### Abstract

We consider inverse chromatic number problems in interval graphs having the following form: we are given an integer K and an interval graph G = (V,E), associated with n = |V| intervals I _{i} = ]a _{i} ,b _{i} [ (1 ≤ i ≤ n), each having a specified length s(I _{i} ) = b _{i} - a _{i} , a (preferred) starting time a _{i} and a completion time b _{i} . The intervals are to be newly positioned with the least possible discrepancies from the original positions in such a way that the related interval graph can be colorable with at most K colors. We propose a model involving this problem called inverse booking problem.We show that inverse booking problems are hard to approximate within O(n ^{1 - ε} ), ε > 0 in the general case with no constraints on lengths of intervals, even though a ratio of n can be achieved by using a result of [13]. This result answers a question recently formulated in [12] about the approximation behavior of the unweighted case of single machine just-in-time scheduling problem with earliness and tardiness costs. Moreover, this result holds for some restrictive cases.

Original language | English |
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Title of host publication | WALCOM |

Subtitle of host publication | Algorithms and Computation - Second International Workshop, WALCOM 2008, Proceedings |

Pages | 180-187 |

Number of pages | 8 |

DOIs | |

Publication status | Published - 2008 Aug 27 |

Event | 2nd International Workshop on Algorithms and Computation, WALCOM 2008 - Dhaka, Bangladesh Duration: 2008 Feb 7 → 2008 Feb 8 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 4921 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 2nd International Workshop on Algorithms and Computation, WALCOM 2008 |
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Country | Bangladesh |

City | Dhaka |

Period | 08/2/7 → 08/2/8 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*WALCOM: Algorithms and Computation - Second International Workshop, WALCOM 2008, Proceedings*(pp. 180-187). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4921 LNCS). https://doi.org/10.1007/978-3-540-77891-2_17