Hypergraph Research Articles

A parameterized view on the complexity of dependence and independence logic

Abstract In this paper, we investigate the parameterized complexity of model checking for Dependence and Independence logic, which are well studied logics in the area of Team Semantics. We start with a list of nine immediate parameterizations for this problem, namely the number of disjunctions (i.e. splits)/(free) variables/universal quantifiers, formula-size, the tree-width of the Gaifman graph of the input structure, the size of the universe/team and the arity of dependence atoms. We present a comprehensive picture of the parameterized complexity of model checking and obtain a division of the problem into tractable and various intractable degrees. Furthermore, we also consider the complexity of the most important variants (data and expression complexity) of the model checking problem by fixing parts of the input.

Lexicographically maximal edges of dual hypergraphs and Nash-solvability of tight game forms

We prove a new property of dual hypergraphs and derive from it Nash-solvability of the corresponding (tight) game forms. This result is known since 1975, but its new proof is much simpler.

Efficient iterated greedy for the two-dimensional bandwidth minimization problem

Graph layout problems are a family of combinatorial optimization problems that consist of finding an embedding of the vertices of an input graph into a host graph such that an objective function is optimized. Within this family of problems falls the so-called Two-Dimensional Bandwidth Minimization Problem (2DBMP). The 2DBMP aims to minimize the maximum distance between each pair of adjacent vertices of the input graph when it is embedded into a grid host graph. In this paper, we present an efficient heuristic algorithm based on the Iterated Greedy (IG) framework hybridized with a new local search strategy to tackle the 2DBMP. Particularly, we propose different designs for the main IG procedures (i.e., construction, destruction, and reconstruction) based on the trade-off between intensification and diversification. Additionally, the improvement method incorporates three advanced strategies: an efficient way to evaluate the objective function of neighbor solutions, a tiebreak criterion to deal with “flat landscapes”, and a neighborhood reduction technique. Extensive experimentation was carried out to assess the IG performance over state-of-the-art methods, emerging our approach as the most competitive algorithm. Specifically, IG finds the best solutions for all instances considered in considerably less execution time. Statistical tests corroborate the merit of our proposal.

Rigidity of Random Subgraphs and Eigenvalues of Stiffness Matrices

In the random subgraph model we consider random subgraphs $G(t)$ of a graph $G$ obtained as follows: for each edge in $G$ we independently decide to retain the edge with probability $t$ and discard the edge with probability $1-t$ for some $0\leq t\leq 1$. A special case of this model is the Erdös--Rényi random graph model, where the host graph is the complete graph $K_n$. In this paper we analyze the rigidity properties of random subgraphs and give new upper bounds on the threshold $t_0$ for which $G(t)$ is asymptotically almost surely (a.a.s.) rigid or globally rigid when $t\geq t_0$. By specializing our results to complete host graphs we obtain, among others, that an Erdös--Rényi random graph is a.a.s. globally rigid in $\mathbb{R}^d$ if $t\geq \frac{C_d\log n}{n}$ for some constant $C_d$. We also consider random subframeworks of (bar-and-joint) frameworks, which are geometric realizations of our graphs. Our bounds for the rigidity threshold of random subgraphs are in terms of the smallest nonzero eigenvalue of the stiffness matrix of the framework, which is the Gramian of its normalized rigidity matrix. Motivated by this connection, we introduce the concept of $d$-dimensional algebraic connectivity of graphs and provide upper and lower bounds for this value of several fundamental graph classes. The case $d=1$ corresponds to the well-known algebraic connectivity, that is, the second smallest Laplacian eigenvalue of the graph. We also consider the rigidity threshold in random molecular graphs, also called bond-bending networks, which are used in the study of rigidity properties of molecules. In this model we are concerned with the rigidity of the square graph of some graph $G$. We give an upper bound for the rigidity threshold of the square of random subgraphs in terms of the algebraic connectivity of the host graph. This enables us to derive an upper bound for the rigidity threshold for sparse host graphs.

