Densité, VC-dimension et étiquetages de graphes

Sébastien Ratel 1, 2
1 ACRO - Algorithmique, Combinatoire et Recherche Opérationnelle
LIS - Laboratoire d'Informatique et Systèmes
2 DALGO - Algorithmique Distribuée
LIS - Laboratoire d'Informatique et Systèmes
Abstract : its vertices. Such a scheme consists in an encoding function that constructs a (short) binary label for every vertex, and in a decoding function that can answer (quickly) a predefined query using only the local information contained in two labels (with no further knowledge of the graph). Constructing such distributed representations constituted the initial motivation of most of the results of this document. However, this manuscript concerns problem of more general interest such as bounding the density of graphs, studying the VC-dimension of set families, or investigating on metric and structural properties of graphs. As a first contribution, we upper bound the density of the subgraphs of Cartesian products of graphs, and of the subgraphs of halved-cubes. To do so, we extend the classical notion of VC-dimension (already used in 1994 by Haussler, Littlestone, and Warmuth to upper bound the density of the subgraphs of hypercubes). From our results, we deduce upper bounds on the size of labels used by an adjacency labeling scheme on these graph classes. We then investigate on distance and routing labeling schemes for two important families of metric graph theory: median graphs and bridged graphs. We first show that the class of cube-free median graphs on n vertices enjoys distance and routing labeling schemes both using labels of O(log^3 n) bits. These labels can be decoded in constant time to respectively return the exact distance between two vertices, or a port to take from a source vertex in order to get (strictly) closer to a target one. We then describe an approximate distance labeling scheme for the family of K_4-free bridged graphs on n vertices. This scheme also uses labels of size O(log^3 n) that can be decoded in constant time to return a value of at most four time the exact distance between two vertices.
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Sébastien Ratel. Densité, VC-dimension et étiquetages de graphes. Mathématique discrète [cs.DM]. Aix-Marseile Université, 2019. Français. ⟨tel-02436255⟩

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