求具有遗传性质的点集的参数化复杂性
The Parameterized Complexity of Finding Point Sets with Hereditary Properties
摘要：
我们考虑的问题，输入是一组点在平面上和一个整数$k$，任务是找到一个子集$S$的输入点的大小$k$，使$S$满足一些属性。我们专注于属性，只依赖于点的顺序类型，是单调下点删除。我们发现，并不是所有这样的问题是固定参数易处理参数$k$，表现出一个属性定义的三个禁止模式，找到一个$k$点子集的属性是$\mathrm{W}[1]$ -完成，（假设指数时间假设）不能解决的时间$n^{o(k/\log k)}$。然而，我们发现，这种类型的问题是固定参数的所有属性，包括所有共线点集，属性，排除至少一个凸多边形，和属性定义的一个单一的禁止模式。
英文摘要：
We consider problems where the input is a set of points in the plane and an integer $k$, and the task is to find a subset $S$ of the input points of size $k$ such that $S$ satisfies some property. We focus on properties that depend only on the order type of the points and are monotone under point removals. We show that not all such problems are fixed-parameter tractable parameterized by $k$, by exhibiting a property defined by three forbidden patterns for which finding a $k$-point subset with the property is $\mathrm{W}[1]$-complete and (assuming the exponential time hypothesis) cannot be solved in time $n^{o(k/\log k)}$. However, we show that problems of this type are fixed-parameter tractable for all properties that include all collinear point sets, properties that exclude at least one convex polygon, and properties defined by a single forbidden pattern.
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