On the Complexity of Generating Maximal Frequent and Minimal Infrequent Sets

Abstract

Abstract. Let A be an m×n binary matrix, t ∈ {1, . . . , m} be a threshold, and ε > 0 be a positive parameter. We show that given a family of O(n ε ) maximal t-frequent column sets for A, it is NP-complete to decide whether A has any further maximal t-frequent sets, or not, even when the number of such additional maximal t-frequent column sets may be exponentially large. In contrast, all minimal t-infrequent sets of columns of A can be enumerated in incremental quasi-polynomial time. The proof of the latter result follows from the inequality α ≤ (m − t + 1)β, where α and β are respectively the numbers of all maximal t-frequent and all minimal t-infrequent sets of columns of the matrix A. We also discuss the complexity of generating all closed t-frequent column sets for a given binary matrix.

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]). …”

Extending the Balas-Yu bounds on the number of maximal independent sets in graphs to hypergraphs and lattices

“…Consequently, for products of lattices with bounded width, this set can be generated in incremental quasi-polynomial time by Theorem 5. The special case of the above result for databases D of binary attributes can be found in [5,6].…”

Matroid Intersections, Polymatroid Inequalities, and Related Problems

“…Since the function f (X) def = |{H ∈ H | H ⊇ X}| is polymatroid of range |H|, Theorems 3 and 2 imply respectively that the number of maximal frequent sets can be bounded by a quasi-polynomial in the number of minimal infrequent sets and the sizes of V, H, and that the minimal infrequent sets can be generated in quasi-polynomial time. In fact, the bound of Theorem 4 can be strengthened to a sharp linear bound in this case, see [7].…”

“…It is worth mentioning that in all of the above examples, generating all maximal infeasible sets for (1) turns out to be NP-hard, see [7,11,15].…”