### Abstract

This paper is concerned with finding a level ideal (LI) of a partially ordered set (poset): given a finite poset P, a level of each element p ∈ P is defined as the number of ideals which do not include p, then the problem is to find an ideal consisting of elements whose levels are less than a given integer i. We call the ideal as the i-th LI. The concept of the level ideal is naturally derived from the generalized median stable matching, that is a fair stable marriage introduced by Teo and Sethuraman (1998). Cheng (2008) showed that finding the i-th LI is #P-hard when i=Θ(N), where N is the total number of ideals of P. This paper shows that finding the i-th LI is #P-hard even if i=Θ(N^{1/c}) where c≥1 is an arbitrary constant. Meanwhile, we give a polynomial time exact algorithm when i=O((logN)^{c′}) where c′ is an arbitrary positive constant. We also devise two randomized approximation schemes using an oracle of almost uniform sampler for ideals of a poset.

Original language | English |
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Title of host publication | Computing and Combinatorics - 15th Annual International Conference, COCOON 2009, Proceedings |

Pages | 317-327 |

Number of pages | 11 |

DOIs | |

Publication status | Published - 2009 |

Event | 15th Annual International Conference on Computing and Combinatorics, COCOON 2009 - Niagara Falls, NY, United States Duration: Jul 13 2009 → Jul 15 2009 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 5609 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 15th Annual International Conference on Computing and Combinatorics, COCOON 2009 |
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Country | United States |

City | Niagara Falls, NY |

Period | 7/13/09 → 7/15/09 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Computing and Combinatorics - 15th Annual International Conference, COCOON 2009, Proceedings*(pp. 317-327). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5609 LNCS). https://doi.org/10.1007/978-3-642-02882-3_32