圆上点的红蓝点分离
Red-Blue Point Separation for Points on a Circle
摘要：
给定平面上的一组红点R和一组蓝点B，红蓝点分离问题询问是否最多有k条线将R和B分开，也就是说，由解的线诱导的每个单元是空的或单色的（只包含一种颜色的点）。该问题的一个常见变体是当要求线轴平行时。已知该问题对于这两种情况都是NP完全的，并且W[1]-在前一种情况下由k硬参数化，在后一种情况下由FPT硬参数化。我们证明了一个多项式时间算法的特殊情况下，当点位于一个圆。此外，我们还证明了在轴平行设置的相关问题的W-硬度，其中的问题是，如果有p个水平和q个垂直线，从B分离R。这里的硬度以参数p表示。
英文摘要：
Given a set R of red points and a set B of blue points in the plane, the Red-Blue point separation problem asks if there are at most k lines that separate R from B, that is, each cell induced by the lines of the solution is either empty or monochromatic (containing points of only one color). A common variant of the problem is when the lines are required to be axis-parallel. The problem is known to be NP-complete for both scenarios, and W[1]-hard parameterized by k in the former setting and FPT in the latter. We demonstrate a polynomial-time algorithm for the special case when the points lie on a circle. Further, we also demonstrate the W-hardness of a related problem in the axis-parallel setting, where the question is if there are p horizontal and q vertical lines that separate R from B. The hardness here is shown in the parameter p.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
