Terms Of Hypergraphs Research Articles

Свойства геометрических гиперграфов беспроводных компьютерных сетей

Mathematical models of wireless computer networks are considered, reflecting two types of interaction between nodes of the same network — broadcast and routing. A natural form of representation of such a network is a hypergraph, in which the direct links between nodes that do not require routing are specified by the hyper-edges. The routes are chains of hyper-edges. The concepts of geometric and random hypergraphs are introduced, the dependences of the probabilistic characteristics of random geometric hypergraphs on the number of vertices and radii of reliable reception / transmission of radio signals by network nodes are studied — the mathematical expectation and standard deviation of the number of hyperedges, degrees of hyperedges. The properties of dynamic geometric hypergraphs are discussed, their influence on the connectivity of a computer network, and the requirements for algorithms for managing connectivity in terms of hypergraphs are formulated.

Self-dual Planar Hypergraphs and Exact Bond Percolation Thresholds

A generalized star-triangle transformation and a concept of triangle-duality have been introduced recently in the physics literature to predict exact percolation threshold values of several lattices. Conditions for the solution of bond percolation models are investigated, and an infinite class of lattice graphs for which exact bond percolation thresholds may be rigorously determined as the solution of a polynomial equation are identified. This class is naturally described in terms of hypergraphs, leading to definitions of planar hypergraphs and self-dual planar hypergraphs. There exist infinitely many self-dual planar 3-uniform hypergraphs, and, as a consequence, there exist infinitely many real numbers $a \in [0,1]$ for which there are infinitely many lattices that have bond percolation threshold equal to $a$.

Arithmetical rank of squarefree monomial ideals of small arithmetic degree

In this paper, we prove that the arithmetical rank of a squarefree monomial ideal I is equal to the projective dimension of R/I in the following cases: (a) I is an almost complete intersection; (b) arithdeg?I=reg?I; (c) arithdeg?I=indeg?I+1. We also classify all almost complete intersection squarefree monomial ideals in terms of hypergraphs, and use this classification in the proof in case (c).

Relative definability of boolean functions via hypergraphs

The aim of this work is to show how hypergraphs can be used as a systematic tool in the classification of continuous boolean functions according to their degree of parallelism. Intuitively f is “less parallel” than g if it can be defined by a sequential program using g as its only free variable. It turns out that the poset induced by this preorder is (as for the degrees of recursion) a sup-semilattice. Although hypergraphs have already been used in Bucciarelli (Theoret. Comput. Sci., to appear) as a tool for studying degrees of parallelism, no general result relating the former to the latter has been proved in that work. We show that the sup-semilattice of degrees has a categorical counterpart: we define a category of hypergraphs such that every object “represents” a monotone boolean function; finite coproducts in this category correspond to lubs of degrees. Unlike degrees of recursion, where every set has a recursive upper bound, monotone boolean functions may have no sequential upper bound. However the ones which do have a sequential upper bound can be nicely characterised in terms of hypergraphs. These subsequential functions play a major role in the proof of our main result, namely that f is less parallel than g if there exists a morphism between their associated hypergraphs.

On greedy algorithms in coding theory

We study a wide class of problems in coding theory for which we consider two different formulations: in terms of incidence matrices and in terms of hypergraphs. These problems are dealt with using a greedy algorithm due to Stein (1974) and Lovasz (1975). Some examples, including constructing covering codes, codes for conflict resolution, separating systems, source encoding with distortion, etc., are given a unified treatment. Under certain conditions derandomization can be performed, leading to an essential reduction in the complexity of the constructions.

How to Analyze the Results of Linear Programs—Part 2: Price Interpretation

In the second part of a four-part series, I consider an exercise in analysis that arises in many applications: Why is the price of 〈some commodity〉 equal to 〈whatever its solution value〉? The ability to interpret dual prices in a linear programming solution is part of economic analysis, and the mathematical basis is as old as linear programming, itself. New approaches, however, go beyond the usual duality arguments in answering this question in more practical terms. One of these new approaches is path tracing, which seeks a portion of the linear program that accounts for the row's price by activity costs from sources to the row. In some cases this is a simple path in a network problem. More generally, in ordinary network terms, it can be a tree or involve embedded cycles, but it is regarded as a path in hypergraph terms.

Classification de certaines matrices 0–1

This paper is a survey of the main properties of special 0–1 matrices. All these properties are presented in a unified framework (in terms of matrices and not in terms of hypergraphs). The paper contains no original results and no proof.