### Abstract

The Satisfactory Bisection problem means to decide whether a given graph has a partition of its vertex set into two parts of the same cardinality such that each vertex has at least as many neighbors in its part as in the other part. A related variant of this problem, called Co-Satisfactory Bisection, requires that each vertex has at most as many neighbors in its part as in the other part. A vertex satisfying the degree constraint above in a partition is called 'satisfied' or 'co-satisfied,' respectively. After stating the NP-completeness of both problems, we study approximation results in two directions. We prove that maximizing the number of (co-)satisfied vertices in a bisection has no polynomial-time approximation scheme (unless P = NP), whereas constant approximation algorithms can be obtained in polynomial time. Moreover, minimizing the difference of the cardinalities of vertex classes in a bipartition that (co-)satisfies all vertices has no polynomial-time approximation scheme either.

Original language | English |
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Pages (from-to) | 875-883 |

Number of pages | 9 |

Journal | Journal of Computer and System Sciences |

Volume | 74 |

Issue number | 5 |

DOIs | |

Publication status | Published - Aug 2008 |

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### Keywords

- Approximation algorithm
- Complexity
- Degree constraints
- Graph
- NP-complete
- Vertex partition

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Networks and Communications
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*Journal of Computer and System Sciences*,

*74*(5), 875-883. https://doi.org/10.1016/j.jcss.2007.12.001