Search

A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general graphs when parameterized by the solution size. By the meta-theorem of Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating Set on planar graphs with at most $410k$ vertices, where $k$ is the size of the solution. This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominating Set, Edge Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue Dominating Set., Comment: 33 pages, 13 figures

In this paper we consider a variation of a recoloring problem, called the Color-Fixing. Let us have some non-proper $r$-coloring $\varphi$ of a graph $G$. We investigate the problem of finding a proper $r$-coloring of $G$, which is "the most similar" to $\varphi$, i.e. the number $k$ of vertices that have to be recolored is minimum possible. We observe that the problem is NP-complete for any $r \geq 3$, even for bipartite planar graphs. On the other hand, the problem is fixed-parameter tractable, when parameterized by the number of allowed transformations $k$. We provide an $2^n \cdot n^{\mathcal{O}(1)}$ algorithm for the problem (for any fixed $r$) and a linear algorithm for graphs with bounded treewidth. We also show several lower complexity bounds, using standard complexity assumptions. Finally, we investigate the {\em fixing number} of a graph $G$. It is the maximum possible distance (in the number of transformations) between some non-proper coloring of $G$ and a proper one., Comment: An extended abstract of this paper was presented on the conference SOFSEM 2015

Due to a classical result of Berge, it is known that a matching of any graph can be turned into a maximum matching by repeatedly augmenting alternating paths whose ends are not covered. In a recent work, Nisse, Salch and Weber considered the influence, on this process, of augmenting paths with length at most k only. Given a graph G, an initial matching M ⊆ E(G) and an odd integer k, the problem is to find a longest sequence of augmenting paths of length at most k that can be augmented sequentially from M. They proved that, when only paths of length at most k = 3 can be augmented, computing such a longest sequence can be done in polynomial time for any graph, while the same problem for any k ≥ 5 is NP-hard. Although the latter result remains true for bipartite graphs, the status of the complexity of the same problem for trees is not known. This work is dedicated to the complexity of this problem for trees. On the positive side, we first show that it can be solved in polynomial time for more classes of trees, namely bounded-degree trees (via a dynamic programming approach), caterpillars and trees where the nodes with degree at least 3 are sufficiently far apart. On the negative side, we show that, when only paths of length exactly k can be augmented, the problem becomes NP-hard already for k = 3, in the class of planar bipartite graphs with maximum degree 3 and arbitrary large girth. We also show that the latter problem is NP-hard in trees when k is part of the input.

