Transversals In Hypergraphs Research Articles

Deterministic Dynamic Matching in O(1) Update Time

We consider the problems of maintaining an approximate maximum matching and an approximate minimum vertex cover in a dynamic graph undergoing a sequence of edge insertions/deletions. Starting with the seminal work of Onak and Rubinfeld (in: Proceedings of the ACM symposium on theory of computing (STOC), 2010), this problem has received significant attention in recent years. Very recently, extending the framework of Baswana et al. (in: Proceedings of the IEEE symposium on foundations of computer science (FOCS), 2011) , Solomon (in: Proceedings of the IEEE symposium on foundations of computer science (FOCS), 2016) gave a randomized dynamic algorithm for this problem that has an approximation ratio of 2 and an amortized update time of O(1) with high probability. This algorithm requires the assumption of an oblivious adversary, meaning that the future sequence of edge insertions/deletions in the graph cannot depend in any way on the algorithm’s past output. A natural way to remove the assumption on oblivious adversary is to give a deterministic dynamic algorithm for the same problem in O(1) update time. In this paper, we resolve this question. We present a new deterministic fully dynamic algorithm that maintains a O(1)-approximate minimum vertex cover and maximum fractional matching, with an amortized update time of O(1). Previously, the best deterministic algorithm for this problem was due to Bhattacharya et al. (in: Proceedings of the ACM-SIAM symposium on discrete algorithms (SODA), 2015); it had an approximation ratio of (2+varepsilon ) and an amortized update time of O(log n/varepsilon ^2). Our result can be generalized to give a fully dynamic O(f^3)-approximate algorithm with O(f^2) amortized update time for the hypergraph vertex cover and fractional hypergraph matching problem, where every hyperedge has at most f vertices.

Decomposing 1-Sperner Hypergraphs

A hypergraph is Sperner if no hyperedge contains another one. A Sperner hypergraph is equilizable (resp., threshold) if the characteristic vectors of its hyperedges are the (minimal) binary solutions to a linear equation (resp., inequality) with positive coefficients. These combinatorial notions have many applications and are motivated by the theory of Boolean functions and integer programming. We introduce in this paper the class of $1$-Sperner hypergraphs, defined by the property that for every two hyperedges the smallest of their two set differences is of size one. We characterize this class of Sperner hypergraphs by a decomposition theorem and derive several consequences from it. In particular, we obtain bounds on the size of $1$-Sperner hypergraphs and their transversal hypergraphs, show that the characteristic vectors of the hyperedges are linearly independent over the reals, and prove that $1$-Sperner hypergraphs are both threshold and equilizable. The study of $1$-Sperner hypergraphs is motivated also by their applications in graph theory, which we present in a companion paper.

Query-oriented text summarization based on hypergraph transversals

The rise in the amount of textual resources available on the Internet has created the need for tools of automatic document summarization. The main challenges of query-oriented extractive summarization are (1) to identify the topics of the documents and (2) to recover query-relevant sentences of the documents that together cover these topics. Existing graph- or hypergraph-based summarizers use graph-based ranking algorithms to produce individual scores of relevance for the sentences. Hence, these systems fail to measure the topics jointly covered by the sentences forming the summary, which tends to produce redundant summaries. To address the issue of selecting non-redundant sentences jointly covering the main query-relevant topics of a corpus, we propose a new method using the powerful theory of hypergraph transversals. First, we introduce a new topic model based on the semantic clustering of terms in order to discover the topics present in a corpus. Second, these topics are modeled as the hyperedges of a hypergraph in which the nodes are the sentences. A summary is then produced by generating a transversal of nodes in the hypergraph. Algorithms based on the theory of submodular functions are proposed to generate the transversals and to build the summaries. The proposed summarizer outperforms existing graph- or hypergraph-based summarizers by at least 6% of ROUGE-SU4 F-measure on DUC 2007 dataset. It is moreover cheaper than existing hypergraph-based summarizers in terms of computational time complexity.

