Intersection Hypergraph Research Articles

The Spectral Radii of Intersecting Uniform Hypergraphs

The celebrated Erdős–Ko–Rado theorem states that given $$n\geqslant 2k,$$ every intersecting k-uniform hypergraph G on n vertices has at most $$\left( {\begin{array}{c}n-1\\ k-1\end{array}}\right)$$ edges. This paper states spectral versions of the Erdős–Ko–Rado theorem: let G be an intersecting k-uniform hypergraph on n vertices with $$n\geqslant2k.$$ Then, the sharp upper bounds for the spectral radius of $$\mathcal {A}_{\alpha }(G)$$ and $$\mathcal {Q}^{*}(G)$$ are presented, where $$\mathcal {A}_{\alpha }(G)=\alpha \mathcal {D}(G)+(1-\alpha ) \mathcal {A}(G)$$ is a convex linear combination of the degree diagonal tensor $$\mathcal {D}(G)$$ and the adjacency tensor $$\mathcal {A}(G)$$ for $$0\leqslant \alpha < 1,$$ and $$\mathcal {Q}^{*}(G)$$ is the incidence $$\mathcal {Q}$$ -tensor, respectively. Furthermore, when $$n>2k,$$ the extremal hypergraphs which attain the sharp upper bounds are characterized. The proof mainly relies on the Perron–Frobenius theorem for nonnegative tensor and the property of the maximizing connected hypergraphs.

Constructing Planar Support for Non-Piercing Regions

Given a hypergraph $$\mathcal {H}=(X,{\mathcal {S}})$$, a planar support for $$\mathcal {H}$$ is a planar graph G with vertex set X, such that for each hyperedge $$S\in \mathcal {S}$$, the subgraph of G induced by the vertices in S is connected. Planar supports for hypergraphs have found several algorithmic applications, including several packing and covering problems, hypergraph coloring, and in hypergraph visualization. The main result proved in this paper is the following: given two families of regions R and B in the plane, each of which consists of connected, non-piercing regions, the intersection hypergraph$${\mathcal {H}}_R(B)=(B,\{B_{r}\}_{r\in R})$$, where $$B_r=\{b\in B:b\cap r\ne \emptyset \}$$ has a planar support. Further, such a planar support can be computed in time polynomial in |R|, |B|, and the number of vertices in the arrangement of the regions in $$R\cup B$$. Special cases of this result include the setting where either the family R, or the family B is a set of points. Our result unifies and generalizes several previous results on planar supports, PTAS’s for packing and covering problems on non-piercing regions in the plane and coloring of intersection hypergraph of non-piercing regions.

Coloring Intersection Hypergraphs of Pseudo-Disks

We prove that the intersection hypergraph of a family of n pseudo-disks with respect to another family of pseudo-disks admits a proper coloring with four colors and a conflict-free coloring with O(log n) colors. Along the way we prove that the respective Delaunay-graph is planar. We also prove that the intersection hypergraph of a family of n regions with linear union complexity with respect to a family of pseudo-disks admits a proper coloring with constantly many colors and a conflict-free coloring with O(log n) colors. Our results serve as a common generalization and strengthening of many earlier results, including ones about proper and conflict-free coloring points with respect to pseudo-disks, coloring regions of linear union complexity with respect to points and coloring disks with respect to disks.

The Finite Projective Plane and the 5-Uniform Linear Intersecting Hypergraphs with Domination Number Four

The matching number \(\alpha '(H)\) of a hypergraph H is the size of a largest matching in H, where a matching is a set of pairwise disjoint edges in H. A dominating set in H is a subset D of vertices of H such that for every \(v\in V(H)\setminus D\) there exists \(u\in D\) such that u and v lie in an edge of H, and the domination number of H, denoted by \(\gamma (H)\), is the minimum cardinality of a dominating set in H. It was shown that a Ryser-like inequality \(\gamma (H)\le (r-1)\alpha '(H)\) holds for hypergraphs H of rank r. In particular, for intersecting hypergraphs H of rank r, \(\gamma (H)\le r-1\), since \(\alpha '(H)=1\). The linear intersecting hypergraphs of rank \(2\le r\le 4\) achieving the equality \(\gamma (H)=r-1\) have been characterized. In this paper we show that all the 5-uniform linear intersecting hypergraphs H with equality \(\gamma (H)=r-1\) are generated by the finite projective plane of order three.

