Tag: networks
Boolean dimension of boolean lattices
summary: Partially ordered sets (or posets for short) are structures that arise naturally in mathematics in the study of binary relations between objects. A poset consists of a ground set -- a set of elements -- and a partial order among them, and are usually visualled with directed acyclic graphs (DAGs). Indeed ...
Efficient maximal cliques enumeration in weakly closed graphs
summary: We show that the algorithm presented in [J. Fox, T. Roughgarden, C. Seshadhri, F. Wei, and N. Wein. Finding cliques in social networks: A new distribution-free model. SIAM journal on computing, 49(2):448-464, 2020] can be modified to have enumeration time complexity αO(n·poly(c)). Here parameter ...
Auction mechanisms for allocation and pricing in Edge Computing
summary: Soon, billions of devices will be connected to the Internet, and most of them will need external resources to store and process their data. Edge Computing can benefit these devices by making these resources available at the network's edges to be closer to the user. However, these offered ...
A Closure Lemma for tough graphs and Hamiltonian degree conditions
summary: A graph G is Hamiltonian if there exists a cycle in G containing all vertices of G exactly once. A graph G is t-tough if, for all subsets of vertices S, the number of connected components in G−S is at most |S|/t. We extended the theorem of ...
Memory-Optimization for Self-Stabilizing Distributed Algorithms
summary: Self-stabilization is a suitable paradigm for distributed systems, particularly prone to transient faults. Errors such as memory or messages corruption, break of a communication link, can put the system in an inconsistent state. A protocol is self-stabilizing if, whatever the initial state of the system, it guarantees that it ...
Complexity of positional games
summary: Positional games are two-player games played on a hypergraph. The players alternate selecting vertices of the hypergraph, and the winning conditions depend solely on the filling of the hyperedges. Tic-tac-toe is a famous example of a positional game, with the rows, columns, and diagonals forming the hyperedges and the ...
Proper vertex coloring with odd occurrence - Probabilistic approach
summary: In graph theory, a graph coloring is an assignment of colors to elements of a graph subject to certain constraints. A vertex coloring is said to be proper if any pair of two adjacent vertices are assigned distinct colors. For a graph G, the chromatic number χ(G) is ...
Some results on directed coloring
summary: Proper coloring of undirected graphs lies among the most studied
problems in graph theory. It asks to color vertices while giving
different colors to adjacent ones.
In 1982, Neumann-Lara introduced a generalization of this problem to
directed graphs. When walking in an undirected graph, an undirected edge
between two ...
Density of sphere packings: from coins to oranges
summary: How to stack an infinite number of oranges to maximize the proportion of the covered space? Kepler conjectured that the "cannonball" packing is an optimal way to do it. This conjecture took almost 400 years to prove and the proof of Hales and Ferguson consists of 6 papers and ...
An introduction to twin-width in graphs via the study of highly inapproximable problems
summary: The goal of this seminar is to introduce the graph parameter "twin-width", defined by Bonnet et al. in 2020. The collection of graphs with bounded twin-width generalizes many well-known families of graphs : planar, bounded treewidth and cliquewidth, unit interval,... The first motivation behind this parameter was that FO-expressible NP-hard ...