Q-Rung Orthopair Fuzzy Hypergraphs with Applications

The concept of q-rung orthopair fuzzy sets generalizes the notions of intuitionistic fuzzy sets and Pythagorean fuzzy sets to describe complicated uncertain information more effectively. Their most dominant attribute is that the sum of the q th power of the truth-membership and the q th power of the falsity-membership must be equal to or less than one, so they can broaden the space of uncertain data. This set can adjust the range of indication of decision data by changing the parameter q, q ≥ 1 . In this research study, we design a new framework for handling uncertain data by means of the combinative theory of q-rung orthopair fuzzy sets and hypergraphs. We define q-rung orthopair fuzzy hypergraphs to achieve the advantages of both theories. Further, we propose certain novel concepts, including adjacent levels of q-rung orthopair fuzzy hypergraphs, ( α , β ) -level hypergraphs, transversals, and minimal transversals of q-rung orthopair fuzzy hypergraphs. We present a brief comparison of our proposed model with other existing theories. Moreover, we implement some interesting concepts of q-rung orthopair fuzzy hypergraphs for decision-making to prove the effectiveness of our proposed model.

Upper transversals in hypergraphs

A set S of vertices in a hypergraph H is a transversal if it has a nonempty intersection with every edge of H. For k≥1, if H is a hypergraph with every edge of size at least k, then a k-transversal in H is a transversal that intersects every edge of H in at least k vertices. In particular, a 1-transversal is a transversal. The upper k-transversal number ϒk(H) of H is the maximum cardinality of a minimal k-transversal in H. We obtain asymptotically best possible lower bounds on ϒk(H) for uniform hypergraphs H. More precisely, we show that for r≥2 and for every integer k∈[r], if H is a connected r-uniform hypergraph with n vertices, then ϒk(H)>23nr−k+1. For r>k≥1 and ε>0, we show that there exist infinitely many r-uniform hypergraphs, Hr,k∗, of order n and a function f(r,k) of r and k satisfying ϒk(Hr,k∗)<(1+ε)⋅f(r,k)⋅nr−k+1.

Counting minimal transversals of β-acyclic hypergraphs

We prove that one can count in polynomial time the number of minimal transversals of β-acyclic hypergraphs. In consequence, we can count in polynomial time the number of minimal dominating sets of strongly chordal graphs, continuing the line of research initiated in M.M. Kanté and T. Uno (2017) [18].

LP/SDP Hierarchy Lower Bounds for Decoding Random LDPC Codes

Random $(d_{v},d_{c})$ - regular low-density parity-check (LDPC) codes, where each variable is involved in $d_{v}$ parity checks and each parity check involves $d_{c}$ variables are well-known to achieve the Shannon capacity of the binary symmetric channel, for sufficiently large $d_{v}$ and $d_{c}$ , under exponential time decoding. However, polynomial time algorithms are only known to correct a much smaller fraction of errors. One of the most powerful polynomial-time algorithms with a formal analysis is the linear programming (LP) decoding algorithm of Feldman et al., which is known to correct an $\Omega (1/d_{c})$ fraction of errors. In this paper, we show that fairly powerful extensions of LP decoding, based on the Sherali–Adams and Lasserre hierarchies, fail to correct much more errors than the basic LP-decoder. In particular, we show that: 1) for any values of $d_{v}$ and $d_{c}$ , a linear number of rounds of the Sherali–Adams LP hierarchy cannot correct more than an $O(1/d_{c})$ fraction of errors on a random $(d_{v},d_{c})$ -regular LDPC code; and 2) for any value of $d_{v}$ and infinitely many values of $d_{c}$ , a linear number of rounds of the Lasserre SDP hierarchy cannot correct more than an $O(1/d_{c})$ fraction of errors on a random $(d_{v},d_{c})$ -regular LDPC code. Our proofs use a new stretching and collapsing technique that allows us to leverage recent progress in the study of the limitations of LP/SDP hierarchies for Maximum Constraint Satisfaction Problems (Max-CSPs). The problem then reduces to the construction of special balanced pairwise independent distributions for Sherali–Adams and special cosets of balanced pairwise independent subgroups for Lasserre. Our (algebraic) construction for the Lasserre hierarchy is based on designing sets of points in ${\mathbb F}_{q}^{d}$ (for $q$ any power of 2 and $d = 2,3$ ) with special hyperplane-incidence properties—constructions that may be of independent interest. An intriguing consequence of our work is that expansion seems to be both the strength and the weakness of random regular LDPC codes. Some of our techniques are more generally applicable to a large class of Boolean CSPs called Min-Ones. In particular, for $k$ -Hypergraph Vertex Cover, we obtain an improved integrality gap of $k-1-\epsilon $ that holds after a linear number of rounds of the Lasserre hierarchy, for any $k = q+1$ with $q$ an arbitrary prime power. The best previous gap for a linear number of rounds was equal to $2-\epsilon $ and due to Schoenebeck.

