Alain Gély

Enumeration aspects of maximal cliques and bicliques

We present a general framework to study enumeration algorithms for maximal cliques and maximal bicliques of a graph. Given a graph G, we introduce the notion of the transition graph T(G) whose vertices are maximal cliques of G and arcs are transitions between cliques. We show that T(G) is a strongly connected graph and characterize a rooted cover tree of T(G) which appears implicitly in [D.S. Johnson, M. Yannakakis, C.H. Papadimitriou, On generating all maximal independent sets, Information Processing Letters 27 (1988) 119-123; S. Tsukiyama, M. Ide, M. Aiyoshi, I. Shirawaka, A new algorithm for generating all the independent sets, SIAM Journal on Computing 6 (1977) 505-517]. When G is a bipartite graph, we show that the Galois lattice of G is a partial graph of T(G) and we deduce that algorithms based on the Galois lattice are a particular search of T(G). Moreover, we show that algorithms in [G. Alexe, S. Alexe, Y. Crama, S. Foldes, P.L. Hammer, B. Simeone, Consensus algorithms for the generation of all maximal bicliques, Discrete Applied Mathematics 145 (1) (2004) 11-21; L. Nourine, O. Raynaud, A fast algorithm for building lattices, Information Processing Letters 71 (1999) 199-204] generate maximal bicliques of a bipartite graph in O(n^2) per maximal biclique, where n is the number of vertices in G. Finally, we show that under some specific numbering, the transition graph T(G) has a hamiltonian path for chordal and comparability graphs.

Uncovering and reducing hidden combinatorics in guigues-duquenne bases

Mannila and Raiha [5] have shown that minimum implicational bases can have an exponential number of implications. Aim of our paper is to understand how and why this combinatorial explosion arises and to propose mechanisms which reduce it.

A generic algorithm for generating closed sets of a binary relation

In this paper we propose a “divide and conquer” based generating algorithm for closed sets of a binary relation. We show that some existing algorithms are particular instances of our algorithm. This allows us to compare those algorithms and exhibit that the practical efficiency relies on the number of invalid closed sets generated. This number strongly depends on a choice function and the structure of the lattice. We exhibit a class of lattices for which no invalid closed sets are generated and thus reduce time complexity for such lattices. We made several tests which illustrate the impact of the choice function in practical efficiency.

Representing lattices using many-valued relations

This paper provides a representation theorem of lattices using many-valued relations. We show that any many-valued relation can be associated to a unique lattice which is a meet-sublattice of a product of chains. Conversely, to any lattice we can associate a many-valued relation such that its associated lattice is isomorphic to the initial one. Thereby, we obtain a representation theorem of lattices using many-valued relations. Moreover, since several many-valued relations might have the same associated lattice, we give a characterization of the minimal many-valued relation that can be associated to a lattice. We then sketch a polynomial time algorithm which computes such a minimal relation from either a lattice or an arbitrary relation. This representation presents several advantages: it is smaller than the usual binary representation; all known reconstruction algorithms working on binary relation can be used without loss of efficiency; it can be used by existing data mining processes.

Crown-free Lattices and Their Related Graphs

This paper investigates links between some classes of graphs and some classes of lattices. We show that a co-atomic lattice is crown-free (i.e. dismantlable) if and only if it is a maximal clique lattice of a strongly chordal graph. We also prove that each crown-free lattice that is not a chain contains at least two incomparable doubly-irreducible elements x1 and x2 such that ↑ x1 and ↑ x2 are chains.

About the enumeration algorithms of closed sets

This paper presents a review of enumeration technics used for the generation of closed sets. A link is made between classical enumeration algorithms of objects in graphs and algorithms for the enumeration of closed sets. A unified framework, the transition graph, is presented. It allows to better explain the behavior of the enumeration algorithms and to compare them independently of the data structures they use.

About the family of closure systems preserving non-unit implications in the guigues-duquenne base

Consider a Guigues-Duquenne base

DC Programming and DCA for Transmit Beamforming and Power Allocation in Multicasting Relay Network

\Sigma^{\mathcal{F}} = \Sigma^{\mathcal{F}}_{\mathcal{J}} \cup \Sigma^{\mathcal{F}}_{\downarrow}

Parsimonious cluster systems

of a closure system

DC Programming and DCA for Enhancing Physical Layer Security via Relay Beamforming Strategies

\mathcal{F}