Computational aspects of monotone dualization: A brief survey

Dualization of a monotone Boolean function represented by a conjunctive normal form (CNF) is a problem which, in different disguise, is ubiquitous in many areas including Computer Science, Artificial Intelligence, and Game Theory to mention some of them. It is also one of the few problems whose prec...

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Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 156; no. 11; pp. 2035 - 2049
Main Authors: Eiter, Thomas, Makino, Kazuhisa, Gottlob, Georg
Format: Journal Article
Language:English
Published: Lausanne Elsevier B.V 06.06.2008
Amsterdam Elsevier
New York, NY
Subjects:
Algorithmics. Computability. Computer arithmetics
Applied sciences
Artificial intelligence
Combinatorial enumeration
Combinatorics
Combinatorics. Ordered structures
Computer science; control theory; systems
Dualization
Exact sciences and technology
Graph theory
Hitting sets
Hypergraphs
Independent sets
Limited nondeterminism
Mathematics
Monotone Boolean functions
Order, lattices, ordered algebraic structures
Output-polynomial algorithms
Polynomial-total time
Quasi-polynomial time
Sciences and techniques of general use
Self-duality
Set coverings
Theoretical computing
Transversals
Set coverings
Hitting sets
Transversals
Dualization
Monotone Boolean functions
Combinatorial enumeration
Self-duality
Independent sets
Polynomial-total time
Limited nondeterminism
Quasi-polynomial time
Hypergraphs
Output-polynomial algorithms
Hypergraph
Duality
Research
Implementation
Non determinism
Optimization
Complexity
Independent set
Set covering
Combinatorics
Learning algorithm
Monotonic function
Enumeration
Computer theory
Boolean function
Computer games
Game theory
Polynomial time
Survey
Experimental result
Area
Parallel computation
Artificial intelligence
Solvability
ISSN:0166-218X, 1872-6771
Online Access:Get full text
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Summary:Dualization of a monotone Boolean function represented by a conjunctive normal form (CNF) is a problem which, in different disguise, is ubiquitous in many areas including Computer Science, Artificial Intelligence, and Game Theory to mention some of them. It is also one of the few problems whose precise tractability status (in terms of polynomial-time solvability) is still unknown, and now open for more than 25 years. In this paper, we briefly survey computational results for this problem, where we focus on the famous paper by Fredman and Khachiyan [On the complexity of dualization of monotone disjunctive normal forms, J. Algorithms 21 (1996) 618–628], which showed that the problem is solvable in quasi-polynomial time (and thus most likely not co-NP-hard), as well as on follow-up works. We consider computational aspects including limited nondeterminism, probabilistic computation, parallel and learning-based algorithms, and implementations and experimental results from the literature. The paper closes with open issues for further research.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2007.04.017