17 research outputs found

    Defensive Alliances in Signed Networks

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    The analysis of (social) networks and multi-agent systems is a central theme in Artificial Intelligence. Some line of research deals with finding groups of agents that could work together to achieve a certain goal. To this end, different notions of so-called clusters or communities have been introduced in the literature of graphs and networks. Among these, defensive alliance is a kind of quantitative group structure. However, all studies on the alliance so for have ignored one aspect that is central to the formation of alliances on a very intuitive level, assuming that the agents are preconditioned concerning their attitude towards other agents: they prefer to be in some group (alliance) together with the agents they like, so that they are happy to help each other towards their common aim, possibly then working against the agents outside of their group that they dislike. Signed networks were introduced in the psychology literature to model liking and disliking between agents, generalizing graphs in a natural way. Hence, we propose the novel notion of a defensive alliance in the context of signed networks. We then investigate several natural algorithmic questions related to this notion. These, and also combinatorial findings, connect our notion to that of correlation clustering, which is a well-established idea of finding groups of agents within a signed network. Also, we introduce a new structural parameter for signed graphs, signed neighborhood diversity snd, and exhibit a parameterized algorithm that finds a smallest defensive alliance in a signed graph

    Study of the Gromov hyperbolicity constant on graphs

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    The concept of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space and Riemannian manifolds of negative sectional curvature. It is remarkable that a simple concept leads to such a rich general theory. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of any geodesic metric space is equivalent to the hyperbolicity of a graph related to it. In this Ph. D. Thesis we characterize the hyperbolicity constant of interval graphs and circular-arc graphs. Likewise, we provide relationships between dominant sets and the hyperbolicity constant. Finally, we study the invariance of the hyperbolicity constant when the graphs are transformed by several operators.Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Domingo de Guzmán Pestana Galván.- Secretaria: Ana Portilla Ferreira.- Vocal: Eva Tourís Loj

    Treewidth in Non-Ground Answer Set Solving and Alliance Problems in Graphs

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    To solve hard problems efficiently via answer set programming (ASP), a promising approach is to take advantage of the fact that real-world instances of many hard problems exhibit small treewidth. Algorithms that exploit this have already been proposed -- however, they suffer from an enormous overhead. In the thesis, we present improvements in the algorithmic methodology for leveraging bounded treewidth that are especially targeted toward problems involving subset minimization. This can be useful for many problems at the second level of the polynomial hierarchy like solving disjunctive ground ASP. Moreover, we define classes of non-ground ASP programs such that grounding such a program together with input facts does not lead to an excessive increase in treewidth of the resulting ground program when compared to the treewidth of the input. This allows ASP users to take advantage of the fact that state-of-the-art ASP solvers perform better on ground programs of small treewidth. Finally, we resolve several open questions on the complexity of alliance problems in graphs. In particular, we settle the long-standing open questions of the complexity of the Secure Set problem and whether the Defensive Alliance problem is fixed-parameter tractable when parameterized by treewidth

    Complexity of Secure Sets

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    A secure set SS in a graph is defined as a set of vertices such that for any X⊆SX\subseteq S the majority of vertices in the neighborhood of XX belongs to SS. It is known that deciding whether a set SS is secure in a graph is co-NP-complete. However, it is still open how this result contributes to the actual complexity of deciding whether for a given graph GG and integer kk, a non-empty secure set for GG of size at most kk exists. In this work, we pinpoint the complexity of this problem by showing that it is Σ2P\Sigma^P_2-complete. Furthermore, the problem has so far not been subject to a parameterized complexity analysis that considers structural parameters. In the present work, we prove that the problem is W[1]W[1]-hard when parameterized by treewidth. This is surprising since the problem is known to be FPT when parameterized by solution size and "subset problems" that satisfy this property usually tend to be FPT for bounded treewidth as well. Finally, we give an upper bound by showing membership in XP, and we provide a positive result in the form of an FPT algorithm for checking whether a given set is secure on graphs of bounded treewidth.Comment: 28 pages, 9 figures, short version accepted at WG 201
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