Let $P$ be a set of $n$ points in the plane. We consider the problem of partitioning $P$ into two subsets $P_1$ and $P_2$ such that the sum of the perimeters of $\text{CH}(P_1)$ and $\text{CH}(P_2)$ is minimized, where $\text{CH}(P_i)$ denotes the convex hull of $P_i$. The problem was first studied by Mitchell and Wynters in 1991 who gave an $O(n^2)$ time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in $O(n \log^2 n)$ time and a $(1+\varepsilon)$-approximation algorithm running in $O(n + 1/\varepsilon^2\cdot\log^2(1/\varepsilon))$ time.

翻译：我们考虑的是将P$分成两个子集(P_1美元和P_2美元)的问题,以便最大限度地减少美元(P_1美元)和美元(text{CH})(P_2美元)的周界总和,其中$(text{CH}(P_i)美元)代表的是美元(P_i)的圆柱体。这个问题最初由Mitchell和Wynters在1991年研究过,他们给出了美元(n_2美元)的时间算法。尽管在相关问题上取得了很大进展,但迄今为止还没有找到该问题的次赤道时间算法。我们提出了一个精确的算法,用美元(n)的时间和美元(1 ⁇ 瓦列普西隆)的美元-辅助算法,用美元(n + 1/ 1/ varepsilon%2\ cdot\log_2(2/\ varepsilon) 运行。