Given a set $S$ of $n$ points in the plane, we study the two-line-center problem: finding two lines that minimize the maximum distance from each point in $S$ to its closest line. We present a $(1+\varepsilon)$-approximation algorithm for the two-line-center problem that runs in $O((n/\varepsilon) \log (1/\varepsilon))$ time, which improves the previously best $O(n\log n + ({n}/{\varepsilon^2}) \log ({1}/{\varepsilon}) + (1/\varepsilon^3)\log ({1}/{\varepsilon}))$-time algorithm. We also consider three variants of this problem, in which the orientations of the two lines are restricted: (1) the orientation of one of the two lines is fixed, (2) the orientations of both lines are fixed, and (3) the two lines are required to be parallel. For each of these three variants, we give the first $(1+\varepsilon)$-approximation algorithm that runs in linear time. In particular, for the variant where the orientation of one of the two lines is fixed, we also give an improved exact algorithm that runs in $O(n \log n)$ time and show that it is optimal.

翻译：给定平面上一个包含 $n$ 个点的集合 $S$，我们研究双线中心问题：寻找两条直线，使得 $S$ 中每个点到其最近直线的最大距离最小。我们提出了一种求解双线中心问题的 $(1+\varepsilon)$-近似算法，其运行时间为 $O((n/\varepsilon) \log (1/\varepsilon))$，这改进了先前最佳的 $O(n\log n + ({n}/{\varepsilon^2}) \log ({1}/{\varepsilon}) + (1/\varepsilon^3)\log ({1}/{\varepsilon}))$ 时间算法。我们还考虑了该问题的三个变体，其中两条直线的方向受到限制：(1) 两条直线中有一条的方向固定，(2) 两条直线的方向均固定，(3) 两条直线要求平行。对于这三个变体中的每一个，我们都给出了首个在**线性时间**内运行的 $(1+\varepsilon)$-近似算法。特别地，对于两条直线中有一条方向固定的变体，我们还给出了一种改进的精确算法，其运行时间为 $O(n \log n)$，并证明了该算法是最优的。