Sommets interdits et obligatoires

International audienceUne instance du problème est un graphe, un ensemble F de sommets interdits et un ensemble R de sommets obligatoires. Nous montrons que construire un vertex cover, connexe ou pas, de taille minimale, contenant tous les sommets de R et aucun sommet de F, peut être 2-approché (s'il existe). Nous montrons aussi que décider s'il existe ou pas un ensemble dominant indépendant contenant tout R et aucun sommet de F est NP-complet

On Iterated Dominance, Matrix Elimination, and Matched Paths

We study computational problems arising from the iterated removal of weakly dominated actions in anonymous games. Our main result shows that it is NP-complete to decide whether an anonymous game with three actions can be solved via iterated weak dominance. The two-action case can be reformulated as a natural elimination problem on a matrix, the complexity of which turns out to be surprisingly difficult to characterize and ultimately remains open. We however establish connections to a matching problem along paths in a directed graph, which is computationally hard in general but can also be used to identify tractable cases of matrix elimination. We finally identify different classes of anonymous games where iterated dominance is in P and NP-complete, respectively.Comment: 12 pages, 3 figures, 27th International Symposium on Theoretical Aspects of Computer Science (STACS

The Complexity of Connectivity Problems in Forbidden-Transition Graphs And Edge-Colored Graphs

The notion of forbidden-transition graphs allows for a robust generalization of walks in graphs. In a forbidden-transition graph, every pair of edges incident to a common vertex is permitted or forbidden; a walk is compatible if all pairs of consecutive edges on the walk are permitted. Forbidden-transition graphs and related models have found applications in a variety of fields, such as routing in optical telecommunication networks, road networks, and bio-informatics. We initiate the study of fundamental connectivity problems from the point of view of parameterized complexity, including an in-depth study of tractability with regards to various graph-width parameters. Among several results, we prove that finding a simple compatible path between given endpoints in a forbidden-transition graph is W[1]-hard when parameterized by the vertex-deletion distance to a linear forest (so it is also hard when parameterized by pathwidth or treewidth). On the other hand, we show an algebraic trick that yields tractability when parameterized by treewidth of finding a properly colored Hamiltonian cycle in an edge-colored graph; properly colored walks in edge-colored graphs is one of the most studied special cases of compatible walks in forbidden-transition graphs

Stability of networks of delay-coupled delay oscillators

Dynamical networks with time delays can pose a considerable challenge for mathematical analysis. Here, we extend the approach of generalized modeling to investigate the stability of large networks of delay-coupled delay oscillators. When the local dynamical stability of the network is plotted as a function of the two delays then a pattern of tongues is revealed. Exploiting a link between structure and dynamics, we identify conditions under which perturbations of the topology have a strong impact on the stability. If these critical regions are avoided the local stability of large random networks can be well approximated analytically

Vers des réseaux optiques efficaces et tolérants aux pannes : complexité et algorithmes

We study in this thesis optimization problems with application in optical networks. The problems we consider are related to fault-tolerance and efficient resource allocation and the results we obtain are mainly related to the computational complexity of these problems. The first part of this thesis is devoted to finding paths and disjoint paths. Finding a path is crucial in all types of networks in order to set up connections and finding disjoint paths is a common approach used to provide some degree of protection against failures in networks. We study these problems under different settings. We first focus on finding paths and node or link-disjoint paths in networks with asymmetric nodes, which are nodes with restrictions on their internal connectivity. Afterwards, we consider networks with star Shared Risk Link Groups (SRLGs) which are groups of links that might fail simultaneously due to a localized event. In these networks, we investigate the problem of finding SRLG-disjoint paths. The second part of this thesis focuses on the problem of Routing and Spectrum Assignment (RSA) in Elastic Optical Networks (EONs). EONs are proposed as the new generation of optical networks and they aim at an efficient and flexible use of the optical resources. RSA is the key problem in EONs and it deals with allocating resources to requests under multiple constraints. We first study the static version of RSA in tree networks. Afterwards, we examine a dynamic version of RSA in which a non-disruptive spectrum defragmentation technique is used. Finally, we present in the appendix another problem that has been studied during this thesis.Nous étudions dans cette thèse des problèmes d’optimisation avec applications dans les réseaux optiques. Les problèmes étudiés sont liés à la tolérance aux pannes et à l’utilisation efficace des ressources. Les résultats obtenus portent principalement sur la complexité de calcul de ces problèmes. La première partie de cette thèse est consacrée aux problèmes de trouver des chemins et des chemins disjoints. La recherche d’un chemin est essentielle dans tout type de réseaux afin d’y établir des connexions et la recherche de chemins disjoints est souvent utilisée pour garantir un certain niveau de protection contre les pannes dans les réseaux. Nous étudions ces problèmes dans des contextes différents. Nous traitons d’abord les problèmes de trouver un chemin et des chemins lien ou nœud- disjoints dans des réseaux avec nœuds asymétriques, c’est-à-dire des nœuds avec restrictions sur leur connectivité interne. Ensuite, nous considérons les réseaux avec des groupes de liens partageant un risque (SRLG) en étoile : ensembles de liens qui peuvent tomber en panne en même temps suite à un événement local. Dans ce type de réseaux, nous examinons le problème de recherche des chemins SRLG-disjoints. La deuxième partie de cette thèse est consacrée au problème de routage et d’allocation de spectre (RSA) dans les réseaux optiques élastiques (EONs). Les EONs sont proposés comme la nouvelle génération des réseaux optiques et ils visent une utilisation plus efficace et flexible des ressources optiques. Le problème RSA est central dans les EONs. Il concerne l’allocation de ressources aux requêtes sous plusieurs contraintes

Regular Languages meet Prefix Sorting

Indexing strings via prefix (or suffix) sorting is, arguably, one of the most successful algorithmic techniques developed in the last decades. Can indexing be extended to languages? The main contribution of this paper is to initiate the study of the sub-class of regular languages accepted by an automaton whose states can be prefix-sorted. Starting from the recent notion of Wheeler graph [Gagie et al., TCS 2017]-which extends naturally the concept of prefix sorting to labeled graphs-we investigate the properties of Wheeler languages, that is, regular languages admitting an accepting Wheeler finite automaton. Interestingly, we characterize this family as the natural extension of regular languages endowed with the co-lexicographic ordering: when sorted, the strings belonging to a Wheeler language are partitioned into a finite number of co-lexicographic intervals, each formed by elements from a single Myhill-Nerode equivalence class. Moreover: (i) We show that every Wheeler NFA (WNFA) with $n$ states admits an equivalent Wheeler DFA (WDFA) with at most $2n-1-|\Sigma|$ states that can be computed in $O(n^3)$ time. This is in sharp contrast with general NFAs. (ii) We describe a quadratic algorithm to prefix-sort a proper superset of the WDFAs, a $O(n\log n)$-time online algorithm to sort acyclic WDFAs, and an optimal linear-time offline algorithm to sort general WDFAs. By contribution (i), our algorithms can also be used to index any WNFA at the moderate price of doubling the automaton's size. (iii) We provide a minimization theorem that characterizes the smallest WDFA recognizing the same language of any input WDFA. The corresponding constructive algorithm runs in optimal linear time in the acyclic case, and in $O(n\log n)$ time in the general case. (iv) We show how to compute the smallest WDFA equivalent to any acyclic DFA in nearly-optimal time.Comment: added minimization theorems; uploaded submitted version; New version with new results (W-MH theorem, linear determinization), added author: Giovanna D'Agostin