Simple Hypergraph Research Articles

Model‐based clustering in simple hypergraphs through a stochastic blockmodel

AbstractWe propose a model to address the overlooked problem of node clustering in simple hypergraphs. Simple hypergraphs are suitable when a node may not appear multiple times in the same hyperedge, such as in co‐authorship datasets. Our model generalizes the stochastic blockmodel for graphs and assumes the existence of latent node groups and hyperedges are conditionally independent given these groups. We first establish the generic identifiability of the model parameters. We then develop a variational approximation Expectation‐Maximization algorithm for parameter inference and node clustering, and derive a statistical criterion for model selection. To illustrate the performance of our R package HyperSBM, we compare it with other node clustering methods using synthetic data generated from the model, as well as from a line clustering experiment and a co‐authorship dataset.

Some Discrete Tomography Problems in Hypergraph Model Interpretation

Abstract In this paper we consider discrete tomography problems with an additional requirement of non-repeatability of rows of the binary matrix to be reconstructed; as well as discrete tomography problems with given pairwise projections. Representing the problems in the hypergraph model and pointing out their equivalence to the basic definitions, we state the following results: (i) nonconvexity of the set of hypergraphic sequences of simple hypergraphs with $$n$$ vertices and $$m$$ hyperedges in the $$n$$-dimensional $$m + 1$$-valued lattice, (ii) characterization of monotone Boolean functions associated with degree sequences of 3-/(n–3)-uniform hypergraphs, (iii) formulation of discrete tomography problems with paired projections, their connection to hypergraph degree sequence problem with generalized degrees, a solution for a particular case.

2-Step nilpotent L∞-algebras and hypergraphs

We describe a procedure to attach a nilpotent strong homotopy Lie algebra to every simple hypergraph and prove that two hypergraphs are isomorphic if and only if the corresponding strong homotopy Lie algebras are isomorphic. As an application, we characterize hypergraphs admitting a system of distinct representatives in terms of symplectic forms on the corresponding strong homotopy Lie algebra. We conclude with a combinatorial description of the cohomology of these strong homotopy Lie algebras in low degree.

New bounds on the size of nearly perfect matchings in almost regular hypergraphs

AbstractLet be a ‐uniform ‐regular simple hypergraph on vertices. Based on an analysis of the Rödl nibble, in 1997, Alon, Kim and Spencer proved that if , then contains a matching covering all but at most vertices, and asked whether this bound is tight. In this paper we improve their bound by showing that for all , contains a matching covering all but at most vertices for some , when and are sufficiently large. Our approach consists of showing that the Rödl nibble process not only constructs a large matching but it also produces many well‐distributed ‘augmenting stars’ which can then be used to significantly improve the matching constructed by the Rödl nibble process. Based on this, we also improve the results of Kostochka and Rödl from 1998 and Vu from 2000 on the size of matchings in almost regular hypergraphs with small codegree. As a consequence, we improve the best known bounds on the size of large matchings in combinatorial designs with general parameters. Finally, we improve the bounds of Molloy and Reed from 2000 on the chromatic index of hypergraphs with small codegree (which can be applied to improve the best known bounds on the chromatic index of Steiner triple systems and more general designs).

New lower bound for the minimal number of edges of simple uniform hypergraph without the property Bk

Abstract A hypergraph H = (V, E) has the property Bk if there exists an assignment of two colors to V such that each edge contains at least k vertices of each color. A hypergraph is called simple if every two edges of it have at most one common vertex. We obtain a new lower bound for the minimal number of edges of n-uniform simple hypergraph without the property Bk .

On the Reconstruction of 3-Uniform Hypergraphs from Degree Sequences of Span-Two

A nonnegative integer sequence is k-graphic if it is the degree sequence of a k-uniform simple hypergraph. The problem of deciding whether a given sequence pi is 3-graphic has recently been proved to be NP-complete, after years of studies. Thus, it acquires primary relevance to detect classes of degree sequences whose graphicality can be tested in polynomial time in order to restrict the NP-hard core of the problem and design algorithms that can also be useful in different research areas. Several necessary and few sufficient conditions for pi to be k-graphic, with kge 3, appear in the literature. Frosini et al. defined a polynomial time algorithm to reconstruct k-uniform hypergraphs having regular or almost regular degree sequences. Our study fits in this research line providing a combinatorial characterization of span-two sequences, i.e., sequences of the form pi =(d,ldots ,d,d-1,ldots ,d-1,d-2,ldots ,d-2), dge 2, which are degree sequences of some 3-uniform hypergraphs. Then, we define a polynomial time algorithm to reconstruct one of the related 3-uniform hypergraphs. Our results are likely to be easily generalized to k ge 4 and to other families of degree sequences having simple characterization, such as gap-free sequences.

