Lexique de la théorie des graphes

Résumé

NOTOC A ; Acyclique : graphe ne contenant pas de cycle. ; Adjacence : une liste d'adjacence est une structure de données constituée d'un tableau dont le i-ème élément correspond à la liste des voisins du i-ème sommet. ; Adjacence : une matrice d'adjacence est une matrice carrée usuellement notée A, de dimensions |V| \times |V|, dont chaque élément A_{ij} est égal au nombre d'arêtes incidentes (ayant pour extrémités) aux sommets d'indices i et j (pour un graphe simple non pondéré, A_{ij} \in {0, 1}). Dans le cas d'un graphe pondéré, chaque élément est égal à la somme du poids des arêtes incidentes. ; Adjacence : une relation d'adjacence propriété de deux sommets d'être connectés par la même arête (dits sommets adjacents) ou propriété de deux arêtes de présenter une extrémité commune (dites arêtes adjacentes). Également appelé relation de voisinage. ; Adjoint : un g

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https://fr.wikipedia.org/wiki/Lexique_de_la_th%C3%A9orie_des_graphes

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Unambiguous DNFs and Alon-Saks-Seymour

Mika Tapani Göös, Siddhartha Jain

We exhibit an unambiguous k-DNF formula that requires CNF width (Omega) over tilde (k(2)), which is optimal up to logarithmic factors. As a consequence, we get a near-optimal solution to the Alon-Saks-Seymour problem in graph theory (posed in 1991), which asks: How large a gap can there be between the chromatic number of a graph and its biclique partition number? Our result is also known to imply several other improved separations in query and communication complexity.

IEEE COMPUTER SOC2022

Chargement

Consider the family of bounded degree graphs in any minor-closed family (such as planar graphs). Let d be the degree bound and n be the number of vertices of such a graph. Graphs in these classes have hyperfinite decompositions, where, one removes a small fraction of edges of the graph controlled by a proximity parameter to get connected components of size independent of n. An important tool for sublinear algorithms and property testing for such classes is the partition oracle, introduced by the seminal work of Hassidim-Kelner-Nguyen-Onak (FOCS 2009). A partition oracle is a local procedure that gives consistent access to a hyperfinite decomposition, without any preprocessing. Given a query vertex v, the partition oracle outputs the component containing v in time independent of n. All the answers are consistent with a single hyperfinite decomposition. The partition oracle of Hassidim et al. runs in time exponential in the proximity parameter per query. They pose the open problem of whether partition oracles which run in time polynomial in reciprocal of proximity parameter can be built. Levi-Ron (ICALP 2013) give a refinement of the previous approach, to get a partition oracle that runs in quasipolynomial time per query. In this paper, we resolve this open problem and give polynomial time partition oracles (in reciprocal of proximity parameter) for bounded degree graphs in any minor-closed family. Unlike the previous line of work based on combinatorial methods, we employ techniques from spectral graph theory. We build on a recent spectral graph theoretical toolkit for minor-closed graph families, introduced by the authors to develop efficient property testers. A consequence of our result is an efficient property tester for any monotone and additive with running time property of minor-closed families (such as bipartite planar graphs). Our result also gives query efficient algorithms for additive approximations for problems such as maximum matching, minimum vertex cover, maximum independent set, and minimum dominating set for these graph families.

IEEE COMPUTER SOC2022

Chargement

Space-Efficient Representations of Graphs

Jakab Tardos

With the increasing prevalence of massive datasets, it becomes important to design algorithmic techniques for dealing with scenarios where the input to be processed does not fit in the memory of a single machine. Many highly successful approaches have emerged in recent decades, such as processing the data in a stream, parallel processing, and data compression. The aim of this thesis is to apply these techniques to various important graph theoretical problems. Our contributions can be broadly classified into two categories: spectral graph theory, and maximum matching.Spectral Graph Theory. Spectral sparsification is a technique of rendering an arbitrary graph sparse, while approximately preserving the quadratic form of the Laplacian matrix. In this thesis, we extend the result of (Kapralov et al.), and propose a sketch and corresponding decoding algorithm for constructing a spectral sparsifier from a dynamic stream of edge insertions and deletions. The size of the resulting sparsifier, the size of the sketch, and the decoding time are all nearly linear in the number of vertices, and consequently nearly optimal.The concept of spectral sparsification has recently been generalized to hypergraphs (Soma and Yoshida) -- an analogue of graphs for modeling higher order relationships. As one of the main contributions of the thesis, we prove for the first time the existence of nearly-linear sized spectral sparsifiers for arbitrary hypergraphs, and provide a corresponding nearly-linear time algorithm for constructing them. Through a lower bound construction, we show that our sparsifiers achieve nearly-optimal compression of the hypergraph spectral structure.On the more applied side of spectral graph theory, we present a fully scalable MPC (massively parallel computation) algorithm which is capable of simulating a large number of independent random walks of length L from an arbitrary starting distribution in O(log(L)) rounds.Maximum Matching. We propose a novel randomized composable coreset for the problem of maximum matching, called the matching skeleton. The coreset achieves a 1/2 approximation, while having fewer than n edges.We also propose a new, highly space-efficient variant of a peeling algorithm for maximum matching. With this, we are able to approximate the maximum matching size of a graph to within a constant factor, using a stream of m uniformly random edges (where m is the total number of edges), in as little as O(log^2(n)) space. Conversely, we show that significantly fewer (that is m^(1-Omega(1))) samples do not suffice, even with unlimited space. Finally, we design a Local Computation Algorithm, which implicitly construct a constant-approximate maximum matching in time and space that is nearly linear in the maximum degree.

EPFL2022

Chargement

Space-Efficient Representations of Graphs

Jakab Tardos

EPFL2022

IEEE COMPUTER SOC2022