Covering Points with Rectangular Boundaries

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.