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On the dualization problem in graphs, hypergraphs, and lattices

Abstract : This thesis focuses on graphs, hypergraphs, and lattices. We study the complexity of the dualization of monotone Boolean functions, and its generalizations, through the many shapes it takes on these structures: minimal dominating sets enumeration, minimal transversals enumeration, lattice dualization, and meet-irreducible enumeration. Both tractable and intractable results are obtained, and future research directions are proposed. The thesis is organized as follows. A first part is devoted to the enumeration of minimal dominating sets in graphs. We obtain new output-polynomial time algorithms in graph classes related to Kt-free graphs and to posets of bounded dimension. A second part is devoted to generalizations of this problem in lattices. One generalization concerns the dualization in lattices given by implicational bases, the other deals with the enumeration of meet-irreducible elements. Both tractability and intractability results are obtained under various restrictions concerning width, acyclicity, and premises’ size in the implicational base. The two parts are sprinkled with hypergraph transversals enumeration and related notions.
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Submitted on : Wednesday, December 2, 2020 - 6:33:06 PM
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Oscar Defrain. On the dualization problem in graphs, hypergraphs, and lattices. Other [cs.OH]. Université Clermont Auvergne [2017-2020], 2020. English. ⟨NNT : 2020CLFAC022⟩. ⟨tel-03036782⟩



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