Dimension Incremental Feature Selection Approach for Vertex Cover of Hypergraph Using Rough Sets

The minimum vertex cover problem is a well-known optimization problem; it has been used in a wide variety of applications. This paper focuses on rough set-based approach for the minimum vertex cover problem of the dynamic and static hypergraphs. First, we demonstrate the relationship between the attribute reduction of decision table and the minimum vertex cover of hypergraph, and the minimum vertex cover problem is converted to an attribute reduction problem based on this relationship. Then, we discuss the update mechanism of minimum vertex cover from the perspective of attribute reduction, and two types of incremental attribute reduction algorithms are proposed, one is the dynamic increase of single vertex and the other is the dynamic increase of multiple vertices. Our algorithms can quickly update the minimum vertex cover in a dynamic hypergraph and improve the rough sets-based method for the minimum vertex cover problem of a static hypergraph in terms of the computational time and the solution quality. The experimental results show the advantages and limitations of the proposed algorithms compared with the existing algorithms.

ON THE DENSE FAMILIES IN THE RELATIONAL DATAMODEL

In this paper, dense families of relation schemes are introduced. We characterize minimal keys of relation schemes in terms of dense families. Note that, the dense families of database relations were introduced by Jarvinen [6]. We prove that the set of all minimal keys of a relation scheme s= (U, F) is the transversal hypergraphs of a hypergraph D– {∅}, where Dis any s-dense family. We give a necessary and sufficient condition for an abitrary family to be s-dense family. We also present some dense families of relation schemes. Furthermore, in this paper, we also study antikeys by means of dense families. We present connections between antikeys and a dense family of relation schemes. Finally, we study the time complexity of the problem finding antikeys.

Transversals of m-polar fuzzy hypergraphs with applications

Transversals of m-polar fuzzy hypergraphs with applications

A note on fractional disjoint transversals in hypergraphs

A transversal in a hypergraph H is a subset of vertices that has a nonempty intersection with every edge of H. A transversal family F of H is a family of (not necessarily distinct) transversals of H. The effective transversal-ratio of the family F is the ratio of the number of sets in F over the maximum times rF any element appears in F. The fractional disjoint transversal number FDT(H) is the supremum of the effective transversal-ratio taken over all transversal families. That is, FDT(H)=supF|F|∕rF. Using a connection with not-all-equal 3-SAT, we prove that if H is a 3-regular 3-uniform hypergraph, then FDT(H)≥2, which proves a known conjecture. Using probabilistic arguments, we prove that for all k≥3, if H is a k-regular k-uniform hypergraph, then FDT(H)≥1∕(1−(k−1k)1k1k−1), and that this bound is essentially best possible.

(Total) Domination in Prisms

Using hypergraph transversals it is proved that $\gamma_t(Q_{n+1}) = 2\gamma(Q_n)$, where $\gamma_t(G)$ and $\gamma(G)$ denote the total domination number and the domination number of $G$, respectively, and $Q_n$ is the $n$-dimensional hypercube. More generally, it is shown that if $G$ is a bipartite graph, then $\gamma_t(G \square K_2) = 2\gamma(G)$. Further, we show that the bipartiteness condition is essential by constructing, for any $k \ge 1$, a (non-bipartite) graph $G$ such that $\gamma_t(G\square K_2) = 2\gamma(G) - k$. Along the way several domination-type identities for hypercubes are also obtained.

