Balanced Interval Coloring

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 $n$ 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 $d \ge 2$ and any $k \ge 2$ 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.