### Abstract

This study is concerned with finding a level ideal (LI) of a partially ordered set (poset). Given a finite poset P, the level of each element p ε P is defined as the number of ideals that do not include p, then the problem is to find the ith LI-the ideal consisting of elements whose levels are less than a given integer i. The concept of a level ideal is naturally derived from the generalized median stable matchings, introduced by Teo and Sethuraman [Teo, C. P., J. Sethuraman. 1998. The geometry of fractional stable matchings and its applications. Math. Oper. Res. 23(4) 874-891] in the context of "fairness" of matchings in a stable marriage problem. Cheng [Cheng, C. T. 2010. Understanding the generalized median stable matchings. Algorithmica 58(1) 34-51] showed that finding the ith LI is #P-hard when i = Θ4N5, where N is the total number of ideals of P. This paper shows that finding the ith LI is #P-hard even if i = Θ4N1/c5, where c is an arbitrary constant at least one. Meanwhile, we present a polynomial time exact algorithm when i - O44logN5c0 5, where c0 is an arbitrary positive constant. We also devise two randomized approximation schemes for the ideals of a poset, by using an oracle of an almost-uniform sampler.

Original language | English |
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Pages (from-to) | 356-371 |

Number of pages | 16 |

Journal | Mathematics of Operations Research |

Volume | 37 |

Issue number | 2 |

DOIs | |

Publication status | Published - May 1 2012 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Computer Science Applications
- Management Science and Operations Research

### Cite this

**On randomized approximation for finding a level ideal of a poset and the generalized median stable matchings.** / Kijima, Shuji; Nemoto, Toshio.

Research output: Contribution to journal › Review article

*Mathematics of Operations Research*, vol. 37, no. 2, pp. 356-371. https://doi.org/10.1287/moor.1110.0526

}

TY - JOUR

T1 - On randomized approximation for finding a level ideal of a poset and the generalized median stable matchings

AU - Kijima, Shuji

AU - Nemoto, Toshio

PY - 2012/5/1

Y1 - 2012/5/1

N2 - This study is concerned with finding a level ideal (LI) of a partially ordered set (poset). Given a finite poset P, the level of each element p ε P is defined as the number of ideals that do not include p, then the problem is to find the ith LI-the ideal consisting of elements whose levels are less than a given integer i. The concept of a level ideal is naturally derived from the generalized median stable matchings, introduced by Teo and Sethuraman [Teo, C. P., J. Sethuraman. 1998. The geometry of fractional stable matchings and its applications. Math. Oper. Res. 23(4) 874-891] in the context of "fairness" of matchings in a stable marriage problem. Cheng [Cheng, C. T. 2010. Understanding the generalized median stable matchings. Algorithmica 58(1) 34-51] showed that finding the ith LI is #P-hard when i = Θ4N5, where N is the total number of ideals of P. This paper shows that finding the ith LI is #P-hard even if i = Θ4N1/c5, where c is an arbitrary constant at least one. Meanwhile, we present a polynomial time exact algorithm when i - O44logN5c0 5, where c0 is an arbitrary positive constant. We also devise two randomized approximation schemes for the ideals of a poset, by using an oracle of an almost-uniform sampler.

AB - This study is concerned with finding a level ideal (LI) of a partially ordered set (poset). Given a finite poset P, the level of each element p ε P is defined as the number of ideals that do not include p, then the problem is to find the ith LI-the ideal consisting of elements whose levels are less than a given integer i. The concept of a level ideal is naturally derived from the generalized median stable matchings, introduced by Teo and Sethuraman [Teo, C. P., J. Sethuraman. 1998. The geometry of fractional stable matchings and its applications. Math. Oper. Res. 23(4) 874-891] in the context of "fairness" of matchings in a stable marriage problem. Cheng [Cheng, C. T. 2010. Understanding the generalized median stable matchings. Algorithmica 58(1) 34-51] showed that finding the ith LI is #P-hard when i = Θ4N5, where N is the total number of ideals of P. This paper shows that finding the ith LI is #P-hard even if i = Θ4N1/c5, where c is an arbitrary constant at least one. Meanwhile, we present a polynomial time exact algorithm when i - O44logN5c0 5, where c0 is an arbitrary positive constant. We also devise two randomized approximation schemes for the ideals of a poset, by using an oracle of an almost-uniform sampler.

UR - http://www.scopus.com/inward/record.url?scp=84861377081&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84861377081&partnerID=8YFLogxK

U2 - 10.1287/moor.1110.0526

DO - 10.1287/moor.1110.0526

M3 - Review article

AN - SCOPUS:84861377081

VL - 37

SP - 356

EP - 371

JO - Mathematics of Operations Research

JF - Mathematics of Operations Research

SN - 0364-765X

IS - 2

ER -