### Abstract

The SATISFACTORY PARTITION problem consists of deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part. This problem was introduced by Gerber and Kobler in 1998 and further studied by other authors but its complexity remained open until now. We prove in this paper that SATISFACTORY PARTITION, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. We also study approximation results for the latter problem, showing that it has no polynomial-time approximation scheme, whereas a constant approximation can be obtained in polynomial time. Similar results hold for balanced partitions where each vertex is required to have at most as many neighbors in its part as in the other part.

Original language | English |
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Title of host publication | Lecture Notes in Computer Science |

Editors | L. Wang |

Pages | 829-838 |

Number of pages | 10 |

Volume | 3595 |

Publication status | Published - 2005 |

Event | 11th Annual International Conference on Computing and Combinatorics, COCOON 2005 - Kunming, China Duration: aug. 16 2005 → aug. 29 2005 |

### Other

Other | 11th Annual International Conference on Computing and Combinatorics, COCOON 2005 |
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Country | China |

City | Kunming |

Period | 8/16/05 → 8/29/05 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science (miscellaneous)
- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science*(Vol. 3595, pp. 829-838)

**Complexity and approximation of satisfactory partition problems.** / Bazgan, Cristina; Tuza, Z.; Vanderpooten, Daniel.

Research output: Conference contribution

*Lecture Notes in Computer Science.*vol. 3595, pp. 829-838, 11th Annual International Conference on Computing and Combinatorics, COCOON 2005, Kunming, China, 8/16/05.

}

TY - GEN

T1 - Complexity and approximation of satisfactory partition problems

AU - Bazgan, Cristina

AU - Tuza, Z.

AU - Vanderpooten, Daniel

PY - 2005

Y1 - 2005

N2 - The SATISFACTORY PARTITION problem consists of deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part. This problem was introduced by Gerber and Kobler in 1998 and further studied by other authors but its complexity remained open until now. We prove in this paper that SATISFACTORY PARTITION, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. We also study approximation results for the latter problem, showing that it has no polynomial-time approximation scheme, whereas a constant approximation can be obtained in polynomial time. Similar results hold for balanced partitions where each vertex is required to have at most as many neighbors in its part as in the other part.

AB - The SATISFACTORY PARTITION problem consists of deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part. This problem was introduced by Gerber and Kobler in 1998 and further studied by other authors but its complexity remained open until now. We prove in this paper that SATISFACTORY PARTITION, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. We also study approximation results for the latter problem, showing that it has no polynomial-time approximation scheme, whereas a constant approximation can be obtained in polynomial time. Similar results hold for balanced partitions where each vertex is required to have at most as many neighbors in its part as in the other part.

UR - http://www.scopus.com/inward/record.url?scp=26844501707&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=26844501707&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:26844501707

VL - 3595

SP - 829

EP - 838

BT - Lecture Notes in Computer Science

A2 - Wang, L.

ER -