# On Dualization over Distributive Lattices

@article{Elbassioni2020OnDO, title={On Dualization over Distributive Lattices}, author={Khaled M. Elbassioni}, journal={ArXiv}, year={2020}, volume={abs/2006.15337} }

Given a partially order set (poset) $P$, and a pair of families of ideals $\cI$ and filters $\cF$ in $P$ such that each pair $(I,F)\in \cI\times\cF$ has a non-empty intersection, the dualization problem over $P$ is to check whether there is an ideal $X$ in $P$ which intersects every member of $\cF$ and does not contain any member of $\cI$. Equivalently, the problem is to check for a distributive lattice $L=L(P)$, given by the poset $P$ of its set of joint-irreducibles, and two given antichains…

## References

SHOWING 1-10 OF 18 REFERENCES

### On the dualization in distributive lattices and related problems

- Mathematics, Computer ScienceDiscret. Appl. Math.
- 2021

### Dualization in lattices given by implicational bases

- Mathematics, Computer ScienceICFCA
- 2019

It is shown using hypergraph dualization that the problem can be solved in output quasi-polynomial time whenever the implicational base has bounded independent-width, defined as the size of a maximum set of implications having independent conclusions.

### Lattices, closures systems and implication bases: A survey of structural aspects and algorithms

- Computer ScienceTheor. Comput. Sci.
- 2018

### On Generating the Irredundant Conjunctive and Disjunctive Normal Forms of Monotone Boolean Functions

- MathematicsDiscret. Appl. Math.
- 1999

### On the complexity of monotone dualization and generating minimal hypergraph transversals

- MathematicsDiscret. Appl. Math.
- 2008

### Complexity of Identification and Dualization of Positive Boolean Functions

- Computer Science, MathematicsInf. Comput.
- 1995

It is shown that the existence of an incrementally polynomial algorithm for this problem is equivalent to the exist of the following algorithms, where ƒ and g are positive Boolean functions.

### Generating all Maximal Independent Sets: NP-Hardness and Polynomial-Time Algorithms

- Computer Science, MathematicsSIAM J. Comput.
- 1980

It is shown that it is possible to apply ideas of Paull and Unger and of Tsukiyama et al. to obtain polynomial-time algorithms for a number of special cases, e.g. the efficient generation of all maximal feasible solutions to a knapsack problem.

### The joy of implications, aka pure Horn formulas: Mainly a survey

- Computer ScienceTheor. Comput. Sci.
- 2017

### Dual-Bounded Generating Problems: All Minimal Integer Solutions for a Monotone System of Linear Inequalities

- MathematicsSIAM J. Comput.
- 2002

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.