We provide an exposition of the hypergraph models for parallel sparse matrix-vector multiplies based on one-dimensional (1D) matrix partitioning. Our aim is to emphasize the expressive power of the hypergraph models. We first set forth an elementary hypergraph model in which vertices represent the data elements of a matrix-vector multiply operation and nets encode data dependencies. We then app...

In this paper we investigate the sensitivity complexity of hypergraph properties. We present a k-uniform hypergraph property with sensitivity complexity O(n⌈k/3⌉) for any k ≥ 3, where n is the number of vertices. Moreover, we can do better when k ≡ 1 (mod 3) by presenting a k-uniform hypergraph property with sensitivity O(n⌈k/3⌉−1/2). This result disproves a conjecture of Babai [1], which conje...

A hypergraph is color-critical if deleting any edge or vertex reduces the chromatic number; a color-critical hypergraph with chromatic number k is k-critical. Every k-chromatic hypergraph contains a k-critical hypergraph, so one can study chromatic number by studying the structure of k-critical (hyper)graphs. There is vast literature on k-critical graphs and hypergraphs. Many references can be ...

Graph and hypergraph partitioning are important problems with applications in several areas including parallel processing, VLSI design automation, physical mapping of chromosomes and data classification. The problem is to divide the vertices of a hypergraph into k similar–sized blocks such that a given cost function defined over the hypergraph is optimized. The intent of this paper is to expose...

A secret sharing scheme is a method to distribute a secret among a set of participants, such that only qualified subsets of the participants can recover the secret. A secret sharing scheme is perfect if any unqualified subset obtains no information regarding the master key. The collection of qualified subsets is called access structure. In a hypergraph, if the size of edges is equal to r, the h...

The upper chromatic number χ(H) of a hypergraph H is the maximum number of colors in a coloring avoiding a polychromatic edge. The stability number α(H) of a hypergraph H is the cardinality of the largest set of vertices of H which does not contain an edge. A hypergraph is k-uniform if the sizes of all its edges are k. A hypergraph H is co-perfect if χ(H ′) = α(H ′) for each induced subhypergra...

Some classical models of clustering (hierarchies, pyramids, . . . ) are related to interval hypergraphs. In this paper we study clustering models related to hypertrees which are an extension of interval hypergraphs. We first prove that a hypertree can be characterized by an order on its vertices, this order allowing to find one of its underlying vertex trees. We then focus on clustering models ...

In this paper, we propose a novel hypergraph based method (called HF) to fit and segment multi-structural data. The proposed HF formulates the geometric model fitting problem as a hypergraph partition problem based on a novel hypergraph model. In the hypergraph model, vertices represent data points and hyperedges denote model hypotheses. The hypergraph, with large and “data-determined” degrees ...