Balanced Interval Coloring
Antonios Antoniadis, Falk H\"uffner, Pascal Lenzner, Carsten, Moldenhauer, Alexander Souza

TL;DR
This paper studies a coloring problem for intervals that minimizes the maximum difference between colors, providing a constructive algorithm for the one-dimensional case and proving NP-completeness for higher dimensions.
Contribution
It introduces a polynomial-time algorithm for balanced interval coloring in one dimension and proves NP-completeness for higher dimensions, highlighting the problem's computational complexity.
Findings
Existence of a coloring with max difference at most one for intervals
Efficient algorithm with O(n log n + kn log k) complexity for construction
NP-completeness of the problem in higher dimensions
Abstract
We consider the discrepancy problem of coloring intervals with 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 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 tasks with given start- and endtimes have to be distributed among servers. Our results imply that this can be done ideally balanced. When generalizing to -dimensional boxes (instead of intervals), a solution with difference at most one is not always possible. We show that for any and any it is NP-complete to…
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