用矩形边界覆盖点
Covering Points with Rectangular Boundaries
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中文总结 AI 辅助
研究了用轴对齐矩形边界覆盖点集的问题，证明了连续情况下的固定参数可解性，并展示了NP难性。
中文摘要 AI 辅助
几何覆盖问题要求找到一个小的几何对象家族，其并集能够覆盖给定的点集。我们研究了更为严格的边界覆盖变种，其中每个点必须位于所选对象的边界上。受Langerman和Morin[Discret.\ Comput.\ Geom., 2005]对超球体框架的启发，我们开始研究由轴对齐矩形构成的边界覆盖。我们首先考虑离散情况，其中矩形必须从给定的家族中选择。我们定义了\bcdaprfull（\bcdaprshort）：给定一个点集$P\subseteq\mathbb{R}^2$，一个轴对齐矩形的家族$\mathcal{R}$，以及一个整数$k$，判断是否可以使用至多$k$个来自$\mathcal{R}$的矩形的边界覆盖$P$。我们证明\bcdaprshort参数化于$k$是$\mathrm{W}[1]$难的。然后我们研究连续变种\prbcfull（\prbcshort），其中矩形可以自由放置。给定$P\subseteq\mathbb{R}^2$和$k$，目标是判断是否可以使用至多$k$个轴对齐矩形的边界覆盖$P$。与离散情况不同，我们证明\prbcshort是固定参数可解的，运行时间为$2^{\cO(k\log k)}\cdot n^{\cO(1)}$，其中$n=|P|$。我们的算法依赖于$k$个矩形如何与点集相互作用的结构分析，将\prbcshort减少到至多$2^{\cO(k\log k)}$个\ddmtcsp实例，每个实例可以在多项式时间内解决。在难度方面，我们证明了使用轴对齐$L$形的边界覆盖是NP难的，并利用此减少证明\prbcshort的NP难性。
英文摘要
Geometric covering problems ask for a small family of geometric objects whose union covers a given point set. We study the more restrictive \emph{boundary covering} variant, where every point must lie on the boundary of a chosen object. Motivated by the framework of Langerman and Morin\,[Discret.\ Comput.\ Geom., 2005] for hyperspheres, we initiate the study of boundary covering by axis-parallel rectangles. We first consider the \emph{discrete} setting, where rectangles must be selected from a given family. We define \bcdaprfull\ (\bcdaprshort): given a point set \(P\subseteq\mathbb{R}^2\), a family \(\mathcal{R}\) of axis-parallel rectangles, and an integer \(k\), decide whether \(P\) can be covered by the boundaries of at most \(k\) rectangles from \(\mathcal{R}\). We prove that \bcdaprshort\ is \(\mathrm{W}[1]\)-hard parameterized by \(k\). We then study the \emph{continuous} variant, \prbcfull\ (\prbcshort), where rectangles may be placed freely. Given \(P\subseteq\mathbb{R}^2\) and \(k\), the goal is to decide whether \(P\) can be covered by the boundaries of at most \(k\) axis-parallel rectangles. In contrast to the discrete case, we show that \prbcshort\ is fixed-parameter tractable, with running time \(2^{\cO(k\log k)}\cdot n^{\cO(1)}\), where \(n=|P|\). Our algorithm relies on a structural analysis of how \(k\) rectangles interact with the point set, reducing \prbcshort\ to at most \(2^{\cO(k\log k)}\) instances of \ddmtcsp, each solvable in polynomial time. On the hardness side, we prove NP-completeness for boundary covering by axis-aligned \(L\)-shapes and use this reduction to establish NP-completeness of \prbcshort.
