## Abstract

We study the parameterized complexity of several NP-Hard graph completion problems: The MINIMUM FILL-IN problem is to decide if a graph can be triangulated by adding at most k edges. We develop an O(k^{5}mn + f(k)) algorithm for the problem on a graph with n vertices and m edges. In particular, this implies that the problem is fixed parameter tractable (FPT). PROPER INTERVAL GRAPH COMPLETION problems, motivated by molecular biology, ask for adding edges in order to obtain a proper interval graph, so that a parameter in that graph does not exceed k. We show that the problem is FPT when k is the number of added edges. For the problem where k is the clique size, we give an O(f(k)n^{k-1}) algorithm, so it is polynomial for fixed k. on the other hand, we prove its hardness in the parameterized hierarchy, so it is probably not FPT. Those results are obtained even when a set of edges which should not be added is given. That set can be given either explicitly or by a proper vertex coloring which the added edges should respect.

Original language | English |
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Pages (from-to) | 780-791 |

Number of pages | 12 |

Journal | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |

DOIs | |

State | Published - 1994 |

Event | Proceedings of the 35th IEEE Annual Symposium on Foundations of Computer Science - Santa Fe, NM, USA Duration: 20 Nov 1994 → 22 Nov 1994 |

### Funding

Funders | Funder number |
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Minstry of Science and the Arts, Israel | |

Of- fice of Naval Research | |

National Science Foundation | CCR-8920505 |

Office of Naval Research | N00014-91-J-1463 |

Princeton University |