On the complexity of monotone dualization and generating minimal hypergraph transversals

@article{Elbassioni2008OnTC,
  title={On the complexity of monotone dualization and generating minimal hypergraph transversals},
  author={Khaled M. Elbassioni},
  journal={Discret. Appl. Math.},
  year={2008},
  volume={156},
  pages={2109-2123}
}
In 1994 Fredman and Khachiyan established the remarkable result that the duality of a pair of monotone Boolean functions, in disjunctive normal forms, can be tested in quasi-polynomial time, thus putting the problem, most likely, somewhere between polynomiality and coNP-completeness. We strengthen this result by showing that the duality testing problem can in fact be solved in polylogarithmic time using a quasi-polynomial number of processors (in the PRAM model). While our decomposition… Expand
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References

SHOWING 1-10 OF 58 REFERENCES
New Results on Monotone Dualization and Generating Hypergraph Transversals
TLDR
It is shown that duality of two monotone CNFs can be disproved with limited nondeterminism and feasible in polynomial time with O(log2 n/\log log n) suitably guessed bits. Expand
Exact Transversal Hypergraphs and Application to Boolean µ-Functions
  • Thomas Eiter
  • Computer Science, Mathematics
  • J. Symb. Comput.
  • 1994
TLDR
It is shown that hypergraphs are recognizable in polynomial time and that their minimal transversals as well as their maximal independent sets can be generated in lexicographic order withPolynomial delay between subsequent outputs, which is impossible in the general case unless P= NP. Expand
Computational aspects of monotone dualization: A brief survey
TLDR
This paper focuses on the famous paper by Fredman and Khachiyan, which showed that the dualization of monotone disjunctive normal forms is solvable in quasi-polynomial time (and thus most likely not co-NP-hard), as well as on follow-up works. Expand
Monotone Boolean dualization is in co-NP[log2n]
TLDR
A modified version of Fredman and Khachiyan's algorithm requires deterministic polynomial time plus O( log2 n) nondeterministic guesses, thus placing the problem in the class co-NP[log2 n]. Expand
Self-Duality of Bounded Monotone Boolean Functions and Related Problems
TLDR
The equivalence between the problem of determining self-duality of a boolean function in DNF and a special type of satisfiability problem called NAESPI is shown and it is shown that c-bounded NAesPI can be solved in polynomial time when c is some constant. Expand
Transversal Hypergraphs and Families of Polyhedral Cones
We discuss the complexity of generating certain geometric configurations related to the classical theorems of Caratheodory and Helly. Given a set K of rational cones in n dimensions and a rationalExpand
Dual subimplicants of positive Boolean functions
Given a positive Boolean function fand a subset δ of its variables, we give a combinatorial condition characterizing the existence of a prime implicant Dˆof the Boolean dual f d of f having theExpand
Dual-Bounded Generating Problems: All Minimal Integer Solutions for a Monotone System of Linear Inequalities
TLDR
The results imply, in particular, that the problem of incrementally generating all minimal integer solutions to a monotone system of linear inequalities can be done in quasi-polynomial time. Expand
A global parallel algorithm for the hypergraph transversal problem
TLDR
A new decomposition technique is given for solving the problem of finding all minimal transversals of a hypergraph H@?2^V, given by an explicit list of its hyperedges with the following advantages: global parallelism, and new results on the complexity of generating minimalTransversals for new classes of hypergraphs. Expand
On the Complexity of the Multiplication Method for Monotone CNF/DNF Dualization
TLDR
The first non-trivial upper-bounds on the complexity of the (generalized) multiplication method are presented, showing that if the grouping of the clauses is done in an output-independent way, then multiplication can be performed in sub-exponential time (n|ψ|) O( √ |Φ|) | Φ| O(log n) . Expand
...
1
2
3
4
5
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