of a chordal graph is at most 2 is NP-hard. We also prove that deciding if the surveillance number of a DAG is at most 4 is PSPACE-complete. Moreover, we show that the problem of computing the surveillance number is NP-hard in split graphs. On the other hand, we provide polynomial-time algorithms computing surveillance numbers of trees and interval graphs. Moreover, in the case of trees, we establish a combinatorial characterization of the surveillance number.

Tree-decompositions

Minimum Size Tree-Decompositions

Tree-Decompositions are the corner-stone of many dynamic programming algorithms for solving graph problems. Since the complexity of such algorithms generally depends exponentially on the width (size of the bags) of the decomposition, much work has been devoted to compute tree-decompositions with small width. However, practical algorithms computing tree-decompositions only exist for graphs with treewidth less than 4. In such graphs, the time-complexity of dynamic programming algorithms based on tree-decompositions is dominated by the size (number of bags) of the tree-decompositions. It is then interesting to try to minimize the size of the tree-decompositions. In [42] , [60] , we consider the problem of computing a tree-decomposition of a graph with width at most $k$ and minimum size. More precisely, we focus on the following problem: given a fixed $k\ge 1$, what is the complexity of computing a tree-decomposition of width at most $k$ with minimum size in the class of graphs with treewidth at most $k$? We prove that the problem is NP-complete in planar graphs for any fixed $k\ge 4$ and polynomial for $k\le 2$. We also show that for $k=3$ the problem can be solved in polynomial time in the class of trees and 2-connected outerplanar graphs.

Exclusive Graph Searching vs. Pathwidth

In Graph Searching, a team of searchers aims at capturing an invisible fugitive moving arbitrarily fast in a graph. Equivalently, the searchers try to clear a contaminated network. The problem is to compute the minimum number of searchers required to accomplish this task. Several variants of Graph Searching have been studied mainly because of their close relationship with the pathwidth of a graph. Blin et al. defined the Exclusive Graph Searching where searchers cannot "jump" and no node can be occupied by more than one searcher. In [61] , we study the complexity of this new variant. We show that the problem is NP-hard in planar graphs with maximum degree 3 and it can be solved in linear time in the class of cographs. We also show that monotone Exclusive Graph Searching is NP-complete in split graphs where Pathwidth is known to be solvable in polynomial time. Moreover, we prove that monotone Exclusive Graph Searching is in P in a subclass of star-like graphs where Pathwidth is known to be NP-hard. Hence, the computational complexities of monotone Exclusive Graph Searching and Pathwidth cannot be compared. This is the first variant of Graph Searching for which such a difference is proved.

Diameter of Minimal Separators in Graphs

In [49] , we establish general relationships between the topological properties of graphs and their metric properties. For this purpose, we upper-bound the diameter of the minimal separators in any graph by a function of their sizes. More precisely, we prove that, in any graph $G$, the diameter of any minimal separator $S$ in $G$ is at most $\lfloor \frac{\ell \left(G\right)}{2}\rfloor ·\left(\right|S|-1)$ where $\ell \left(G\right)$ is the maximum length of an isometric cycle in $G$. We refine this bound in the case of graphs admitting a distance preserving ordering for which we prove that any minimal separator $S$ has diameter at most $2\left(\right|S|-1)$. Our proofs are mainly based on the property that the minimal separators in a graph $G$ are connected in some power of $G$.

Our result easily implies that the treelength $tl\left(G\right)$ of any graph $G$ is at most $\lfloor \frac{\ell \left(G\right)}{2}\rfloor$ times its treewidth $tw\left(G\right)$. In addition, we prove that, for any graph $G$ that excludes an apex graph $H$ as a minor, $tw\left(G\right)\le c_H·tl\left(G\right)$ for some constant $c_H$ only depending on $H$. We refine this constant when $G$ has bounded genus. As a consequence, we obtain a very simple $O\left(\ell \right(G\left)\right)$-approximation algorithm for computing the treewidth of $n$-node $m$-edge graphs that exclude an apex graph as a minor in $O\left(nm\right)$-time.

