Computing metric hulls in graphs

We prove that, given a closure function the smallest preimage of a closed set can be calculated in polynomial time in the number of closed sets. This confirms a conjecture of Albenque and Knauer and implies that there is a polynomial time algorithm to compute the convex hull-number of a graph, when all its convex subgraphs are given as input. We then show that computing if the smallest preimage of a closed set is logarithmic in the size of the ground set is LOGSNP-complete if only the ground set is given. A special instance of this problem is computing the dimension of a poset given its linear extension graph, that was conjectured to be in P. The intent to show that the latter problem is LOGSNP-complete leads to several interesting questions and to the definition of the isometric hull, i.e., a smallest isometric subgraph containing a given set of vertices $S$. While for $|S|=2$ an isometric hull is just a shortest path, we show that computing the isometric hull of a set of vertices is NP-complete even if $|S|=3$. Finally, we consider the problem of computing the isometric hull-number of a graph and show that computing it is $\Sigma^P_2$ complete.Comment: 13 pages, 3 figure

Weighted Coloring in Trees

A proper coloring of a graph is a partition of its vertex set into stable sets, where each part corresponds to a color. For a vertex-weighted graph, the weight of a color is the maximum weight of its vertices. The weight of a coloring is the sum of the weights of its colors. Guan and Zhu (1997) defined the weighted chromatic number of a vertex-weighted graph G as the smallest weight of a proper coloring of G. If vertices of a graph have weight 1, its weighted chromatic number coincides with its chromatic number. Thus, the problem of computing the weighted chromatic number, a.k.a. Max Coloring Problem, is NP-hard in general graphs. It remains NP-hard in some graph classes as bipartite graphs. Approximation algorithms have been designed in several graph classes, in particular, there exists a PTAS for trees. Surprisingly, the time-complexity of computing this parameter in trees is still open. The Exponential Time Hypothesis (ETH) states that 3-SAT cannot be solved in sub-exponential time. We show that, assuming ETH, the best algorithm to compute the weighted chromatic number of n-node trees has time-complexity n O(log(n)). Our result mainly relies on proving that, when computing an optimal proper weighted coloring of a graph G, it is hard to combine colorings of its connected components

Brief Announcement: Distributed Exclusive and Perpetual Tree Searching

International audienceWe tackle a practical version of the well known {\it graph searching} problem, where a team of robots aims at capturing an intruder in a graph. The robots and the intruder move along the edges of the graph. The intruder is invisible, arbitrary fast, and omniscient. It is caught whenever it stands on a node occupied by a robot, and cannot escape to a neighboring node. We study graph searching in the CORDA model of mobile computing: robots are asynchronous, and they perform cycles of {\it Look-Compute-Move} actions. Moreover, motivated by physical constraints, we consider the \emph{exclusive} property, stating that no two or more robots can occupy the same node at the same time. In addition, we assume that the network and the robots are anonymous. Finally, robots are \emph{oblivious}, i.e., each robot performs its move actions based only on its current ''vision'' of the positions of the other robots. Our objective is to characterize, for a graph $G$, the set of integers $k$ such that graph searching can be achieved by a team of $k$ robots starting from \emph{any} $k$ distinct nodes in $G$. Our main result consists in a full characterization of this set, for any asymmetric tree. Towards providing a characterization for all trees, including trees with non-trivial automorphisms, we have also provides a set of positive and negative results, including a full characterization for any line. All our positive results are based on the design of algorithms enabling \emph{perpetual} graph searching to be achieved with the desired number of robots

Spy-Game on Graphs

We define and study the following two-player game on a graph G. Let k in N^*. A set of k guards is occupying some vertices of G while one spy is standing at some node. At each turn, first the spy may move along at most s edges, where s in N^* is his speed. Then, each guard may move along one edge. The spy and the guards may occupy same vertices. The spy has to escape the surveillance of the guards, i.e., must reach a vertex at distance more than d in N (a predefined distance) from every guard. Can the spy win against k guards? Similarly, what is the minimum distance d such that k guards may ensure that at least one of them remains at distance at most d from the spy? This game generalizes two well-studied games: Cops and robber games (when s=1) and Eternal Dominating Set (when s is unbounded). We consider the computational complexity of the problem, showing that it is NP-hard and that it is PSPACE-hard in DAGs. Then, we establish tight tradeoffs between the number of guards and the required distance d when G is a path or a cycle. Our main result is that there exists beta>0 such that Omega(n^{1+beta}) guards are required to win in any n*n grid