Graph Coloring via Clique Search with Symmetry Breaking

It is known that the problem of proper coloring of the nodes of a given graph can be reduced to finding cliques in a suitably constructed auxiliary graph. In this work, we explore the possibility of reducing the search space by exploiting the symmetries present in the auxiliary graph. The proposed method can also be used for efficient exact coloring of hyper graphs. We also precondition the auxiliary graph in order to further reduce the search space. We carry out numerical experiments to assess the practicality of these proposals. We solve some hard cases and prove a new lower limit of seven for the mycielski7 graph with the aid of the proposed technique.

Regular Turán numbers and some Gan–Loh–Sudakov‐type problems

AbstractMotivated by a Gan–Loh–Sudakov‐type problem, we introduce the regular Turán numbers, a natural variation on the classical Turán numbers where we restrict ourselves to the class of regular graphs. Among other results, we prove a striking supersaturation version of Mantel's theorem in the case of a regular host graph of odd order. We also characterize the graphs for which the regular Turán numbers behave classically or otherwise.

MAKER–BREAKER GAMES ON AND

AbstractWe investigate Maker–Breaker games on graphs of size $\aleph _1$ in which Maker’s goal is to build a copy of the host graph. We establish a firm dependence of the outcome of the game on the axiomatic framework. Relating to this, we prove that there is a winning strategy for Maker in the $K_{\omega ,\omega _1}$ -game under ZFC+MA+ $\neg $ CH and a winning strategy for Breaker under ZFC+CH. We prove a similar result for the $K_{\omega _1}$ -game. Here, Maker has a winning strategy under ZF+DC+AD, while Breaker has one under ZFC+CH again.

Rainbow trees in uniformly edge‐colored graphs

AbstractWe obtain sufficient conditions for the emergence of spanning and almost‐spanning bounded‐degree rainbow trees in various host graphs, having their edges colored independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform coloring of , using a palette of size , a.a.s. admits a rainbow copy of any given bounded‐degree tree on at most vertices, where is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon et al. pertaining to the embedding of bounded‐degree almost‐spanning prescribed trees in , where is independent of . Given an ‐vertex graph with minimum degree at least , where is fixed, we use our aforementioned result in order to prove that a uniform coloring of the randomly perturbed graph , using colors, where is arbitrarily small yet fixed, a.a.s. admits a rainbow copy of any given bounded‐degree spanning tree. This can be viewed as a rainbow variant of a result by Krivelevich et al. who proved that , where is independent of , a.a.s. admits a copy of any given bounded‐degree spanning tree. Finally, and with as above, we prove that a uniform coloring of using colors a.a.s. admits a rainbow spanning tree. Put another way, the trivial lower bound on the size of the palette required for supporting a rainbow spanning tree is also sufficient, essentially as soon as the random perturbation a.a.s. has edges.

Anti-Ramsey numbers for cycles in the generalized Petersen graphs

For H⊆G, the anti-Ramsey number ar(G,H) is the maximum number of colors in an edge-coloring of G such that each subgraph isomorphic to H has at least two edges in the same color. The study of anti-Ramsey number ar(Kn,H) was introduced by Erdős et al. in 1973, and plentiful results were researched for some special graph H. Later, the problem was extended to ar(G,H) when replacing Kn by other graph G such as hypergraph, complete split graph, regular bipartite graph, triangulation and so on. In this paper, we consider a generalized Petersen graph Pn,k as the host graph and determine the exact anti-Ramsey numbers for cycles C5 and C6 in Pn,k, respectively.

Semi-supervised multi-view clustering with dual hypergraph regularized partially shared non-negative matrix factorization

Semi-supervised multi-view clustering with dual hypergraph regularized partially shared non-negative matrix factorization

A variable neighborhood search approach for cyclic bandwidth sum problem

In this paper, we tackle the Cyclic Bandwidth Sum Problem, consisting in minimizing the sum of the bandwidth of the edges of an input graph computed in a cycle-structured host graph. This problem has been widely studied in the literature due to its multiple real-world applications, such as circuit design, migration of telecommunication networks, or graph drawing, among others. Particularly, we tackle this problem by proposing a multistart procedure whose main components are a new greedy constructive algorithm and an intensification strategy based on the Variable Neighborhood Search metaheuristic. The constructive procedure introduces two different greedy criteria to determine each step of the construction phase, which can be used for other related problems. Additionally, we illustrate how to perform an efficient exploration of the neighborhood structure by using an alternative neighborhood. Our algorithmic proposal is evaluated over a set of 40 instances previously studied in the literature and over a new proposed set of 66 well-known instances introduced in this paper. The obtained results have been satisfactory compared with the ones obtained by the best previous algorithm in the state of the art. The statistical tests performed indicate that the differences between the methods are significant.