Duality approach to quantum annealing of the 3-variable exclusive-or satisfiability problem (3-XORSAT)

Classical models with complex energy landscapes represent a perspective avenue for the near-term application of quantum simulators. Until now, many theoretical works studied the performance of quantum algorithms for models with a unique ground state. However, when the classical problem is in a so-called clustering phase, the ground state manifold is highly degenerate. As an example, we consider a 3-XORSAT model defined on simple hypergraphs. The degeneracy of classical ground state manifold translates into the emergence of an extensive number of $Z_2$ symmetries, which remain intact even in the presence of a quantum transverse magnetic field. We establish a general duality approach that restricts the quantum problem to a given sector of conserved $Z_2$ charges and use it to study how the outcome of the quantum adiabatic algorithm depends on the hypergraph geometry. We show that the tree hypergraph which corresponds to a classically solvable instance of the 3-XORSAT problem features a constant gap, whereas the closed hypergraph encounters a second-order phase transition with a gap vanishing as a power-law in the problem size. The duality developed in this work provides a practical tool for studies of quantum models with classically degenerate energy manifold and reveals potential connections between glasses and gauge theories.

The treewidth of 2-section of hypergraphs

Let $H=(V,F)$ be a simple hypergraph without loops. $H$ is called linear if $|f\cap g|\le 1$ for any $f,g\in F$ with $f\not=g$. The $2$-section of $H$, denoted by $[H]_2$, is a graph with $V([H]_2)=V$ and for any $ u,v\in V([H]_2)$, $uv\in E([H]_2)$ if and only if there is $ f\in F$ such that $u,v\in f$. The treewidth of a graph is an important invariant in structural and algorithmic graph theory. In this paper, we consider the treewidth of the $2$-section of a linear hypergraph. We will use the minimum degree, maximum degree, anti-rank and average rank of a linear hypergraph to determine the upper and lower bounds of the treewidth of its $2$-section. Since for any graph $G$, there is a linear hypergraph $H$ such that $[H]_2\cong G$, we provide a method to estimate the bound of treewidth of graph by the parameters of the hypergraph.

Sampling hypergraphs with given degrees

There is a well-known connection between hypergraphs and bipartite graphs, obtained by treating the incidence matrix of the hypergraph as the biadjacency matrix of a bipartite graph. We use this connection to describe and analyse a rejection sampling algorithm for sampling simple uniform hypergraphs with a given degree sequence. Our algorithm uses, as a black box, an algorithm A for sampling bipartite graphs with given degrees, uniformly or nearly uniformly, in (expected) polynomial time. The expected runtime of the hypergraph sampling algorithm depends on the (expected) runtime of the bipartite graph sampling algorithm A, and the probability that a uniformly random bipartite graph with given degrees corresponds to a simple hypergraph. We give some conditions on the hypergraph degree sequence which guarantee that this probability is bounded below by a positive constant.

Increasing paths in countable graphs

In this paper we study variations of an old result by Müller, Reiterman, and the last author stating that a countable graph has a subgraph with infinite degrees if and only if in any labeling of the vertices (or edges) of this graph by positive integers one can always find an infinite increasing path. We study corresponding questions for simple hypergraphs and paths of infinite as well as of arbitrary finite length. For example we show that the condition that a simple hypergraph contains a subhypergraph with infinite degrees is equivalent to the condition that any vertex labeling contains an infinite increasing loose path. We also find an equivalent condition for a graph to have a property that any vertex labeling with positive integers contains a path of arbitrary finite length.

