• Bulletin of the Korean Mathematical Society
• /
• v.55 no.5
• /
• pp.1523-1528
• /
• 2018

For $r{\geq}2$, let ${\mathcal{H}}$ be an r-uniform hypergraph with n vertices and m hyperedges. Let R be a random vertex set obtained by choosing each vertex of ${\mathcal{H}}$ independently with probability p. Let ${\mathcal{H}}[R]$ be the subhypergraph of ${\mathcal{H}}$ induced on R. We obtain an upper bound on the matching number ${\nu}({\mathcal{H}}[R])$ and a lower bound on the independence number ${\alpha}({\mathcal{H}}[R])$ of ${\mathcal{H}}[R]$. First, we show that if $mp^r{\geq}{\log}\;n$, then ${\nu}(H[R]){\leq}2e^{\ell}mp^r$ with probability at least $1-1/n^{\ell}$ for each positive integer ${\ell}$. It is best possible up to a constant factor depending only on ${\ell}$ if $m{\leq}n/r$. Next, we show that if $mp^r{\geq}{\log}\;n$, then ${\alpha}({\mathcal{H}}[R]){\geq}np-{\sqrt{3{\ell}np\;{\log}\;n}-2re^{\ell}mp^r$ with probability at least $1-3/n^{\ell}$.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.4
• /
• pp.1027-1039
• /
• 2019

The concept of jump concerns the distribution of $Tur{\acute{a}}n$ densities. A number ${\alpha}\;{\in}\;[0,1)$ is a jump for r if there exists a constant c > 0 such that if the $Tur{\acute{a}}n$ density of a family $\mathfrak{F}$ of r-uniform graphs is greater than ${\alpha}$, then the $Tur{\acute{a}}n$ density of $\mathfrak{F}$ is at least ${\alpha}+c$. To determine whether a number is a jump or non-jump has been a challenging problem in extremal hypergraph theory. In this paper, we give a way to generate non-jumps for hypergraphs. We show that if ${\alpha}$, ${\beta}$ are non-jumps for $r_1$, $r_2{\geq}2$ respectively, then $\frac{{\alpha}{\beta}(r_1+r_2)!r_1^{r_1}r_2^{r_2}}{r_1!r_2!(r_1+R_2)^{r_1+r_2}}$ is a non-jump for $r_1+r_2$. We also apply the Lagrangian method to determine the $Tur{\acute{a}}n$ density of the extension of the (r - 3)-fold enlargement of a 3-uniform matching.

• Journal of KIISE:Computer Systems and Theory
• /
• v.26 no.12
• /
• pp.1548-1555
• /
• 1999

본 논문에서는 하이퍼그래프의 {{{{k분 분할을 위한 서열화(vertex ordering) 알고리즘의 효율을 개선하기 위한 후처리 알고리즘인 재서열법을 소개한다. 제안된 알고리즘은 {{{{k분 분할을 위한 다양한 알고리즘에 쉽게 적용될 수 있다. 보통 초기 분할은 서열화를 기반으로 하는 알고리즘에 의해 형성된다. 그 후 제안된 알고리즘은 클러스터와 정점을 재배열하여 분할하는 과정을 반복함으로써 분할의 효율을 향상시켜간다. 이 방법을 여러 가지 그래프에 적용하여 향상된 결과를 얻었다.Abstract This paper addresses the post-processing algorithm for {{{{k-way hypergraph partitioning by using a cluster and vertex reordering method. The proposed algorithm applies to several {{{{k-way partitioning algorithm. Generally, the initial partition generating method is based on a vertex ordering algorithm. Our reordering algorithm construct an enhanced partitioning by iteratively partition the reodered clusters and vertices. Experimental results on several graphs demonstrate that reodering provides substantial enhancement.

