Balanced Interval Coloring
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
531-542
Regular Paper
Antonios
Antoniadis
Antonios Antoniadis
Falk
Hueffner
Falk Hueffner
Pascal
Lenzner
Pascal Lenzner
Carsten
Moldenhauer
Carsten Moldenhauer
Alexander
Souza
Alexander Souza
10.4230/LIPIcs.STACS.2011.531
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license
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