### Abstract

Finding a d-regular spanning subgraph (or d-factor) of a graph is easy by Tutte’s reduction to the matching problem. By the same reduction, it is easy to find a minimal or maximal d-factor of a graph. However, if we require that the d-factor is connected, these problems become NP-hard – finding a minimal connected 2-factor is just the traveling salesman problem (TSP).
Given a complete graph with edge weights that satisfy the triangle inequality, we consider the problem of finding a minimal connected d-factor. We give a 3-approximation for all d and improve this to an (r + 1)-approximation for even d, where r is the approximation ratio of the TSP. This yields a 2.5-approximation for even d. The same algorithm yields an (r + 1)-approximation for the directed version of the problem, where r is the approximation ratio of the asymmetric TSP. We also show that none of these minimization problems can be approximated better than the corresponding TSP.
Finally, for the decision problem of deciding whether a given graph contains a connected d-factor, we extend known hardness results.

Original language | Undefined |
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Title of host publication | Proceedings of the 11th Workshop on Approximation and Online Algorithms (WAOA 2013) |

Editors | C. Kaklamanis, K. Pruhs |

Place of Publication | Berlin, Germany |

Publisher | Springer |

Pages | 120-131 |

Number of pages | 12 |

ISBN (Print) | 978-3-319-08000-0 |

DOIs | |

Publication status | Published - 2014 |

Event | 11th Workshop on Approximation and Online Algorithms 2013 - Sophia Antipolis, France Duration: 5 Sep 2013 → 6 Sep 2013 http://algo2013.inria.fr/waoa.shtml |

### Publication series

Name | Lecture Notes in Computer Science |
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Publisher | Springer International Publishing |

Volume | 8447 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 11th Workshop on Approximation and Online Algorithms 2013 |
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Abbreviated title | WAOA 2013 |

Country | France |

City | Sophia Antipolis |

Period | 5/09/13 → 6/09/13 |

Internet address |

### Keywords

- EWI-24832
- Graph factors
- METIS-305910
- Approximation algorithms
- IR-91426

## Cite this

Cornelissen, K., Hoeksma, R. P., Manthey, B., Narayanaswamy, N. S., & Rahul, C. S. (2014). Approximability of Connected Factors. In C. Kaklamanis, & K. Pruhs (Eds.),

*Proceedings of the 11th Workshop on Approximation and Online Algorithms (WAOA 2013)*(pp. 120-131). (Lecture Notes in Computer Science; Vol. 8447). Berlin, Germany: Springer. https://doi.org/10.1007/978-3-319-08001-7_11