Fully polynomial FPT algorithms for some classes of bounded clique-width graphs

Parameterized complexity theory has enabled a refined classification of the difficulty of NP-hard optimization problems on graphs with respect to key structural properties, and so to a better understanding of their true difficulties. More recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as: Strong Exponential Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms, nor even truly subcubic algorithms (Williams and Williams, FOCS 2010 and Abboud, Grandoni, Williams, SODA 2015). A central technique used to tackle the difficulty of the above mentioned problems is fixed-parameter algorithms for polynomial-time problems with polynomial dependency in the fixed parameter (P-FPT). This technique was introduced by Abboud, Williams and Wang in SODA 2016 and continued by Husfeldt (IPEC 2016) and Fomin et al. (SODA 2017), using the treewidth as a parameter. Applying this technique to clique-width, another important graph parameter, remained to be done. In this paper we study several graph theoretic problems for which hardness results exist such as cycle problems (triangle detection, triangle counting, girth, diameter), distance problems (diameter, eccentricities, Gromov hyperbolicity, betweenness centrality) and maximum matching. We provide hardness results and fully polynomial FPT algorithms, using clique-width and some of its upper-bounds as parameters (split-width, modular-width and $P\_4$-sparseness). We believe that our most important result is an ${\cal O}(k^4 \cdot n + m)$-time algorithm for computing a maximum matching where $k$ is either the modular-width or the $P\_4$-sparseness. The latter generalizes many algorithms that have been introduced so far for specific subclasses such as cographs, $P\_4$-lite graphs, $P\_4$-extendible graphs and $P\_4$-tidy graphs. Our algorithms are based on preprocessing methods using modular decomposition, split decomposition and primeval decomposition. Thus they can also be generalized to some graph classes with unbounded clique-width

Grooming

chapter VI.27International audienceState-of-the-art on traffic grooming with a design theory approac

How Graph Theory can help Communications Engineering

International audienceWe give an overview of different aspects of graph theory which can be applied in communication engineering, not trying to present immediate results to be applied neither a complete survey of results, but to give a flavor of how graph theory can help this field. We deal in this paper with network topologies, resource competition, state transition diagrams and specific models for optical networks

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)

A distributed algorithm for computing and updating the process number of a forest

In this paper, we present a distributed algorithm to compute various parameters of a tree such as the process number, the edge search number or the node search number and so the pathwidth. This algorithm requires n steps, an overall computation time of O(n log(n)), and n messages of size log_3(n)+3. We then propose a distributed algorithm to update the process number (or the node search number, or the edge search number) of each component of a forest after adding or deleting an edge. This second algorithm requires O(D) steps, an overall computation time of O(D log(n)), and O(D) messages of size log_3(n)+3, where D is the diameter of the modified connected component. Finally, we show how to extend our algorithms to trees and forests of unknown size using messages of less than 2a+4+e bits, where a is the parameter to be determined and e=1 for updates algorithms

OTIS-Based Multi-Hop Multi-OPS Lightwave Networks

International audienceAdvances in optical technology, such as low loss Optical Passive Star couplers (OPS) and the possibility of building tunable optical transmitters and receivers have increased the interest for multiprocessor architectures based on lightwave networks because of the vast bandwidth available. Many research have been done at both technological and theoretical level. An essential effort has to be done in linking those results. In this paper we propose optical designs for two multi-OPS networks: the single-hop POPS network and the multi-hop stack-Kautz network; using the Optical Transpose Interconnecting System (OTIS) architecture, from the Optoelectronic Computing Group of UCSD. In order to achieve our result, we also provide the optical design of a generalization of the Kautz digraph, using OTIS

Diving angle of great cormorants

Seabirds can maximize the relative time spent at depths where prey occur by minimizing the commuting time taken to reach these depths. One way to achieve this goal is to modify dive angle, but there are few measures of dive-angle in free-foraging seabirds. In 2003, we monitored simultaneously the swimming speeds and diving depths of great cormorants (Phalacrocorax carbo) foraging off the Greenland coast, and used these data to reconstruct their descent angle. Both males and females dived on average 12 m. We suggest that birds are able to reduce their descent time for dives beyond this depth by performing pre-dive leaps that allow them to use the momentum of the fall to descend almost vertically and at great speeds. Such pre-dive leaps in shallower dives would be unsuitable because of the proximity of the seabed and the risk of startling prey. Finally, in contrast with deeper divers, descent angles were not steeper when undulations were observed in the depth profile of the previous dive, probably because birds feed on dispersed prey

Space and Time Trade-Off for the k Shortest Simple Paths Problem

The k shortest simple path problem (kSSP) asks to compute a set of top-k shortest simple paths from a vertex s to a vertex t in a digraph. Yen (1971) proposed the first algorithm with the best known theoretical complexity of O(kn(m+n log n)) for a digraph with n vertices and m arcs. Since then, the problem has been widely studied from an algorithm engineering perspective, and impressive improvements have been achieved. In particular, Kurz and Mutzel (2016) proposed a sidetracks-based (SB) algorithm which is currently the fastest solution. In this work, we propose two improvements of this algorithm. We first show how to speed up the SB algorithm using dynamic updates of shortest path trees. We did experiments on some road networks of the 9th DIMAC\u27S challenge with up to about half a million nodes and one million arcs. Our computational results show an average speed up by a factor of 1.5 to 2 with a similar working memory consumption as SB. We then propose a second algorithm enabling to significantly reduce the working memory at the cost of an increase of the running time (up to two times slower). Our experiments on the same data set show, on average, a reduction by a factor of 1.5 to 2 of the working memory

Dynamic Routing and Spectrum Assignment with Non-Disruptive Defragmentation

International audienceL'augmentation du trafic Internet motive une évolution des réseaux WDM traditionnels vers les réseaux optiques élastiques (Elastic Optical Networks, EON). Les EONs sont conçus pour optimiser l'utilisation des ressources optiques; ils permettent l'attribution de bandes de largeurs quelconques et offrent ainsi plus de souplesse que les réseaux WDM avec leurs bandes fixes. Cependant, la gestion du spectre dans les EONs devient plus difficile du fait de la fragmentation. Dans ce papier, nous proposons deux algorithmes qui permettent de résoudre le problème de Routage et d'Allocation de Spectre d'une nouvelle requête en utilisant une technique de défragmentation non-perturbatrice