Inapproximability of $H$-Transversal/Packing

Given an undirected graph $G = (V_G, E_G)$ and a fixed “pattern” graph $H = (V_H, E_H)$ with $k$ vertices, we consider the $H$-Transversal and $H$-Packing problems. The former asks to find the smallest $S \subseteq V_G$ such that the subgraph induced by $V_G \setminus S$ does not have $H$ as a subgraph, and the latter asks to find the maximum number of pairwise disjoint $k$-subsets $S_1, \ldots, S_m \subseteq V_G$ such that the subgraph induced by each $S_i$ has $H$ as a subgraph. We prove that if $H$ is 2-connected, $H$-Transversal and $H$-Packing are almost as hard to approximate as general $k$-Hypergraph Vertex Cover and $k$-Set Packing, so it is NP-hard to approximate them within a factor of $\Omega (k)$ and $\widetilde \Omega (k)$, respectively. We also show that there is a 1-connected $H$ where $H$-Transversal admits an $O(\log k)$-approximation algorithm, so that the connectivity requirement cannot be relaxed from 2 to 1. For a special case of $H$-Transversal where $H$ is a (family of) cycles, we m...

Parameterized ceteris paribus preferences over atomic conjunctions under conservative semantics

We consider a propositional language for describing parameterized ceteris paribus preferences over atomic conjunctions. Such preferences are only required to hold when the alternatives being compared agree on a specified subset of propositional variables. Regarding the expressivity of the language in question, we show that a parameterized preference statement is equivalent to a conjunction of an exponential number of classical, non-parameterized, ceteris paribus statements. Next, we present an inference system for parameterized statements and prove that the problem of checking the semantic consequence relation for such statements is coNP-complete. We propose an approach based on formal concept analysis to learning preferences from data by showing that ceteris paribus preferences valid in a particular model correspond to implications of a special formal context derived from this model. The computation of a complete preference set is then reducible to the computation of minimal hypergraph transversals. Finally, we adapt a polynomial-time algorithm for abduction using Horn clauses represented by their characteristic models to the problem of determining preferences over new alternatives from preferences over given alternatives (with ceteris paribus preferences as the underlying model).

Sufficient conditions for the existence of Nash equilibria in bimatrix games in terms of forbidden $$2 \times 2$$ 2 × 2 subgames

In 1964 Shapley observed that a matrix has a saddle point in pure strategies whenever every its \(2 \times 2\) submatrix has one. In contrast, a bimatrix game may have no pure strategy Nash equilibrium (NE) even when every \(2 \times 2\) subgame has one. Nevertheless, Shapley’s claim can be extended to bimatrix games as follows. We partition all \(2 \times 2\) bimatrix games into fifteen classes \(C = \{c_1, \ldots , c_{15}\}\) depending only on the preferences of two players. A subset \(t \subseteq C\) is called a NE-theorem if a bimatrix game has a NE whenever it contains no subgame from t. We suggest a method to construct all minimal (that is, strongest) NE-theorems based on the procedure of joint generation of transversal hypergraphs given by a special oracle. By this method we obtain all (six) strongest NE-theorems. Let us remark that the suggested approach, which may be called “math-pattern recognition”, is very general: it allows to characterize completely an arbitrary “target” in terms of arbitrary “attributes”.

A transversal hypergraph approach for the frequent itemset hiding problem

We propose a methodology for hiding all sensitive frequent itemsets in a transaction database. Our methodology relies on a novel technique that enumerates the minimal transversals of a hypergraph in order to induce the ideal border between frequent and sensitive itemsets. The ideal border is then utilized to formulate an integer linear program (ILP) that answers whether a feasible sanitized database that attains the ideal border, exists. The solution of the program identifies the set of transactions that need to be modified (sanitized) so that the hiding can be achieved with the maximum accuracy. If no solution exists, we modify the ILP by relaxing the constraints needed to be satisfied so that the sanitized database preserves the privacy with guarantee but with minimum effect in data quality. Experimental evaluation of the proposed approach on a number of real datasets has shown that the produced sanitized databases exhibit higher accuracy when compared with the solutions of other well-known approaches.

