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...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Antonios Antoniadis, Falk Hüffner, Pascal Lenzner, Carsten Moldenhauer, Alexander Souza
• Keywords: CORR 2010, Discrepancy Problems, Education, Load Balancing Scenario, Maximal Difference