On dualization in products of forests, in

Abstract. Let P = P1 ×...×Pn be the product of n partially ordered sets, each with an acyclic precedence graph in which either the in-degree or the out-degree of each element is bounded. Given a subset A⊆P,it is shown that the set of maximal independent elements of A in P can be incrementally generated in quasi-polynomial time. We discuss some applications in data mining related to this dualization problem

On the Complexity of Generating Maximal Frequent and Minimal Infrequent Sets

Let A be an mn binary matrix, t . . . , m} be a threshold, and # > 0 be a positive parameter. We show that given a family of O(n ) 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 # t + 1)#, where # and # 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

Matroid Intersections, Polymatroid Inequalities, and Related Problems

Given m matroids M1 , . . . , Mm on the common ground set V , it is shown that all maximal subsets of V , independent in the m matroids, can be generated in quasi-polynomial time. More generally, given a system of polymatroid inequalities f1 (X) t1 , . . . , fm (X) tm with quasi-polynomially bounded right hand sides t1 , . . . , tm , all minimal feasible solutions X V to the system can be generated in incremental quasi-polynomial time. Our proof of these results is based on a combinatorial inequality for polymatroid functions which may be of independent interest. Precisely, for a polymatroid function f and an integer threshold t 1, let # = #(f, t) denote the number of maximal V satisfying f(X) < t, let # = #(f, t) be the number of minimal t, and let n = |V |

Hypergraph Transversal Computation and Related Problems in Logic and AI

Generating minimal transversals of a hypergraph is an important problem which has many applications in Computer Science. In the present paper, we address this problem and its decisional variant, i.e., the recognition of the transversal hypergraph for another hypergraph