New sufficient conditions on the degree sequences of uniform hypergraphs

The study of the degree sequences of k-uniform hypergraphs, usually called k-sequences, has been a longstanding open problem for the case of k>2, and the corresponding decision version was proved to be NP-complete recently in 2018 [15]. The problem can be formalized as follows: Given a non decreasing sequence of positive integers π=(d1,d2,…,dn), can π be the degree sequence of a k-uniform simple hypergraph? If the answer is positive, then the sequence π is said to be k-graphic. For k=2, that is for simple graphs, Erdös and Gallai [16] provided a characterization of the sequences that are 2-graphic (or simply, graphic). From this characterization, a polynomial time algorithm can be designed to reconstruct the incidence matrix of a graph having a given π as degree sequence (provided this graph exists). Due to the result of [15] and assuming P≠NP, an efficiently computable characterization like the one for k=2 does not even exist for the case of 3-uniform hypergraphs.Necessary or sufficient conditions for π to be k-graphic (k≥3) can be found in the literature. In this paper we prove some different new conditions: first we provide sufficient and also necessary conditions for the case of k-uniform and (almost) regular hypergraphs. Then, for k=3, we prove sufficient conditions in the case where π can be decomposed into π′ and π″, and π′ is graphic.Most of the results are obtained by borrowing tools from discrete tomography, a current research field on discrete mathematics.

Chromatic index of simple hypergraphs

The degree of a vertex v in a hypergraph H is the number of vertices in H adjacent to v, and the maximum degree of H is denoted Δ(H). A hypergraph is called simple if every edge has size at least two and every pair of edges has at most one vertex in common. Let χ′(H) denote the chromatic index of a simple hypergraph H. A well-known conjecture of Erdős, Faber and Lovász says that χ′(H)≤|V(H)|. A generalization of their conjecture states that χ′(H)≤Δ(H)+1. In this paper we show that the conjecture holds when Δ(H)≤5, and that χ′(H)≤⌊74Δ(H)⌋−1.

2-Colorings of Hypergraphs with Large Girth

A hypergraph $$H=(V,E)$$ has property $$B_k$$ if there exists a 2-coloring of the set $$V$$ such that each edge contains at least $$k$$ vertices of each color. We let $$m_{k,g}(n)$$ and $$m_{k,b}(n)$$ , respectively, denote the least number of edges of an $$n$$ -homogeneous hypergraph without property $$B_k$$ which contains either no cycles of length at least $$g$$ or no two edges intersecting in more than $$b$$ vertices. In the paper, upper bounds for these quantities are given. As a consequence, we obtain results for $$m^{*}_k(n)$$ , i.e., for the least number of edges of an $$n$$ -homogeneous simple hypergraph without property $$B_k$$ . Let $$\Delta(H)$$ be the maximal degree of vertices of a hypergraph $$H$$ . By $$\Delta_k(n,g)$$ we denote the minimal degree $$\Delta$$ such that there exists an $$n$$ -homogeneous hypergraph $$H$$ with maximal degree $$\Delta$$ and girth at least $$g$$ but without property $$B_k$$ . In the paper, an upper bound for $$\Delta_k(n,g)$$ is obtained.

In the shadows of a hypergraph: looking for associated primes of powers of square-free monomial ideals

The aim of this paper is to study the associated primes of powers of square-free monomial ideals. Each square-free monomial ideal corresponds uniquely to a finite simple hypergraph via the cover ideal construction, and vice versa. Let H be a finite simple hypergraph and J(H) the cover ideal of H. We define the shadows of hypergraph, H, described as a collection of smaller hypergraphs related to H under some conditions. We then investigate how the shadows of H preserve information about the associated primes of the powers of J(H). Finally, we apply our findings on shadows to study the persistence property of square-free monomial ideals and construct some examples exhibiting failure of containment.

A short nonalgorithmic proof of the containers theorem for hypergraphs

Recently the breakthrough method of hypergraph containers, developed independently by Balogh, Morris, and Samotij as well as Saxton and Thomason, has been used to study sparse random analogs of a variety of classical problems from combinatorics and number theory. The previously known proofs of the containers theorem use the so-called scythe algorithm---an iterative procedure that runs through the vertices of the hypergraph. (Saxton and Thomason have also proposed an alternative, randomized construction in the case of simple hypergraphs.) Here we present the first known deterministic proof of the containers theorem that is not algorithmic, i.e., it does not involve an iterative process. Our proof is less than 4 pages long while being entirely self-contained and conceptually transparent. Although our proof is completely elementary, it was inspired by considering hypergraphs in the setting of nonstandard analysis, where there is a notion of dimension capturing the logarithmic rate of growth of finite sets. Before presenting the proof in full detail, we include a one-page informal outline that refers to this notion of dimension and summarizes the essence of the argument.

