We study a geometric facility location problem under imprecision. Given $n$ unit intervals in the real line, each with one of $k$ colors, the goal is to place one point in each interval such that the resulting \emph{minimum color-spanning interval} is as large as possible. A minimum color-spanning interval is an interval of minimum size that contains at least one point from a given interval of each color. We prove that if the input intervals are pairwise disjoint, the problem can be solved in $O(n)$ time, even for intervals of arbitrary length. For overlapping intervals, the problem becomes much more difficult. Nevertheless, we show that it can be solved in $O(n \log^2 n)$ time when $k=2$, by exploiting several structural properties of candidate solutions, combined with a number of advanced algorithmic techniques. Interestingly, this shows a sharp contrast with the 2-dimensional version of the problem, recently shown to be NP-hard.

翻译：我们研究一种不精确条件下的几何设施选址问题。给定实数轴上的 $n$ 个单位区间，每个区间被赋予 $k$ 种颜色之一，目标是在每个区间内放置一个点，使得生成的 \emph{最小颜色跨越区间} 尽可能大。最小颜色跨越区间是指包含每种颜色至少一个给定区间内点的最小长度区间。我们证明，若输入区间两两不相交，即使区间长度任意，该问题可在 $O(n)$ 时间内求解。对于重叠区间，问题难度显著增加。尽管如此，我们通过利用候选解的若干结构特性，并结合多种先进算法技术，证明了当 $k=2$ 时问题可在 $O(n \log^2 n)$ 时间内求解。有趣的是，这与此问题的二维版本（近期被证明是 NP 难问题）形成了鲜明对比。