Transversals in 5-uniform hypergraphs and total domination in graphs with minimum degree five

In [9] Thomassé and Yeo pose the following problem: Find the minimum ck for which every k-uniform hypergraph with n vertices and n edges has a transversal of size at most ckn. A direct consequence of this result is that every graph of order n with minimum degree at least k has a total dominating set of cardinality at most ckn. It is known that c2 = 2/3, c3 = 1/2, and c4 = 3/7. Thomassé and Yeo show that 4/11 ≤ c5 and conjecture that c5 = 4/11. In this paper we show that c5 ≤ 17/44. Thus, 16/44 ≤ c5 ≤ 17/44. More generally we prove that every 5-uniform hypergraph on n vertices and m edges has a transversal with no more than (10n+7m)/44 vertices. Consequently, every graph on n vertices with minimum degree at least five has total domination number at most 17n/44.

Total Transversals in Hypergraphs and Their Applications

Let $H = (V,E)$ be a hypergraph with vertex set $V$ and edge set $E$ of order ${n_{_H}} = |V|$ and size ${m_{_H}} = |E|$. The hypergraph $H$ is $k$-uniform if every edge of $H$ has size $k$. Two vertices in $H$ are adjacent if they belong to a common edge in $H$. A transversal in $H$ is a subset of vertices in $H$ that has a nonempty intersection with every edge of $H$. A total transversal in $H$ is a transversal $T$ in $H$ with the additional property that every vertex in $T$ is adjacent to some other vertex of $T$. The total transversal number $\tau_t(H)$ of $H$ is the minimum cardinality of a total transversal in $H$. For $k \ge 2$, let $b_k = \sup_{H \in {\cal H}_k} \, {\tau_t}(H) / ({n_{_H}} + {m_{_H}})$, where ${\cal H}_k$ denotes the class of all $k$-uniform hypergraphs containing no isolated vertices or isolated edges or multiple edges. It is known that $b_2 = 2/5$, $b_3 = 1/3$, $b_4 \le 1/3$, and $b_5 \le 2/7$. In this paper, we show that $b_4 = 2/7$ and $b_6 \le 1/4$. Further, for $k \ge 7$, we show that $b_7 \le 2/9$. These results on total transversals have applications in total domination in hypergraphs. A total dominating set in $H$ is a subset of vertices $D \subseteq V$ such that every vertex in $H$ is adjacent to some vertex in $D$. The total domination number $\gamma_t(H)$ is the minimum cardinality of a total dominating set in $H$. The following relationship between the total transversal number and the total domination number of uniform hypergraphs is known: For $k \ge 3$ and $H \in {\cal H}_k$, we have ${\gamma_t}(H) \le ( \max \{ \frac{2 }{k+1}, b_{k-1} \} ) \times {n_{_H}}$. As a consequence of our results on the total transversal number, for $k \in \{2,3,4,5,6,7,8\}$ and a hypergraph $H \in {\cal H}_k$, we have ${\gamma_t}(H) \le 2{n_{_H}}/(k+1)$.

Enumerating minimal dominating sets in chordal bipartite graphs

We show that all minimal dominating sets of a chordal bipartite graph can be generated in incremental polynomial, hence output polynomial, time. Enumeration of minimal dominating sets in graphs is equivalent to enumeration of minimal transversals in hypergraphs. Whether the minimal transversals of a hypergraph can be enumerated in output polynomial time is a well-studied and challenging question that has been open for several decades. With this result we contribute to the known cases having an affirmative reply to this important question.

Application of Hypergraphs in the Prime Implicants Selection Process

The paper presents a new concept of the selection of prime implicants in a two-level logic minimization of Boolean functions. The method is based on the two-level minimization process of the Boolean functions, according to the Quine-McCluskey approach. Initially, the set of prime implicants for the logic function ought to be calculated. Next, the selection process is applied to achieve the minimal formula. Such an operation is a typical covering problem and in general case it has exponential computational complexity. In the paper we propose a new prime implicants selection method. An idea is based on the hypergraph theory. The prime implicants matrix (chart) is formed as a selection hypergraph. If the selection hyper-graph belongs to the Exact Transversal Hypergraph class (xt-class), the solution may be obtained in a polynomial time, which is not possible in a general case. The proposed method is illustrated by an example. All necessary steps will be shown in order to apply the proposed selection algorithm to minimize an exemplary Boolean function.