On the Number of Independent Sets in Simple Hypergraphs

Extremal problems on the number of j-independent sets in uniform simple hypergraphs are studied. Nearly optimal results on the maximum number of independent sets for the class of simple regular hypergraphs and on the minimum number of independent sets for the class of simple hypergraphs with given average degree of vertices are obtained.

Achieving New Upper Bounds for the Hypergraph Duality Problem through Logic

The hypergraph duality problem DUAL is defined as follows: given two simple hypergraphs $\mathcal{G}$ and $\mathcal{H}$, decide whether $\mathcal{H}$ consists precisely of all minimal transversals of $\mathcal{G}$ (in which case we say that $\mathcal{G}$ is the dual of $\mathcal{H}$). This problem is equivalent to deciding whether two given non-redundant monotone DNFs are dual. It is known that non-DUAL, the complementary problem to DUAL, is in $\mathrm{GC}(\log^2 n,\mathrm{PTIME})$, where $\mathrm{GC}(f(n),\mathcal{C})$ denotes the complexity class of all problems that after a nondeterministic guess of $O(f(n))$ bits can be decided (checked) within complexity class $\mathcal{C}$. It was conjectured that non-DUAL is in $\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE})$. In this paper we prove this conjecture and actually place the non-DUAL problem into the complexity class $\mathrm{GC}(\log^2 n,\mathrm{TC}^0)$ which is a subclass of $\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE})$. We here refer to the logtime-uniform version of $\mathrm{TC}^0$, which corresponds to $\mathrm{FO(COUNT)}$, i.e., first order logic augmented by counting quantifiers. We achieve the latter bound in two steps. First, based on existing problem decomposition methods, we develop a new nondeterministic algorithm for non-DUAL that requires to guess $O(\log^2 n)$ bits. We then proceed by a logical analysis of this algorithm, allowing us to formulate its deterministic part in $\mathrm{FO(COUNT)}$. From this result, by the well known inclusion $\mathrm{TC}^0\subseteq\mathrm{LOGSPACE}$, it follows that DUAL belongs also to $\mathrm{DSPACE}[\log^2 n]$. Finally, by exploiting the principles on which the proposed nondeterministic algorithm is based, we devise a deterministic algorithm that, given two hypergraphs $\mathcal{G}$ and $\mathcal{H}$, computes in quadratic logspace a transversal of $\mathcal{G}$ missing in $\mathcal{H}$.

Colourings of Uniform Hypergraphs with Large Girth and Applications

This paper deals with a combinatorial problem concerning colourings of uniform hypergraphs with large girth. We prove that ifHis ann-uniform non-r-colourable simple hypergraph then its maximum edge degree Δ(H) satisfies the inequality$$ \Delta(H)\geqslant c\cdot r^{n-1}\ffrac{n(\ln\ln n)^2}{\ln n} $$for some absolute constantc> 0.As an application of our probabilistic technique we establish a lower bound for the classical van der Waerden numberW(n, r), the minimum naturalNsuch that in an arbitrary colouring of the set of integers {1,. . .,N} withrcolours there exists a monochromatic arithmetic progression of lengthn. We prove that$$ W(n,r)\geqslant c\cdot r^{n-1}\ffrac{(\ln\ln n)^2}{\ln n}. $$

Randomly coloring simple hypergraphs with fewer colors

We study the problem of constructing a (near) uniform random proper q-coloring of a simple k-uniform hypergraph with n vertices and maximum degree Δ. (Proper in that no edge is mono-colored and simple in that two edges have maximum intersection of size one.) We show that if q≥max⁡{Cklog⁡n,500k3Δ1/(k−1)} then the Glauber Dynamics will become close to uniform in O(nlog⁡n) time, given a random (improper) start. This improves on the results in Frieze and Melsted [5].

Qudit hypergraph states

We generalize the class of hypergraph states to multipartite systems of qudits, by means of constructions based on the d-dimensional Pauli group and its normalizer. For simple hypergraphs, the different equivalence classes under local operations are shown to be governed by a greatest common divisor hierarchy. Moreover, the special cases of three qutrits and three ququarts is analysed in detail.