Distributed computing with mobile agents

Stigmergy of Anonymous Agents in Discrete Environments

Communication by stigmergy consists, for agents/robots devoid of other dedicated communication devices, in exchanging information by observing each other's movements, similar to how honeybees use a dance to inform each other on the location of food sources. Stigmergy, while a popular technique in soft computing (e.g., swarm intelligence and swarm robotics), has received little attention from a computational viewpoint, with only one study proposing a method in a continuous environment. An important question is whether there are limits intrinsic to the environment on the feasibility of stigmergy. While it is not the case in a continuous environment, we show that the answer is quite different when the environment is discrete. In [53] , [37] , we consider stigmergy in graphs and identifies classes of graphs in which robots can communicate by stigmergy. We provide two algorithms with different tradeoffs. One algorithm achieves faster stigmergy when the density of robots is low enough to let robots move independently. This algorithm works when the graph contains some particular pairwise-disjoint subgraphs. The second algorithm, while slower solves the problem under an extremely high density of robots assuming that the graph admits some large cycle. Both algorithms are described in a general way, for any graph that admits the desired properties and with identified nodes. We show how the latter assumption can be removed in more specific topologies. Indeed, we consider stigmergy in the grid which offers additional orientation information not available in a general graphs, allowing us to relax some of the assumptions. Given an $N\times M$ anonymous grid, we show that the first algorithm requires $O\left(ℳ\right)$ steps to achieve communication by stigmergy, where $ℳ$ is the maximum length of a communication message, but it works only if the number of robots is less than $⌊\frac{N·M}{9}⌋$. The second algorithm, which requires $O\left(k^2\right)$ steps, where $k$ is the number of robots, on the other hand, works for up to $N·M-5$ robots. In both cases, we consider very weak assumptions on the robots capabilities: i.e., we assume that the robots are anonymous, asynchronous, uniform, and execute deterministic algorithms.

Gathering and Exclusive Searching on Rings under Minimal Assumptions

Consider a set of mobile robots with minimal capabilities placed over distinct nodes of a discrete anonymous ring. Asynchronously, each robot takes a snapshot of the ring, determining which nodes are either occupied by robots or empty. Based on the observed configuration, it decides whether to move to one of its adjacent nodes or not. In the first case, it performs the computed move, eventually. The computation also depends on the required task. In [38] , we solve both the well-known Gathering and Exclusive Searching tasks. In the former problem, all robots must simultaneously occupy the same node, eventually. In the latter problem, the aim is to clear all edges of the graph. An edge is cleared if it is traversed by a robot or if both its endpoints are occupied. We consider the exclusive searching where it must be ensured that two robots never occupy the same node. Moreover, since the robots are oblivious, the clearing is perpetual, i.e., the ring is cleared infinitely often. In the literature, most contributions are restricted to a subset of initial configurations. Here, we design two different algorithms and provide a characterization of the initial configurations that permit the resolution of the problems under minimal assumptions.

Enhancing the Web's Transparency

Today's Web services – such as Google, Amazon, and Facebook – leverage user data for varied purposes, including personalizing recommendations, targeting advertisements, and adjusting prices. At present, users have little insight into how their data is being used. Hence, they cannot make informed choices about the services they choose.

To increase transparency, we developed XRay [40] , the first fine-grained, robust, and scalable personal data tracking system for the Web. XRay predicts which data in an arbitrary Web account (such as emails, searches, or viewed products) is being used to target which outputs (such as ads, recommended products, or prices). XRay's core functions are service agnostic and easy to instantiate for new services, and they can track data within and across services. To make predictions independent of the audited service, XRay relies on the following insight: by comparing outputs from different accounts with similar, but not identical, subsets of data, one can pinpoint targeting through correlation. We show both theoretically, and through experiments on Gmail, Amazon, and YouTube, that XRay achieves high precision and recall by correlating data from a surprisingly small number of extra accounts.

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  [63] . 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.

Consider a set of oligomers listing the subunits involved in sub-complexes of a macro-molecular assembly, obtained e.g. using native mass spectrometry or affinity purification. Given these oligomers, connectivity inference (CI) consists of finding the most plausible contacts between these subunits, and minimum connectivity inference (MCI) is the variant consisting of finding a set of contacts of smallest cardinality. MCI problems avoid speculating on the total number of contacts, but yield a subset of all contacts and do not allow exploiting a priori information on the likelihood of individual contacts. In this context, we present in [43] two novel algorithms, ALGO-MILP-W and ALGO-MILP-WB. The former solves the minimum weight connectivity inference (MWCI), an optimization problem whose criterion mixes the number of contacts and their likelihood. The latter uses the former in a bootstrap fashion, to improve the sensitivity and the specificity of solution sets. Experiments on the yeast exosome, for which both a high resolution crystal structure and a large set of oligomers is known, show that our algorithms predict contacts with high specificity and sensitivity, yielding a very significant improvement over previous work. The software accompanying this paper is made available, and should prove of ubiquitous interest whenever connectivity inference from oligomers is faced.