Experimental Evaluation of a Branch and Bound Algorithm for computing Pathwidth

International audienceIt 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 this paper, 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).Les décompositions en chemin de graphes sont très importants pour la conception d'algorithmes de programmation dynamique pour résoudre de nombreux problèmes NP-difficiles. Calculer la pathwidth et la décomposition en chemin correspondante sont donc d'un grand intérêt tant d'un point de vue théorique que pratique. Dans ce papier, nous proposons un algorithme de Branch and Bound qui calcule la pathwidth et une décomposition. Notre contribution principale réside dans les techniques que nous prouvons pour réduire la taille du graphe donné en entrée (prétraitement) et réduire la taille de l'espace d'exploration de la phase de recherche de l'algorithme. Nous évaluons expérimentalement notre algorithme en le comparant aux algorithmes proposés dans la littérature. Les simulations montrent que notre algorithme apporte un gain significatif par rapport aux algorithmes existants. Il est capable de calculer la valeur exacte de la pathwidth de tout graphe composé d'au plus 60 sommets en un temps raisonnable (moins de 10 minutes). De plus, notre algorithme montre de bonnes performances lorsqu'il est utilisé en heuristique (c'est-à-dire lorsqu'il retourne le meilleur résultat trouvé en un temps donné). Notre algorithme n'est pas spécifique au graphes non orientés car il permet de calculer la vertex-separation des digraphes (qui coïncide avec la pathwidth dans le cas des graphes non orientés)

Nettoyage perpétuel de réseaux

International audienceDans le cadre du nettoyage de graphes contaminés ( graph searching), des agents mobiles se déplacent successivement le long des arêtes du graphe afin de les nettoyer. Le but général est le nettoyage en utilisant le moins d'agents possible. Nous plaçons notre étude dans le modèle de calcul distribué CORDA minimaliste. Ce modèle est muni d'hypothèses très faibles : les nœuds du réseau et les agents sont anonymes, n'ont pas de mémoire du passé ni sens commun de l'orientation et agissent par cycles Voir-Calculer-Agir de manière asynchrone. Un intérêt de ce modèle vient du fait que si le nettoyage peut être fait à partir de positions arbitraires des agents (par exemple, après pannes ou recontamination), l'absence de mémoire implique un nettoyage perpétuel et donc fournit une première approche de nettoyage de graphe tolérant aux pannes. Les contraintes dues au modèle CORDA minimaliste nous amènent à définir une nouvelle variante de nettoyage de graphes - le nettoyage sans collision, autrement dit, plusieurs agents ne peuvent occuper simultanément un même sommet. Nous montrons que, dans un contexte centralisé, cette variante ne satisfait pas certaines propriétés classiques de nettoyage comme par exemple la monotonie. Nous montrons qu'interdire les ''collisions'' peut augmenter le nombre d'agents nécessaires d'un facteur au plus $\Delta$ le degré maximum du graphe et nous illustrons cette borne. De plus, nous caractérisons complètement le nettoyage sans collision dans les arbres. Dans le contexte distribué, la question qui se pose est la suivante. Existe-t-il un algorithme qui, étant donné un ensemble d'agents mobiles arbitrairement répartis sur des sommets distincts d'un réseau, permet aux agents de nettoyer perpétuellement le graphe ? Dans le cas des chemins, nous montrons que la réponse est négative si le nombre d'agents est pair dans un chemin d'ordre impair, ou si il y a au plus deux agents dans un chemin d'ordre au moins $3$. Nous proposons un algorithme qui nettoie les chemins dans tous les cas restants, ainsi qu'un algorithme pour nettoyer les arbres lorsqu'un nombre suffisant d'agents est disponible initialement

