Balanced Interval Coloring

Authors Antonios Antoniadis, Falk Hueffner, Pascal Lenzner, Carsten Moldenhauer, Alexander Souza

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Antonios Antoniadis
Falk Hueffner
Pascal Lenzner
Carsten Moldenhauer
Alexander Souza

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Antonios Antoniadis, Falk Hueffner, Pascal Lenzner, Carsten Moldenhauer, and Alexander Souza. Balanced Interval Coloring. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 531-542, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)


We consider the discrepancy problem of coloring n intervals with 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.
  • Load balancing
  • discrepancy theory
  • NP-hardness


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