Local Balance in Graph Decompositions

In an equireplicate graph decomposition, every vertex of the host graph appears in the same number of blocks. We propose the use of colored loops as a framework for unifying various other types of local regularity conditions in graph decompositions. In the basic case where a single graph with colored loops is used as a block, an existence theory for such decompositions follows as a straightforward generalization of previous work on graph decompositions.

Knots and Knot-Hyperpaths in Hypergraphs

This paper deals with some theoretical aspects of hypergraphs related to hyperpaths and hypertrees. In ordinary graph theory, the intersecting or adjacent edges contain exactly one vertex; however, in the case of hypergraph theory, the adjacent or intersecting hyperedges may contain more than one vertex. This fact leads to the intuitive notion of knots, i.e., a collection of explicit vertices. The key idea of this manuscript lies in the introduction of the concept of the knot, which is a subset of the intersection of some intersecting hyperedges. We define knot-hyperpaths and equivalent knot-hyperpaths and study their relationships with the algebraic space continuity and the pseudo-open character of maps. Moreover, we establish a sufficient condition under which a hypergraph is a hypertree, without using the concept of the host graph.

Inverse Turán numbers

For given graphs G and F, the Turán number ex(G,F) is defined to be the maximum number of edges in an F-free subgraph of G. Foucaud, Krivelevich and Perarnau and later independently Briggs and Cox introduced a dual version of this problem wherein for a given number k, one maximizes the number of edges in a host graph G for which ex(G,H)<k.Addressing a problem of Briggs and Cox, we determine the asymptotic value of the inverse Turán number of the paths of length 4 and 5 and provide an improved lower bound for all paths of even length. Moreover, we obtain bounds on the inverse Turán number of even cycles giving improved bounds on the leading coefficient in the case of C4. Finally, we give multiple conjectures concerning the asymptotic value of the inverse Turán number of C4 and Pℓ, suggesting that in the latter problem the asymptotic behavior depends heavily on the parity of ℓ.

An Analysis on the Markovian Model for Production Management and Establishment of New Pharmacies Using Hyper Graph

Stochastic models are mainly used to rectify the real‐life problems in different areas of the busy world such as marketing, trading, networking, communicating media, and challenging biomedical inventions, for the survival of mankind. Any business which runs on the basis of getting more profit with less investment includes all the resources for production and the man power too, during the current century. To bring the equality in business gain and man power, many stochastic models have been introduced since the 1940s. The different states and the steady state probabilities are discussed by the Markov models to overcome the insufficiency of man power. An Iterative Hidden Markov Model has been introduced for effective production by considering the observable states depend on hidden states under the circumstances of real‐world probabilities are known and unknown. This paper mainly deals with the various machines, pharmacies, and different methods, namely, Nonadjacency Search Method and Adjacency Search Method applied through the Hyper graph. Comparison made between Nonadjacency Search Method and Adjacency Search Method through hyper assignments using cliques helps us to obtain tremendous profit through effective production and to start new pharmacies, instead of predictions through computational methods. Results were discussed based on the information obtained from Apollo pharmacies.

Breaking the Hegemony of the Triangle Method in Clique Detection

We consider the fundamental problem of detecting/counting copies of a fixed pattern graph in a host graph. The recent progress on this problem has not included complete pattern graphs, i.e., cliques (and their complements, i.e., edge-free pattern graphs, in the induced setting). The fastest algorithms for the aforementioned patterns are based on a straightforward reduction to triangle detection/counting. We provide an alternative method of detection/counting copies of fixed size cliques based on a multi-dimensional matrix product. It is at least as time efficient as the triangle method in cases of $K_4$ and $K_5.$ The complexity of the multi-dimensional matrix product is of interest in its own rights. We provide also another alternative method for detection/counting $K_r$ copies, again time efficient for $r\in \{ 4, 5 \}$.