Structure vs métrique dans les graphes

International audienceL'émergence de réseaux de très grande taille oblige à repenser de nombreux problèmes sur les graphes : en apparence simples, mais pour lesquels les algorithmes de résolution connus ne passent plus a l'échelle. Une approche possible est de mieux comprendre les propriétés de ces réseaux complexes, et d'en déduire de nouvelles méthodes plus efficaces. C'est dans ce but que nous démontrons des relations générales entre les propriétés structurelles des graphes et leurs propriétés métriques. Nos relations se déduisent de nouvelles bornes serrées sur le diamètre des séparateurs minimaux dans un graphe. Plus précisément , nous prouvons que dans tout graphe G le diamètre d'un séparateur minimal S dans G est au plus (l(G)/2) · (|S| − 1), avec l(G) la plus grande taille d'un cycle isométrique dans G. Nos preuves reposent sur des propriétés de connexité dans les puissances d'un graphe. Une conséquence de nos résultats est que pour tout graphe G, sa longueur arborescente (treelength) est au plus l(G)/2 fois sa largeur arborescente (treewidth). En complément de cette relation, nous bornons la largeur arborescente par une fonction de la longueur arborescente et du genre du graphe. Cette borne se généralise à la famille des graphes qui excluent un apex-graph H comme mineur. Par conséquent , nous obtenons un algorithme très simple qui, étant donné un graphe excluant un apex-graph fixé comme mineur, calcule sa largeur arborescente en temps O(n²) et avec facteur d'approximation O(l(G))

Interconnection network with a shared whiteboard: Impact of (a)synchronicity on computing power

In this work we study the computational power of graph-based models of distributed computing in which each node additionally has access to a global whiteboard. A node can read the contents of the whiteboard and, when activated, can write one message of O(log n) bits on it. When the protocol terminates, each node computes the output based on the final contents of the whiteboard. We consider several scheduling schemes for nodes, providing a strict ordering of their power in terms of the problems which can be solved with exactly one activation per node. The problems used to separate the models are related to Maximal Independent Set, detection of cycles of length 4, and BFS spanning tree constructions

Localization game on geometric and planar graphs

The main topic of this paper is motivated by a localization problem in cellular networks. Given a graph $G$ we want to localize a walking agent by checking his distance to as few vertices as possible. The model we introduce is based on a pursuit graph game that resembles the famous Cops and Robbers game. It can be considered as a game theoretic variant of the \emph{metric dimension} of a graph. We provide upper bounds on the related graph invariant $\zeta (G)$, defined as the least number of cops needed to localize the robber on a graph $G$, for several classes of graphs (trees, bipartite graphs, etc). Our main result is that, surprisingly, there exists planar graphs of treewidth $2$ and unbounded $\zeta (G)$. On a positive side, we prove that $\zeta (G)$ is bounded by the pathwidth of $G$. We then show that the algorithmic problem of determining $\zeta (G)$ is NP-hard in graphs with diameter at most $2$. Finally, we show that at most one cop can approximate (arbitrary close) the location of the robber in the Euclidean plane

Centroidal localization game

One important problem in a network is to locate an (invisible) moving entity by using distance-detectors placed at strategical locations. For instance, the metric dimension of a graph $G$ is the minimum number $k$ of detectors placed in some vertices $\{v_1,\cdots,v_k\}$ such that the vector $(d_1,\cdots,d_k)$ of the distances $d(v_i,r)$ between the detectors and the entity's location $r$ allows to uniquely determine $r \in V(G)$. In a more realistic setting, instead of getting the exact distance information, given devices placed in $\{v_1,\cdots,v_k\}$, we get only relative distances between the entity's location $r$ and the devices (for every $1\leq i,j\leq k$, it is provided whether $d(v_i,r) >$, $<$, or $=$ to $d(v_j,r)$). The centroidal dimension of a graph $G$ is the minimum number of devices required to locate the entity in this setting. We consider the natural generalization of the latter problem, where vertices may be probed sequentially until the moving entity is located. At every turn, a set $\{v_1,\cdots,v_k\}$ of vertices is probed and then the relative distances between the vertices $v_i$ and the current location $r$ of the entity are given. If not located, the moving entity may move along one edge. Let $\zeta^* (G)$ be the minimum $k$ such that the entity is eventually located, whatever it does, in the graph $G$. We prove that $\zeta^* (T)\leq 2$ for every tree $T$ and give an upper bound on $\zeta^*(G\square H)$ in cartesian product of graphs $G$ and $H$. Our main result is that $\zeta^* (G)\leq 3$ for any outerplanar graph $G$. We then prove that $\zeta^* (G)$ is bounded by the pathwidth of $G$ plus 1 and that the optimization problem of determining $\zeta^* (G)$ is NP-hard in general graphs. Finally, we show that approximating (up to any constant distance) the entity's location in the Euclidean plane requires at most two vertices per turn