On Horn envelopes and hypergraph transversals

Horn representation of a concept lattice

“…is an element of R. We continue with an important and well known fact about ∧-closed relations which dates back to [10,15] and can also be found in [9], [11], and [13] in a more general form than the one we prove. Our formulation however is sufficient and more appropriate for our considerations.…”

“…As in general large relations can be equivalent to short Horn formulas, the time complexity of such an approach would not even be polynomial relative to the input and output size. In [13] it is proved that the computation of a Horn formula that is equivalent toR K and takes R K as input is at least as difficult as the generation of all the transversals of a hypergraph. To our knowledge the complexity of this hypergraph problem is still open.…”

Output-Sensitive Algorithms for Enumerating Minimal Transversals for Some Geometric Hypergraphs

“…When V is a finite set of points and each object in F is an arbitrary finite subset of V, we obtain the well-known hypergraph transversal or dualization problem [2], which calls for finding all minimal hitting sets for a given hypergraph G ⊆ 2 V , defined on a finite set of vertices V. Denote by Tr(G) the set of all minimal hitting sets of G, also known as the transversal hypergraph of G. The problem of finding Tr(G) has received considerable attention in the literature (see, e.g., [3,12,13,19,29,31]), since it is known to be polynomially or quasi-polynomially equivalent with many problems in various areas, such as artificial intelligence (e.g., [12,24]), database theory (e.g., [30]), distributed systems (e.g., [23]), machine learning and data mining (e.g., [1,7,20]), mathematical programming (e.g., [5,25]), matroid theory (e.g., [26]), and reliability theory (e.g., [9]). …”

On Berge Multiplication for Monotone Boolean Dualization

“…Equivalently, the problem is to compute, for an explicitly given hypergraph C ⊆ 2 V , the transversal hypergraph D, consisting of all minimal transversals D of H (i.e., all minimal subsets D ⊆ V such that D ∩ C = ∅ for all C ∈ C). This problem has received considerable attention in the literature (see e.g., [4,11,13,29,32]), since it is known to be polynomially or quasi-polynomially equivalent with many problems in various areas, such as artificial intelligence (e.g., [11,22]), database theory (e.g., [31]), distributed systems (e.g., [18,20]), machine learning and data mining (e.g., [1,7,19]), mathematical programming (e.g., [5,24]), matroid theory (e.g., [27,25]), and reliability theory (e.g., [8,33]). …”