Mining association rules between sets of items in large databases
SIGMOD '93 Proceedings of the 1993 ACM SIGMOD international conference on Management of data
Complexity of identification and dualization of positive Boolean functions
Information and Computation
Interior and exterior functions of Boolean functions
Discrete Applied Mathematics
On the complexity of dualization of monotone disjunctive normal forms
Journal of Algorithms
Fast discovery of association rules
Advances in knowledge discovery and data mining
On generating the irredundant conjunctive and disjunctive normal forms of monotone Boolean functions
Discrete Applied Mathematics - Special issue on the satisfiability problem and Boolean functions
The Combinatorics of Network Reliability
The Combinatorics of Network Reliability
Dual-Bounded Generating Problems: Partial and Multiple Transversals of a Hypergraph
SIAM Journal on Computing
On frequent sets of Boolean matrices
Annals of Mathematics and Artificial Intelligence
On Generating All Minimal Integer Solutions for a Monotone System of Linear Inequalities
ICALP '01 Proceedings of the 28th International Colloquium on Automata, Languages and Programming,
On the Complexity of Generating Maximal Frequent and Minimal Infrequent Sets
STACS '02 Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
An inequality for polymatroid functions and its applications
An inequality for polymatroid functions and its applications
Scientific contributions of Leo Khachiyan (a short overview)
Discrete Applied Mathematics
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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) 驴 t1,..., fm(X) 驴 tm with quasi-polynomially bounded right hand sides t1,..., tm all minimal feasible solutions X 驴 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 驴 1, let 驴 = 驴(f, t) denote the number of maximal sets X 驴 V satisfying f(X) t, let 脽 = 脽(f, t) be the number of minimal sets X 驴 V for which f(X) 驴 t, and let n = |V |. We show that 驴 驴 max{n, 脽(log t)/c}, where c = c(n, 脽) is the unique positive root of the equation 2c(nc/ log 脽 - 1) = 1. In particular, our bound implies that 驴 驴 (n脽)log t. We also give examples of polymatroid functions with arbitrarily large t, n, 驴 and 脽 for which 驴 = 脽(1-o(1)) log t/c.