Zarankiewicz's problem for semi-algebraic hypergraphs

Zarankiewicz's problem asks for the largest possible number of edges in a graph that does not contain a Ku,u subgraph for a fixed positive integer u. Recently, Fox, Pach, Sheffer, Sulk and Zahl [12] considered this problem for semi-algebraic graphs, where vertices are points in Rd and edges are defined by some semi-algebraic relations. In this paper, we extend this idea to semi-algebraic hypergraphs. For each k≥2, we find an upper bound on the number of hyperedges in a k-uniform k-partite semi-algebraic hypergraph without Ku1,…,uk for fixed positive integers u1,…,uk. When k=2, this bound matches the one of Fox et al. and when k=3, it isO((mnp)2d2d+1+ε+m(np)dd+1+ε+n(mp)dd+1+ε+p(mn)dd+1+ε+mn+np+pm), where m,n,p are the sizes of the parts of the tripartite hypergraph and ε is an arbitrarily small positive constant. We then present applications of this result to a variant of the unit area problem, the unit minor problem and intersection hypergraphs.

On Ryser’s Conjecture for $t$-Intersecting and Degree-Bounded Hypergraphs

A famous conjecture (usually called Ryser's conjecture) that appeared in the PhD thesis of his student, J. R. Henderson, states that for an $r$-uniform $r$-partite hypergraph $\mathcal{H}$, the inequality $\tau(\mathcal{H})\le(r-1)\!\cdot\! \nu(\mathcal{H})$ always holds. This conjecture is widely open, except in the case of $r=2$, when it is equivalent to Kőnig's theorem, and in the case of $r=3$, which was proved by Aharoni in 2001.Here we study some special cases of Ryser's conjecture. First of all, the most studied special case is when $\mathcal{H}$ is intersecting. Even for this special case, not too much is known: this conjecture is proved only for $r\le 5$ by Gyárfás and Tuza. For $r&gt;5$ it is also widely open.Generalizing the conjecture for intersecting hypergraphs, we conjecture the following. If an $r$-uniform $r$-partite hypergraph $\mathcal{H}$ is $t$-intersecting (i.e., every two hyperedges meet in at least $t&lt;r$ vertices), then $\tau(\mathcal{H})\le r-t$. We prove this conjecture for the case $t&gt; r/4$.Gyárfás showed that Ryser's conjecture for intersecting hypergraphs is equivalent to saying that the vertices of an $r$-edge-colored complete graph can be covered by $r-1$ monochromatic components.Motivated by this formulation, we examine what fraction of the vertices can be covered by $r-1$ monochromatic components of different colors in an $r$-edge-colored complete graph. We prove a sharp bound for this problem.Finally we prove Ryser's conjecture for the very special case when the maximum degree of the hypergraph is two.

On decomposability of Multilinear sets

We consider the Multilinear set $${\mathcal {S}}$$ defined as the set of binary points (x, y) satisfying a collection of multilinear equations of the form $$y_I = \prod _{i \in I} x_i$$ , $$I \in {\mathcal {I}}$$ , where $${\mathcal {I}}$$ denotes a family of subsets of $$\{1,\ldots , n\}$$ of cardinality at least two. Such sets appear in factorable reformulations of many types of nonconvex optimization problems, including binary polynomial optimization. A great simplification in studying the facial structure of the convex hull of the Multilinear set is possible when $${\mathcal {S}}$$ is decomposable into simpler Multilinear sets $${\mathcal {S}}_j$$ , $$j \in J$$ ; namely, the convex hull of $${\mathcal {S}}$$ can be obtained by convexifying each $${\mathcal {S}}_j$$ , separately. In this paper, we study the decomposability properties of Multilinear sets. Utilizing an equivalent hypergraph representation for Multilinear sets, we derive necessary and sufficient conditions under which $${\mathcal {S}}$$ is decomposable into $${\mathcal {S}}_j$$ , $$j \in J$$ , based on the structure of pair-wise intersection hypergraphs. Our characterizations unify and extend the existing decomposability results for the Boolean quadric polytope. Finally, we propose a polynomial-time algorithm to optimally decompose a Multilinear set into simpler subsets. Our proposed algorithm can be easily incorporated in branch-and-cut based global solvers as a preprocessing step for cut generation.

On the Zarankiewicz problem for intersection hypergraphs

Let d and t be fixed positive integers, and let Kt,…,td denote the complete d-partite hypergraph with t vertices in each of its parts, whose hyperedges are the d-tuples of the vertex set with precisely one element from each part. According to a fundamental theorem of extremal hypergraph theory, due to Erdős [6], the number of hyperedges of a d-uniform hypergraph on n vertices that does not contain Kt,…,td as a subhypergraph, is nd−1td−1. This bound is not far from being optimal.We address the same problem restricted to intersection hypergraphs of (d−1)-dimensional simplices in Rd. Given an n-element set S of such simplices, let Hd(S) denote the d-uniform hypergraph whose vertices are the elements of S, and a d-tuple is a hyperedge if and only if the corresponding simplices have a point in common. We prove that if Hd(S) does not contain Kt,…,td as a subhypergraph, then its number of edges is O(n) if d=2, and O(nd−1+ϵ) for any ϵ>0 if d≥3. This is almost a factor of n better than Erdős's above bound. Our result is tight, apart from the error term ϵ in the exponent. In particular, for d=2, we obtain a theorem of Fox and Pach [7], which states that every Kt,t-free intersection graph of n segments in the plane has O(n) edges. The original proof was based on a separator theorem that does not generalize to higher dimensions. The new proof works in any dimension and is simpler: it uses size-sensitive cuttings, a variant of random sampling.

