## Abstract

We consider the discrepancy problem of coloring n intervals with

When generalizing to

In an online scenario, where intervals arrive over time and the color has to be decided upon arrival, the maximal difference in the size of color classes can become arbitrarily high for any online algorithm.

*k*colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is minimal. Somewhat surprisingly, a coloring with maximal difference at most one always exists. Furthermore, we give an algorithm with running time O(*n*log*n*+*kn*log*k*) for its construction. This is in particular interesting because many known results for discrepancy problems are non-constructive. This problem naturally models a load balancing scenario, where*n*tasks with given start- and endtimes have to be distributed among k servers. Our results imply that this can be done ideally balanced.When generalizing to

*d*-dimensional boxes (instead of intervals), a solution with difference at most one is not always possible. We show that for any*d*≥2 and any*k*≥2 it is NP-complete to decide if such a solution exists, which implies also NP-hardness of the respective minimization problem.In an online scenario, where intervals arrive over time and the color has to be decided upon arrival, the maximal difference in the size of color classes can become arbitrarily high for any online algorithm.

Original language | English |
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Title of host publication | 28th International Symposium on Theoretical Aspects of Computer Science |

Subtitle of host publication | STACS’11, March 10–12, 2011, Dortmund, Germany |

Editors | Thomas Schwentick, Christoph Dürr |

Place of Publication | Saarbrücken/Wadern |

Publisher | Dagstuhl |

Pages | 531-542 |

Number of pages | 12 |

ISBN (Print) | 978-3-939897-25-5 |

DOIs | |

Publication status | Published - 2011 |

Externally published | Yes |

Event | 28th International Symposium on Theoretical Aspects of Computer Science, STACS 2011 - Dortmund, Germany Duration: 10 Mar 2011 → 12 Mar 2011 Conference number: 28 |

### Publication series

Name | LIPIcs–Leibniz International Proceedings in Informatics |
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Publisher | Schloss Dagstuhl |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 28th International Symposium on Theoretical Aspects of Computer Science, STACS 2011 |
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Abbreviated title | STACS 2011 |

Country/Territory | Germany |

City | Dortmund |

Period | 10/03/11 → 12/03/11 |