Semisupervised Feature Extraction of Hyperspectral Image Using Nonlinear Geodesic Sparse Hypergraphs

Recently, the sparse representation (SR)-based graph embedding method has been extensively used in feature extraction (FE) tasks, but it is hard to reveal the complex manifold structure and multivariate relationship of samples in the hyperspectral image (HSI). Meanwhile, the small size sample problem in HSI data also limits the performance of the traditional SR approach. To tackle this problem, this article develops a new semisupervised FE algorithm called a geodesic-based sparse manifold hypergraph (GSMH). The presented method first utilizes the geodesic distance to measure the nonlinear similarity between samples lying on manifold space and further constructs the manifold neighborhood of each sample. Then, a geodesic-based neighborhood SR (GNSR) model is designed to explore the multivariate sparse correlations of different manifold neighborhoods. Considering the multivariate sparse manifold correlations among samples, a pair of semisupervised hypergraphs (HGs) is constructed to effectively incorporate the labeled and unlabeled training information in the embedding process and obtain the nonlinear discriminative feature representation for HSI. Experimental results on three HSI datasets indicate that the proposed method not only achieves satisfying FE performance with limited labeled training samples but also shows superiority compared with other state-of-the-art methods.

A Small-Step Operational Semantics for GP 2

The operational semantics of a programming language is said to be small-step if each transition step is an atomic computation step in the language. A semantics with this property faithfully corresponds to the implementation of the language. The previous semantics of the graph programming language GP 2 is not fully small-step because the loop and branching commands are defined in big-step style. In this paper, we present a truly small-step operational semantics for GP 2 which, in particular, accurately models diverging computations. To obtain small-step definitions of all commands, we equip the transition relation with a stack of host graphs and associated operations. We prove that the new semantics is non-blocking in that every computation either diverges or eventually produces a result graph or the failure state. We also show the finite nondeterminism property, viz. that each configuration has only a finite number of direct successors. The previous semantics of GP 2 is neither non-blocking nor does it have the finite nondeterminism property. We also show that, for a program and a graph that terminate, both semantics are equivalent, and that the old semantics can be simulated with the new one.

Generalized Planar Turán Numbers

In a generalized Turán problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $n$-vertex graph not containing $F$ as a subgraph. We consider generalized Turán problems where the host graph is planar. In particular, we obtain the order of magnitude of the maximum number of copies of a fixed tree in a planar graph containing no even cycle of length at most $2\ell$, for all $\ell$, $\ell \geqslant 1$. We also determine the order of magnitude of the maximum number of cycles of a given length in a planar $C_4$-free graph. An exact result is given for the maximum number of $5$-cycles in a $C_4$-free planar graph. Multiple conjectures are also introduced.&#x0D;

Identifying potential association on gene-disease network via dual hypergraph regularized least squares

BackgroundIdentifying potential associations between genes and diseases via biomedical experiments must be the time-consuming and expensive research works. The computational technologies based on machine learning models have been widely utilized to explore genetic information related to complex diseases. Importantly, the gene-disease association detection can be defined as the link prediction problem in bipartite network. However, many existing methods do not utilize multiple sources of biological information; Additionally, they do not extract higher-order relationships among genes and diseases.ResultsIn this study, we propose a novel method called Dual Hypergraph Regularized Least Squares (DHRLS) with Centered Kernel Alignment-based Multiple Kernel Learning (CKA-MKL), in order to detect all potential gene-disease associations. First, we construct multiple kernels based on various biological data sources in gene and disease spaces respectively. After that, we use CAK-MKL to obtain the optimal kernels in the two spaces respectively. To specific, hypergraph can be employed to establish higher-order relationships. Finally, our DHRLS model is solved by the Alternating Least squares algorithm (ALSA), for predicting gene-disease associations.ConclusionComparing with many outstanding prediction tools, DHRLS achieves best performance on gene-disease associations network under two types of cross validation. To verify robustness, our proposed approach has excellent prediction performance on six real-world networks. Our research work can effectively discover potential disease-associated genes and provide guidance for the follow-up verification methods of complex diseases.