Most Probably Intersecting Hypergraphs

The celebrated Erdős-Ko-Rado theorem shows that for $n \ge 2k$ the largest intersecting $k$-uniform set family on $[n]$ has size $\binom{n-1}{k-1}$. It is natural to ask how far from intersecting larger set families must be. Katona, Katona and Katona introduced the notion of most probably intersecting families, which maximise the probability of random subfamilies being intersecting.We consider the most probably intersecting problem for $k$-uniform set families. We provide a rough structural characterisation of the most probably intersecting families and, for families of particular sizes, show that the initial segment of the lexicographic order is optimal.

Matching critical intersection hypergraphs

A matching in a hypergraph H is a set of pairwise vertex disjoint edges in H and the matching number of H is the maximum cardinality of a matching in H. A hypergraph H is an intersecting hypergraph if every two distinct edges of H have a non-empty intersection. Equivalently, H is an intersecting hypergraph if and only if it has matching number one. In this paper, we study intersecting hypergraphs that are matching critical in the sense that the matching number increases under various definitions of criticality.

On Randomly Generated Intersecting Hypergraphs

Let $c$ be a positive constant. We show that if $r=\lfloor{cn^{1/3}}\rfloor$ and the members of ${[n]\choose r}$ are chosen sequentially at random to form an intersecting hypergraph then with limiting probability $(1+c^3)^{-1}$, as $n\to\infty$, the resulting family will be of maximum size ${n-1\choose r-1}$.

Efficient knowledge discovery through the integration of heterogeneous data

It is commonly the case that a distributed database holds data originating from a number of different sources. These heterogeneous data sources may provide different views of the same data, or they may be different samples from the same population. In each case, a methodology is provided for combining data which may be held at different levels of granularity. Variations in granularity arise due to the use of different levels within a concept hierarchy or the use of different concept hierarchies. Data integration is accomplished using the intersection hypergraph to produce the integrated universal classification scheme and to determine the cardinalities of each category within the universal table. In the commonly occurring case of continuous or ordinal data, an explicit and efficient computational algorithm is presented. By integrating heterogeneous data it is often possible to discover new information at a finer level of granularity than that available in any of the contributing data sources. A system may be thus be enabled, so as to induce new rules based on information at a finer level of granularity than that would be possible without integration.

Optimal and efficient integration of heterogeneous summary tables in a distributed database

In any particular combination of domains, although a common understanding of the underlying low-level concepts concerning a domain attribute may exist, different concept hierarchies may have been built at different data-holding sites. A distributed database may therefore hold different views of the same data, or differently classified samples from the same population. Because of this, when statistical functions are applied to generate summary tables, the resulting summary-based partitions may be heterogeneous. In these situations, integration of such summary-based partitions can reveal latent information at a new, and finer, level of granularity. In this paper, the classification schemes are described using a matrix representation of the intersection hypergraph, and efficient numerical algorithms are proposed to determine the optimal granularity of the integrated summary data.

Intersecting fuzzy hypergraphs

Intersecting fuzzy hypergraphs possess a number of attributes which parallel properties of crisp intersecting hypergraphs. We explore this phenomenon in detail. We also develop an interesting dialogue between intersecting fuzzy hypergraphs and fuzzy transversals.

Structure of the high-order Boltzmann machine from independence maps

In this paper we consider the determination of the structure of the high-order Boltzmann machine (HOBM), a stochastic recurrent network for approximating probability distributions. We obtain the structure of the HOBM, the hypergraph of connections, from conditional independences of the probability distribution to model. We assume that an expert provides these conditional independences and from them we build independence maps, Markov and Bayesian networks, which represent conditional independences through undirected graphs and directed acyclic graphs respectively. From these independence maps we construct the HOBM hypergraph. The central aim of this paper is to obtain a minimal hypergraph. Given that different orderings of the variables provide in general different Bayesian networks, we define their intersection hypergraph. We prove that the intersection hypergraph of all the Bayesian networks (N!) of the distribution is contained by the hypergraph of the Markov network, it is more simple, and we give a procedure to determine a subset of the Bayesian networks that verifies this property. We also prove that the Markov network graph establishes a minimum connectivity for the hypergraphs from Bayesian networks.

Query evaluability in statistical databases

The evaluability of queries on a statistical database containing joinable tables connected by an intersection hypergraph is considered. A characterization of evaluable queries is given, which allows one to define polynomial-time procedures both for testing evaluability and for evaluating queries. These results are useful in designing an 'informed query system' for statistical databases which promotes an integrated use of stored information. Such a query system allows the user to formulate a query involving attributes from several joinable tables as if they were all contained in a single universal table. >