Minimal Transversals Research Articles

On the number of minimal transversals in 3-uniform hypergraphs

We prove that the number of minimal transversals (and also the number of maximal independent sets) in a 3-uniform hypergraph with n vertices is at most c n , where c ≈ 1.6702 . The best known lower bound for this number, due to Tomescu, is ad n , where d = 10 1 / 5 ≈ 1.5849 and a is a constant.

On the complexity of monotone dualization and generating minimal hypergraph transversals

In 1994 Fredman and Khachiyan established the remarkable result that the duality of a pair of monotone Boolean functions, in disjunctive normal forms, can be tested in quasi-polynomial time, thus putting the problem, most likely, somewhere between polynomiality and coNP-completeness. We strengthen this result by showing that the duality testing problem can in fact be solved in polylogarithmic time using a quasi-polynomial number of processors (in the PRAM model). While our decomposition technique can be thought of as a generalization of that of Fredman and Khachiyan, it yields stronger bounds on the sequential complexity of the problem in the case when the sizes of f and g are significantly different, and also allows for generating all minimal transversals of a given hypergraph using only polynomial space.

COMPUTING MANY MAXIMAL INDEPENDENT SETS FOR HYPERGRAPHS IN PARALLEL

A hypergraph [Formula: see text] is called uniformly δ-sparse if for every nonempty subset X ⊆ V of vertices, the average degree of the sub-hypergraph of [Formula: see text] induced by X is at most δ. We show that there is a deterministic algorithm that, given a uniformly δ-sparse hypergraph [Formula: see text], and a positive integer k, outputs k or all minimal transversals for [Formula: see text] in O(δ log (1 + k) polylog (δ|V|))-time using |V|O( log δ)kO(δ) processors. Equivalently, the algorithm can be used to compute in parallel k or all maximal independent sets for [Formula: see text].

Operative treatment of periampullary retroperitoneal perforation complicating endoscopic sphincterotomy

Evidence-based strategies are lacking regarding the appropriate management of periampullary retroperitoneal perforations complicating endoscopic retrograde cholangiopancreatography (ERCP) combined with endoscopic sphincterotomy (ES). We propose a transduodenal operative repair of periampullary retroperitoneal perforation. Six patients with duodenal periampullary perforation induced by endoscopic sphincterotomy underwent operation after failure of an attempt of conservative management. After mobilization of the second and the third part of the duodenum, a minimal transversal duodenotomy was carried out, the papilla was exposed, periampullary perforation was readily identified, and was sutured easily as a sphincteroplasty or by 2 or 3 Vicryl 3/0 sutures. Patient outcomes were measured. Periampullary perforation was repaired as sphincteroplasty in 2 cases, and with Vicryl 3/0 sutures in 4 cases. The mean duration of operation was 176 minutes. There were no intraoperative complications. None of the patients required reoperation after transduodenal repair of the perforation. The patients had a normal postoperative course. The median hospital stay was 10.5 days (range, 9 to 20 days) and the mortality rate was nil. There were no delayed complications during a median follow-up of 60 months. The transduodenal operative approach to periampullary perforation after ERCP/ES at an early stage in the clinical evolution of the perforation is a safe and effective procedure. We consider this approach a useful option for the treatment of periampullary perforation after ERCP/ES when initial endoscopic and conservative management do not yield good results within 24 hours.

A global parallel algorithm for the hypergraph transversal problem

We consider the problem of finding all minimal transversals of a hypergraph H ⊆ 2 V , given by an explicit list of its hyperedges. We give a new decomposition technique for solving the problem with the following advantages: (i) Global parallelism: for certain classes of hypergraphs, e.g., hypergraphs of bounded edge size, and any given integer k, the algorithm outputs k minimal transversals of H in time bounded by polylog ( | V | , | H | , k ) assuming poly ( | V | , | H | , k ) number of processors. Except for the case of graphs, none of the previously known algorithms for solving the same problem exhibit this feature. (ii) Using this technique, we also obtain new results on the complexity of generating minimal transversals for new classes of hypergraphs, namely hypergraphs of bounded dual-conformality, and hypergraphs in which every edge intersects every minimal transversal in a bounded number of vertices.

An Efficient Algorithm for the Transversal Hypergraph Generation

The Transversal Hypergraph Generation is the problem of generating, given a hypergraph, the set of its minimal transversals, i.e., the hypergraph whose hyperedges are the minimal hitting sets of the given one. The purpose of this paper is to present an efficient and practical algorithm for solving this problem. We show that the proposed algorithm operates in a way that rules out regeneration and, thus, its memory requirements are polynomially bounded to the size of the input hypergraph. Although no time bound for the algorithm is given, experimental evaluation and comparison with other approaches have shown that it behaves well in practice and it can successfully handle large problem instances.

New Results on Monotone Dualization and Generating Hypergraph Transversals

We consider the problem of dualizing a monotone CNF (equivalently, computing all minimal transversals of a hypergraph) whose associated decision problem is a prominent open problem in NP-completeness. We present a number of new polynomial time, respectively, output-polynomial time results for significant cases, which largely advance the tractability frontier and improve on previous results. Furthermore, we show that duality of two monotone CNFs can be disproved with limited nondeterminism. More precisely, this is feasible in polynomial time with O(log2n /\log log n) suitably guessed bits. This result sheds new light on the complexity of this important problem.

Optimal design of hydraulic support

This paper describes a procedure for optimal determination of two groups of parameters of a hydraulic support employed in the mining industry. The procedure is based on mathematical programming methods. In the first step, the optimal values of some parameters of the leading four-bar mechanism are found in order to ensure the desired motion of the support with minimal transversal displacements. In the second step, maximal tolerances of the optimal values of the leading four-bar mechanism are calculated, so the response of hydraulic support will be satisfying.

