Computing the largest empty rectangle
SIAM Journal on Computing
A note on finding a maximum empty rectangle
Discrete Applied Mathematics
Mining association rules between sets of items in large databases
SIGMOD '93 Proceedings of the 1993 ACM SIGMOD international conference on Management of data
Identifying the Minimal Transversals of a Hypergraph and Related Problems
SIAM Journal on Computing
Complexity of identification and dualization of positive Boolean functions
Information and Computation
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
Data mining, hypergraph transversals, and machine learning (extended abstract)
PODS '97 Proceedings of the sixteenth ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems
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
Mining for Empty Rectangles in Large Data Sets
ICDT '01 Proceedings of the 8th International Conference on Database Theory
Using Decision Tree Induction for Discovering Holes in Data
PRICAI '98 Proceedings of the 5th Pacific Rim International Conference on Artificial Intelligence: Topics in Artificial Intelligence
An inequality for polymatroid functions and its applications
Discrete Applied Mathematics - Submodularity
On the generation of circuits and minimal forbidden sets
Mathematical Programming: Series A and B
Discovering interesting holes in data
IJCAI'97 Proceedings of the Fifteenth international joint conference on Artifical intelligence - Volume 2
An intersection inequality for discrete distributions and related generation problems
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Scientific contributions of Leo Khachiyan (a short overview)
Discrete Applied Mathematics
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We consider monotone ∨, ∧-formulae φ of m atoms, each of which is a monotone inequality of the form fi(x)≥ ti over the integers, where for i = 1,...,m, $f_i : \mathbb{Z}^n \mapsto \mathbb{R}$ is a given monotone function and ti is a given threshold. We show that if the ∨-degree of φ is bounded by a constant, then for linear, transversal and polymatroid monotone inequalities all minimal integer vectors satisfying φ can be generated in incremental quasi-polynomial time. In contrast, the enumeration problem for the disjunction of m inequalities is NP-hard when m is part of the input. We also discuss some applications of the above results in disjunctive programming, data mining, matroid and reliability theory.