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Matrix decompositions and algorithmic applications to (hyper)graphs

Abstract : In the last decades, considerable efforts have been spent to characterize what makes NP-hard problems tractable. A successful approach in this line of research is the theory of parameterized complexity introduced by Downey and Fellows in the nineties.In this framework, the complexity of a problem is not measured only in terms of the input size, but also in terms of a parameter on the input. One of the most well-studied parameters is tree-width, a graph parameter which measures how close a graph is to the topological structure of a tree.It appears that tree-width has numerous structural properties and algorithmic applications.However, only sparse graph classes can have bounded tree-width. But, many NP-hard problems are tractable on dense graph classes. Most of the time, this tractability can be explained by the ability of these graphs to be recursively decomposable along vertex bipartitions (A,B) where the adjacency between A and B is simple to describe. A lot of graph parameters -- called width measures -- have been defined to characterize this ability, the most remarkable ones are certainly clique-width, rank-width, and mim-width. In this thesis, we study the algorithmic properties of these width measures. We provide a framework that generalizes and simplifies the tools developed for tree-width and for problems with a constraint of acyclicity or connectivity such as Connected Vertex Cover, Connected Dominating Set, Feedback Vertex Set, etc. For all these problems, we obtain 2^{O(k)}⋅n^{O(1)}, 2^{O(k log(k))}⋅n^{O(1)}, 2^{O(k²)}⋅n^{O(1)} and n^{O(k)} time algorithms parameterized respectively by clique-width, Q-rank-width, rank-width and mim-width. We also prove that there exists an algorithm solving Hamiltonian Cycle in time n^{O(k)}, when a clique-width decomposition of width k is given. Finally, we prove that we can count in polynomial time the minimal transversals of β-acyclic hypergraphs and the minimal dominating sets of strongly chordal graphs. All these results offer promising perspectives towards a generalization of width measures and their algorithmic applications.
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Benjamin Bergougnoux. Matrix decompositions and algorithmic applications to (hyper)graphs. Other [cs.OH]. Université Clermont Auvergne, 2019. English. ⟨NNT : 2019CLFAC025⟩. ⟨tel-02388683⟩

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