Identifying the Minimal Transversals of a Hypergraph and Related Problems

The paper considers two decision problems on hypergraphs, hypergraph saturation and recognition of the transversal hypergraph, and discusses their significance for several search problems in applied computer science. Hypergraph saturation (i.e., given a hypergraph $\mathcal{H}$, decide if every subset of vertices is contained in or contains some edge of $\mathcal{H}$) is shown to be co-${\bf NP}$-complete. A certain subproblem of hypergraph saturation, the saturation of simple hypergraphs (i.e., Sperner families), is shown to be under polynomial transformation equivalent to transversal hypergraph recognition; i.e., given two hypergraphs $\mathcal{H}_{1}$, $\mathcal{H}_{2}$, decide if the sets in $\mathcal{H}_{2}$ are all the minimal transversals of $\mathcal{H}_{1}$. The complexity of the search problem related to the recognition of the transversal hypergraph, the computation of the transversal hypergraph, is an open problem. This task needs time exponential in the input size; it is unknown whether an output-polynomial algorithm exists. For several important subcases (for instance, if an upper or lower bound is imposed on the edge size or for acyclic hypergraphs) output-polynomial algorithms are presented. Computing or recognizing the minimal transversals of a hypergraph is a frequent problem in practice, which is pointed out by identifying important applications in database theory, Boolean switching theory, logic, and artificial intelligence (AI), particularly in model-based diagnosis.

Exact Transversal Hypergraphs and Application to Boolean μ-Functions

Call an hypergraph, that is a family of subsets (edges) from a finite vertex set, an exact transversal hypergraph iff each of its minimal transversals, i.e., minimal vertex subsets that intersect each edge, meets each edge in a singleton. We show that such hypergraphs are recognizable in polynomial time and that their minimal transversals as well as their maximal independent sets can be generated in lexicographic order with polynomial delay between subsequent outputs, which is impossible in the general case unless P= NP. The results obtained are applied to monotone Boolean μ-functions, that are Boolean functions defined by a monotone Boolean expression (that is, built with ∧, ∨ only) in which no variable occurs repeatedly. We also show that recognizing such functions from monotone Boolean expressions is co-NP-hard, thus complementing Mundici's result that this problem is in co-NP.

Constructible hypergraphs

The class M of finite manuals (i.e. hypergraphs formed by the cliques of finite graphs) is closed under the formation of sums and products. We define the class of constructible hypergraphs to be the smallest subclass of M which contains all finite classical manuals (i.e. hypergraphs having a single edge) and is closed under the formation of sums and products. Constructible hypergraphs have two interesting properties: (1) they do not contain hooks and (2) all their minimal transversals are supports of dispersion-free stochastic functions. Property (1) is known to characterize the constructible members of M . In this paper we show that the same holds true for property (2).

A Class of Polynomially Solvable Set-Covering Problems

A clutterL is a collection of m subsets of a ground set $E( L ) = \{ x_1 , \cdots ,x_n \}$ with the property that for every pair $A_i ,A_j \in L,A_i $ is neither contained nor contains $A_j $. A transversal of L is a subset of $E( L )$ having at least one element in common with each member of L.The problem of finding the minimum weight transversal of a clutter L is equivalent to the well-known set-covering problem.In this paper the class of ideal clutters that properly contains the class of clutters the members of which are the bases of a matroid (matroidal clutters) is introduced.An ideal clutter L has the property that the number of its minimal transversals is bounded by a polynomial in m and n. The properties of ideal clutters are described, and two polynomial algorithms for recognizing them and finding their minimum weight transversal are presented.

An O( mn) algorithm for regular set-covering problems

A clutter L is a collection of m subsets of a ground set E( L) = { x 1,…, x n } with the property that, for every pair A i , A j ϵ L, A i is neither contained nor contains A j , A transversal of L is a subset of E( L) intersecting every member of L. If we associate with each element x j ϵ E( L) a weight c j , the problem of finding a transversal having minimum weight is equivalent to the following set-covering problem min{c Tx|M Lx ⩾ 1 m, x j ϵ {0, 1}, j = 1,…, n} where M L is the matrix whose rows are the incidence vectors of the subsets A i ϵ L and 1 m denotes the vector with m ones. A set-covering problem is regular if there exists an ordering of the variables σ = ( x 1,…, x n ) such that, for every feasible solution x with x i = 1, x j = 0, j < i, the vector x + e j − e i is also a feasible solution, where e i is the ith unit vector. The matrix M of a regular set-covering problem is said to be regular. A regular clutter is any clutter whose incidence matrix is regular. In this paper we describe some properties of regular clutters and propose an algorithm which, in O( mn) steps, generates all the minimal transversals of a regular clutter L and produces the transversal having minimum weight.

Chains, antichains, and fibres

Call a subset of an ordered set a fibre if it meets every maximal antichain. We prove several instances of the conjecture that, in an ordered set P without splitting elements, there is a subset F such that both F and P − F are fibres. For example, this holds in every ordered set without splitting points and in which each chain has at most four elements. As it turns out several of our results can be cast more generally in the language of graphs from which we may derive “complementary” results about cutsets of ordered sets, that is, subsets which meet every maximal chain. One example is this: In a finite graph G every minimal transversal is independent if and only if G contains no path of length three.

Coverings by minimal transversals

In this paper, we give counterexamples to the conjecture: “Every nonempty regular simple graph contains two disjoint maximal independent sets” [2, 7]. For this, we generalize this problem to the following: covering the set of vertices of a graph by minimal transversals. An equivalence of this last problem is given.

Coverings by minimal transversals

Coverings by minimal transversals