平衡区间着色
Balanced Interval Coloring
摘要：
考虑用$k$颜色对$n$区间着色的差异问题，使得在直线上的每一点上，任意两种颜色的区间数的最大差异最小.令人惊讶的是，最大差异最多为1的着色总是存在的。并给出了一个具有运行时间的构造算法$O(n \log n + kn \log k)$.这是特别有趣的，因为许多已知的差异问题的结果是非建设性的。这个问题自然地模拟了负载均衡场景，其中具有给定开始和结束时间的$n$任务必须在$k$服务器之间分布。我们的研究结果表明，这可以做到理想的平衡。当推广到$d$维盒（而不是区间）时，最多相差1的解并不总是可能的。我们表明，对于任何$d \ge 2$和任何$k \ge 2$，它是NP-完全的，以决定是否存在这样的解决方案，这意味着也NP-困难的相应的最小化问题。在在线场景中，间隔随着时间的推移而到达，并且颜色必须在到达时决定，颜色类大小的最大差异对于任何在线算法都可以变得任意高。
英文摘要：
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.
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