### Abstract

In this paper, we consider the complexity of the problem of finding a stable fractional matching in a hypergraphic preference system. Aharoni and Fleiner proved that there exists a stable fractional matching in every hypergraphic preference system. Furthermore, Kintali, Poplawski, Rajaraman, Sundaram, and Teng proved that the problem of finding a stable fractional matching in a hypergraphic preference system is PPAD-complete. In this paper, we consider the complexity of the problem of finding a stable fractional matching in a hypergraphic preference system whose maximum degree is bounded by some constant. The proof by Kintali, Poplawski, Rajaraman, Sundaram, and Teng implies the PPAD-completeness of the problem of finding a stable fractional matching in a hypergraphic preference system whose maximum degree is 5. In this paper, we prove that (i) this problem is PPAD-complete even if the maximum degree is 3, and (ii) if the maximum degree is 2, then this problem can be solved in polynomial time. Furthermore, we prove that the problem of finding an approximate stable fractional matching in a hypergraphic preference system is PPAD-complete.

Original language | English |
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Title of host publication | 29th International Symposium on Algorithms and Computation, ISAAC 2018 |

Editors | Chung-Shou Liao, Der-Tsai Lee, Wen-Lian Hsu |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770941 |

DOIs | |

Publication status | Published - Dec 1 2018 |

Event | 29th International Symposium on Algorithms and Computation, ISAAC 2018 - Jiaoxi, Yilan, Taiwan, Province of China Duration: Dec 16 2018 → Dec 19 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 123 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 29th International Symposium on Algorithms and Computation, ISAAC 2018 |
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Country | Taiwan, Province of China |

City | Jiaoxi, Yilan |

Period | 12/16/18 → 12/19/18 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software

### Cite this

*29th International Symposium on Algorithms and Computation, ISAAC 2018*(Leibniz International Proceedings in Informatics, LIPIcs; Vol. 123). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ISAAC.2018.11

**On the complexity of stable fractional hypergraph matching.** / Ishizuka, Takashi; Kamiyama, Naoyuki.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*29th International Symposium on Algorithms and Computation, ISAAC 2018.*Leibniz International Proceedings in Informatics, LIPIcs, vol. 123, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 29th International Symposium on Algorithms and Computation, ISAAC 2018, Jiaoxi, Yilan, Taiwan, Province of China, 12/16/18. https://doi.org/10.4230/LIPIcs.ISAAC.2018.11

}

TY - GEN

T1 - On the complexity of stable fractional hypergraph matching

AU - Ishizuka, Takashi

AU - Kamiyama, Naoyuki

PY - 2018/12/1

Y1 - 2018/12/1

N2 - In this paper, we consider the complexity of the problem of finding a stable fractional matching in a hypergraphic preference system. Aharoni and Fleiner proved that there exists a stable fractional matching in every hypergraphic preference system. Furthermore, Kintali, Poplawski, Rajaraman, Sundaram, and Teng proved that the problem of finding a stable fractional matching in a hypergraphic preference system is PPAD-complete. In this paper, we consider the complexity of the problem of finding a stable fractional matching in a hypergraphic preference system whose maximum degree is bounded by some constant. The proof by Kintali, Poplawski, Rajaraman, Sundaram, and Teng implies the PPAD-completeness of the problem of finding a stable fractional matching in a hypergraphic preference system whose maximum degree is 5. In this paper, we prove that (i) this problem is PPAD-complete even if the maximum degree is 3, and (ii) if the maximum degree is 2, then this problem can be solved in polynomial time. Furthermore, we prove that the problem of finding an approximate stable fractional matching in a hypergraphic preference system is PPAD-complete.

AB - In this paper, we consider the complexity of the problem of finding a stable fractional matching in a hypergraphic preference system. Aharoni and Fleiner proved that there exists a stable fractional matching in every hypergraphic preference system. Furthermore, Kintali, Poplawski, Rajaraman, Sundaram, and Teng proved that the problem of finding a stable fractional matching in a hypergraphic preference system is PPAD-complete. In this paper, we consider the complexity of the problem of finding a stable fractional matching in a hypergraphic preference system whose maximum degree is bounded by some constant. The proof by Kintali, Poplawski, Rajaraman, Sundaram, and Teng implies the PPAD-completeness of the problem of finding a stable fractional matching in a hypergraphic preference system whose maximum degree is 5. In this paper, we prove that (i) this problem is PPAD-complete even if the maximum degree is 3, and (ii) if the maximum degree is 2, then this problem can be solved in polynomial time. Furthermore, we prove that the problem of finding an approximate stable fractional matching in a hypergraphic preference system is PPAD-complete.

UR - http://www.scopus.com/inward/record.url?scp=85063689090&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85063689090&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ISAAC.2018.11

DO - 10.4230/LIPIcs.ISAAC.2018.11

M3 - Conference contribution

AN - SCOPUS:85063689090

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 29th International Symposium on Algorithms and Computation, ISAAC 2018

A2 - Liao, Chung-Shou

A2 - Lee, Der-Tsai

A2 - Hsu, Wen-Lian

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -