瓶颈凸集：在点集中寻找$k$大凸集
Bottleneck Convex Subsets: Finding $k$ Large Convex Sets in a Point Set
摘要：
Chvátal和Klincsek（1980）给出了一个求解平面上任意给定$n$点集$P$的最大基数凸子集问题的$O(n^3)$ -时间算法。本文探讨了一个推广的问题，瓶颈凸集问题：给定一组$P$的$n$点在平面上和一个正整数$k$，选择$k$两两不相交的凸子集的$P$，使最小的子集的基数最大化。等价地，解最大化具有相等基数的$P$的$k$相互不相交凸子集的基数。当$k$为任意输入参数时，该问题是NP-困难的;当$k$为固定参数时，给出了一个精确求解该问题的算法，其运行时间多项式为$n$;
英文摘要：
Chvátal and Klincsek (1980) gave an $O(n^3)$-time algorithm for the problem of finding a maximum-cardinality convex subset of an arbitrary given set $P$ of $n$ points in the plane. This paper examines a generalization of the problem, the Bottleneck Convex Subsets problem: given a set $P$ of $n$ points in the plane and a positive integer $k$, select $k$ pairwise disjoint convex subsets of $P$ such that the cardinality of the smallest subset is maximized. Equivalently, a solution maximizes the cardinality of $k$ mutually disjoint convex subsets of $P$ of equal cardinality. We show the problem is NP-hard when $k$ is an arbitrary input parameter, we give an algorithm that solves the problem exactly, with running time polynomial in $n$ when $k$ is fixed, and we give a fixed-parameter tractable algorithm parameterized in terms of the number of points strictly interior to the convex hull.
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