83 research outputs found
Neighborhood Inclusions for Minimal Dominating Sets Enumeration: Linear and Polynomial Delay Algorithms in P_7 - Free and P_8 - Free Chordal Graphs
In [M. M. Kant\'e, V. Limouzy, A. Mary, and L. Nourine. On the enumeration of
minimal dominating sets and related notions. SIAM Journal on Discrete
Mathematics, 28(4):1916-1929, 2014] the authors give an delay
algorithm based on neighborhood inclusions for the enumeration of minimal
dominating sets in split and -free chordal graphs. In this paper, we
investigate generalizations of this technique to -free chordal graphs for
larger integers . In particular, we give and delays
algorithms in the classes of -free and -free chordal graphs. As for
-free chordal graphs for , we give evidence that such a technique
is inefficient as a key step of the algorithm, namely the irredundant extension
problem, becomes NP-complete.Comment: 16 pages, 3 figure
Extended Dualization: Application to Maximal Pattern Mining
International audienceThe dualization in arbitrary posets is a well-studied problem in combinatorial enumeration and is a crucial step in many applications in logics, databases, artificial intelligence and pattern mining.The objective of this paper is to study reductions of the dualization problem on arbitrary posets to the dualization problem on boolean lattices, for which output quasi-polynomial time algorithms exist. Quasi-polynomial time algorithms are algorithms which run in no(logn) where n is the size of the input and output. We introduce convex embedding and poset reflection as key notions to characterize such reductions. As a consequence, we identify posets, which are not boolean lattices, for which the dualization problem remains in quasi-polynomial time and propose a classification of posets with respect to dualization.From these results, we study how they can be applied to maximal pattern mining problems. We deduce a new classification of pattern mining problems and we point out how known problems involving sequences and conjunctive queries patterns, fit into this classification. Finally, we explain how to adapt the seminal Dualize & Advance algorithm to deal with such patterns.As far as we know, this is the first contribution to explicit non-trivial reductions for studying the hardness of maximal pattern mining problems and to extend the Dualize & Advance algorithm for complex patterns
Minimum Implicational Basis for -Semidistributive Lattices
Received 13 January 2006, Received in Revised form 21 February 2006, Accepted 21 April 2006International audienceFor a ∧-semidistributive lattice L, we study some particular implicational systems and show that the cardinality of a minimum implicational basis is polynomial in the size of join-irreducible elements of the lattice L. We also provide a polynomial time algorithm to compute a minimum implicational basis for L
Dualization on Partially Ordered Sets: Preliminary Results
International audienceThe dualization problem on arbitrary posets is a crucial step in many applications in logics, databases, artificial intelligence and pattern mining. The objective of this paper is to study reductions of the dualization problem on arbitrary posets to the dualization problem on boolean lattices, for which output quasi-polynomial time algorithms exist. We introduce convex embedding and poset reflection as key notions to characterize such reductions. As a consequence, we identify posets, which are not boolean lattices, for which the dualization problem remains quasi-polynomial and propose a classification of posets with respect to dualization. As far as we know, this is the first contribution to explicit non-trivial reductions for studying the hardness of dualization problems on arbitrary posets
On the Enumeration of Minimal Dominating Sets and Related Notions
A dominating set in a graph is a subset of its vertex set such that each
vertex is either in or has a neighbour in . In this paper, we are
interested in the enumeration of (inclusion-wise) minimal dominating sets in
graphs, called the Dom-Enum problem. It is well known that this problem can be
polynomially reduced to the Trans-Enum problem in hypergraphs, i.e., the
problem of enumerating all minimal transversals in a hypergraph. Firstly we
show that the Trans-Enum problem can be polynomially reduced to the Dom-Enum
problem. As a consequence there exists an output-polynomial time algorithm for
the Trans-Enum problem if and only if there exists one for the Dom-Enum
problem. Secondly, we study the Dom-Enum problem in some graph classes. We give
an output-polynomial time algorithm for the Dom-Enum problem in split graphs,
and introduce the completion of a graph to obtain an output-polynomial time
algorithm for the Dom-Enum problem in -free chordal graphs, a proper
superclass of split graphs. Finally, we investigate the complexity of the
enumeration of (inclusion-wise) minimal connected dominating sets and minimal
total dominating sets of graphs. We show that there exists an output-polynomial
time algorithm for the Dom-Enum problem (or equivalently Trans-Enum problem) if
and only if there exists one for the following enumeration problems: minimal
total dominating sets, minimal total dominating sets in split graphs, minimal
connected dominating sets in split graphs, minimal dominating sets in
co-bipartite graphs.Comment: 15 pages, 3 figures, In revisio
A Polynomial Delay Algorithm for Enumerating Minimal Dominating Sets in Chordal Graphs
An output-polynomial algorithm for the listing of minimal dominating sets in
graphs is a challenging open problem and is known to be equivalent to the
well-known Transversal problem which asks for an output-polynomial algorithm
for listing the set of minimal hitting sets in hypergraphs. We give a
polynomial delay algorithm to list the set of minimal dominating sets in
chordal graphs, an important and well-studied graph class where such an
algorithm was open for a while.Comment: 13 pages, 1 figure, submitte
Reasoning in description logics with variables: preliminary results regarding the EL logic.
International audienceThis paper studies the extension of description logics with variables ranging over infinite domains of concept names and role names.As a preliminary work, we consider more specifically the extension of the logic EL with variables and we investigate in this context two reasoningmechanisms, namely compliance (a kind of matching) and pattern containment. The main technical results are derived by establishing acorrespondance between the EL logic and finite variable automata
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