• Annual Conference on Human and Language Technology
• /
• 2010.10a
• /
• pp.127-129
• /
• 2010

본 논문에서는 다양한 코퍼스를 통해 언어를 학습하는 과정을 모델링하여 무감독학습(Unsupervised learning)으로 문법적 패턴을 습득하는 방법론을 소개한다. 제안 방법에서는 적은 수의 특성 조합으로 잠재적 패턴의 부분만을 표현한 후 표현된 규칙을 조합하여 유의미한 문법적 패턴을 탐색한다. 본 논문에서 제안한 방법은 베이지만 추론(Bayesian Inference)과 MCMC (Markov Chain Mote Carlo) 샘플링에 기반하여 특성 조합을 유의미한 문법적 패턴으로 정제하는 방법으로, 랜덤하이퍼그래프(Random Hypergraph) 모델을 이용하여 많은 수의 하이퍼에지를 생성한 후 생성된 하이퍼에지의 가중치를 조정하여 유의미한 문법적 패턴을 탈색하는 방법론이다. 우리는 본 논문에서 유아용 비디오의 스크립트를 이용하여 다양한 유아용 비디오 스크립트에서 문법적 패턴을 습득하는 방법론을 소개한다.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.3
• /
• pp.569-583
• /
• 2019

Given a graph G, the Motzkin and Straus formulation of the maximum clique problem is the quadratic program (QP) formed from the adjacent matrix of the graph G over the standard simplex. It is well-known that the global optimum value of this QP (called Lagrangian) corresponds to the clique number of a graph. It is useful in practice if similar results hold for hypergraphs. In this paper, we attempt to explore the relationship between the Lagrangian of a hypergraph and the order of its maximum cliques when the number of edges is in a certain range. Specifically, we obtain upper bounds for the Lagrangian of a hypergraph when the number of edges is in a certain range. These results further support a conjecture introduced by Y. Peng and C. Zhao (2012) and extend a result of J. Talbot (2002). We also establish an upper bound of the clique number in terms of Lagrangians for hypergraphs.

• Journal of applied mathematics & informatics
• /
• v.10 no.1_2
• /
• pp.209-216
• /
• 2002

We study correspondences between interval structures and hyperstructures.

• The Transactions of the Korea Information Processing Society
• /
• v.3 no.7
• /
• pp.1659-1668
• /
• 1996

Graph grammars generate graph languages while string grammars generate string languages which are the subset of graph languages. One of the most successful graph grammars models is the NLC grammars, which gen-erate graphs by replacing a node by a node by a graph through node labels. For grammars generating hypergraphs which are the superset of graphs, there are CFHG grammars, which replace a hyperedge by a hypergraph through their preidentified gluing points, an extension of CFHG grammars called HH grammars, which replace a handle by a hypergraph through the rewriting mechanism that can also duplicate or delete the hyperedges surrounding the replaced handle, and finally HRNCE grammars, which replace a handle by a hypergraph through an eNCE way of rewriting, In this paper, we compare the language-generating power of HRNCE grammars with that a graph grammars mentioned above by comparing graph langrages generated by them, respecti vely.

• Journal of KIISE:Software and Applications
• /
• v.36 no.3
• /
• pp.208-216
• /
• 2009

A number of estimation of distribution algorithms have been proposed that do not use explicitly crossover and mutation of traditional genetic algorithms, but estimate the distribution of population for more efficient search. But because it is not easy to discover higher-order correlations of variables, lower-order correlations are estimated most cases under various constraints. In this paper, we propose a new estimation of distribution algorithm that represents higher-order correlations of the data and finds global optimum more efficiently. The proposed algorithm represents the higher-order correlations among variables by building random hypergraph model composed of hyperedges consisting of variables which are expected to be correlated, and generates the next population by Bayesian sampling algorithm Experimental results show that the proposed algorithm can find global optimum and outperforms the simple genetic algorithm and BOA(Bayesian Optimization Algorithm) on decomposable functions with deceptive building blocks.

• Honam Mathematical Journal
• /
• v.39 no.4
• /
• pp.493-501
• /
• 2017

In this paper, we define and enumerate two tree-like hypergraph structures which we call them (d, k)-trees and d-trees, where $d{\geq}2$ and k > 0 are integers. These new definitions generalize traditional and HP-hypertrees.

• Proceedings of the Korean Information Science Society Conference
• /
• 2010.06a
• /
• pp.148-149
• /
• 2010