In the fields of Algorithmic and Complexity, a large area of research is based on the assumption that P ≠ NP(Polynomial time and Non deterministic Polynomial time), which means that there are problems for which a solution can be verified but not constructed in polynomial time. Many natural problems are not in P, which means, that they have no efficient algorithm. In order to tackle such problems, many different branches of Algorithmic have been developed. One of them is called Parametric Complexity. It consists in developing exact algorithms whose complexity is measured as a function of the size of the instance and of a parameter. Such a parameter allows a more precise analysis of the complexity. In this context, an algorithm will be considered to be efficient if it is fixed parameter tractable (fpt), that is, if it has a complexity which is exponential in the parameter and polynomial in the size of the instance. Problems that can be solved by such an algorithm form the FPT class.Kernelisation is a technical that produces fpt algorithms, among others. It can be viewed as a preprocessing of the instance, with a guarantee on the compression of the data. More formally, a kernelisation is a polynomial reduction from a problem to itself, with the additional constraint that the size of the kernel, the reduced instance, is bounded by a function of the parameter. In order to obtain an fpt algorithm, it is sufficient to solve the problem in the reduced instance, by brute-force for example (which has exponential complexity, in the parameter). Hence, the existence of a kernelisiation implies the existence of an fpt algorithm. It holds that the converse is true also. Nevertheless, the existence of an efficient fpt algorithm does not imply a small kernel, meaning a kernel with a linear or polynomial size. Under certain hypotheses, it can be proved that some problems can not have a kernel (that is, are not in FPT) and that some problems in FPT do not have a polynomial kernel.One of the main results in the field of Kernelisation is the construction of a linear kernel for the Dominating Set problem on planar graphs, by Alber, Fellows and Niedermeier.To begin with, the region decomposition method proposed by Alber, Fellows and Niedermeier has been reused many times to develop kernels for variants of Dominating Set on planar graphs. Nevertheless, this method had quite a few inaccuracies, which has invalidated the proofs. In the first part of our thesis, we present a more thorough version of this method and we illustrate it with two examples: Red Blue Dominating Set and Total Dominating Set.Next, the method has been generalised to larger classes of graphs (bounded genus, minor-free, topological-minor-free), and to larger families of problems. These meta-results prove the existence of a linear or polynomial kernel for all problems verifying some generic conditions, on a class of sparse graphs. As a price of generality, the proofs do not provide constructive algorithms and the bound on the size of the kernel is not explicit. In the second part of our thesis, we make a first step to constructive meta-results. We propose a framework to build linear kernels based on principles of dynamic programming and a meta-result of Bodlaender, Fomin, Lokshtanov, Penninkx, Saurabh and Thilikos.; En algorithmique et en complexité, la plus grande part de la recherche se base sur l’hypothèse que P ≠ NP (Polynomial time et Non deterministic Polynomial time), c'est-à-dire qu'il existe des problèmes dont la solution peut être vérifiée mais non construite en temps polynomial. Si cette hypothèse est admise, de nombreux problèmes naturels ne sont pas dans P (c'est-à-dire, n'admettent pas d'algorithme efficace), ce qui a conduit au développement de nombreuses branches de l'algorithmique. L'une d'elles est la complexité paramétrée. Elle propose des algorithmes exacts, dont l'analyse est faite en fonction de la taille de l'instance et d'un paramètre. Ce paramètre permet une granularité plus fine dans l'analyse de la complexité.Un algorithme sera alors considéré comme efficace s'il est à paramètre fixé, c'est-à-dire, lorsque sa complexité est exponentielle en fonction du paramètre et polynomiale en fonction de la taille de l'instance. Ces algorithmes résolvent les problèmes de la classe FPT (Fixed Parameter Tractable).L'extraction de noyaux est une technique qui permet, entre autre, d’élaborer des algorithmes à paramètre fixé. Elle peut être vue comme un pré-calcul de l'instance, avec une garantie sur la compression des données. Plus formellement, une extraction de noyau est une réduction polynomiale depuis un problème vers lui même, avec la contrainte supplémentaire que la taille du noyau (l'instance réduite) est bornée en fonction du paramètre. Pour obtenir l’algorithme à paramètre fixé, il suffit de résoudre le problème dans le noyau, par exemple par une recherche exhaustive (de complexité exponentielle, en fonction du paramètre). L’existence d'un noyau implique donc l'existence d'un algorithme à paramètre fixé, la réciproque est également vraie. Cependant, l’existence d'un algorithme à paramètre fixé efficace ne garantit pas un petit noyau, c'est a dire un noyau dont la taille est linéaire ou polynomiale. Sous certaines hypothèses, il existe des problèmes n’admettant pas de noyau (c'est-à-dire hors de FPT) et il existe des problèmes de FPT n’admettant pas de noyaux polynomiaux.Un résultat majeur dans le domaine des noyaux est la construction d'un noyau linéaire pour le problème Domination dans les graphes planaires, par Alber, Fellows et Niedermeier.Tout d'abord, la méthode de décomposition en régions proposée par Alber, Fellows et Niedermeier, a permis de construire de nombreux noyaux pour des variantes de Domination dans les graphes planaires. Cependant cette méthode comportait un certain nombre d’imprécisions, ce qui rendait les preuves invalides. Dans la première partie de notre thèse, nous présentons cette méthode sous une forme plus rigoureuse et nous l’illustrons par deux problèmes : Domination Rouge Bleue et Domination Totale.Ensuite, la méthode a été généralisée, d'une part, sur des classes de graphes plus larges (de genre borné, sans-mineur, sans-mineur-topologique), d'autre part, pour une plus grande variété de problèmes. Ces méta-résultats prouvent l’existence de noyaux linéaires ou polynomiaux pour tout problème vérifiant certaines conditions génériques, sur une classe de graphes peu denses. Cependant, pour atteindre une telle généralité, il a fallu sacrifier la constructivité des preuves : les preuves ne fournissent pas d'algorithme d'extraction constructif et la borne sur le noyau n'est pas explicite. Dans la seconde partie de notre thèse nous effectuons un premier pas vers des méta-résultats constructifs ; nous proposons un cadre général pour construire des noyaux linéaires en nous inspirant des principes de la programmation dynamique et d'un méta-résultat de Bodlaender, Fomin, Lokshtanov, Penninkx, Saurabh et Thilikos.

A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general graphs when parameterized by the solution size. By the meta-theorem of Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating Set on planar graphs with at most $410k$ vertices, where $k$ is the size of the solution. This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominating Set, Edge Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue Dominating Set., Discrete Mathematics & Theoretical Computer Science ; Vol. 20 no. 1 ; 13658050