Problems in scientific computing, such as distributing large sparse matrix operations, have analogous formulations hypergraph partitioning problems. A is a generalization of traditional graph wherein “hyperedges” may connect any number nodes. As result, an NP-Hard problem to both solve or approximate. State-of-the-art algorithms that this follow the multilevel paradigm, which begins by iterativ...

Hypergraphs are a generalization of graphs where edges (aka nets ) allowed to connect more than two vertices. They have similarly wide range applications as graphs. This article considers the fundamental and intensively studied problem balanced hypergraph partitioning (BHP) , which asks for vertices into k disjoint blocks bounded size while minimizing an objective function over hyperedges. Here...

Let σ be a partition of the positive integer r. A σ-hypergraph H = H(n, r, q|σ) is an r-uniform hypergraph on nq vertices which are partitioned into n classes V1, V2, . . . , Vn each containing q vertices. An r-subset K of vertices is an edge of the hypergraph if the partition of r formed by the non-zero cardinalities |K ∩ Vi|, 1 ≤ i ≤ n, is σ. In earlier works we have considered colourings of ...

A hypergraph is k-irregular if there is no set of k vertices all of which have the same degree. We asymptotically determine the probability that a random uniform hypergraph is k-irregular.

A mixed hypergraph consists of two families of edges: the C-edges and D-edges. In a coloring every C-edge has at least two vertices of the same color, while every D-edge has at least two vertices colored differently. The largest and smallest possible numbers of colors in a coloring are termed the upper and lower chromatic number, χ̄ and χ, respectively. A mixed hypergraph is called uniquely colo...

The capacity of a directed hypergraph is a particular numerical quantity associated with a hypergraph. It is of interest because of certain important connections to longstanding conjectures in theoretical computer science related to fast matrix multiplication and perfect hashing as well as various longstanding conjectures in extremal combinatorics. We give an overview of the concept of the capa...

In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a k-uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval [0, 2], and the real part is zero (respectively two) if and only if the eigenvalue is zero (respectively two). All the H+-eigenvalues of the Laplacian and all the smallest H+-eigenvalues of its sub-t...

In this work we propose a model to represent the web as a directed hypergraph (instead of a graph), where links connect pairs of disjointed sets of pages. The web hypergraph is derived from the web graph by dividing the set of pages into non-overlapping blocks and using the links between pages of distinct blocks to create hyperarcs. A hyperarc connects a block of pages to a single page, in orde...

In a visual cryptography scheme, a secret image is encoded into n shares, in the form of transparencies. The shares are then distributed to n participants. Qualified subsets of participants can recover the secret image by superimposing their transparencies, but non-qualified subsets of participants have no information about the secret image. Pixel expansion, which represents the number of subpi...

Given a square-free monomial ideal $I$ in polynomial ring $R$ over field $\mathbb{K}$, one can associate it with its LCM-lattice and hypergraph. In this short note, we establish the connection between hypergraph, doing so provide sufficient condition for removing higher dimension edges of hypergraph without impacting projective ideal. We also offer algorithms to compute class ideals built using...