rectangle stabbing

main.tex

@@ -14,6 +14,8 @@
 
 For errata and more stuff, see \url{https://sarielhp.org/book/}
 
+Note that unless specifically stated, we always consider the RAM model.
+
 \section{Grid}
 \begin{exercise}\label{ex1.1}
 Let $P$ be a max cardinality point set contained in the $d$-dimensional unit hypercube such that the smallest distance of point pairs in $P$ is 1. Prove that 
@@ -46,4 +48,23 @@ Let $C$ and $P$ be two given set of points such that $k=|C|$ and $n=|P|$. Define
 \end{exercise}
 
 
+\section*{Not in the book}
+\begin{problem}[$d$-dimensional rectangle stabbing \cite{gaur_constant_2002}]
+Given a set $R$ of $n$ axis-parallel rectangles and a set $L$ of axis-parallel real lines, find the minimum subset of $L$ such that every rectangle is stabbed by at least one line in the subset.
+\end{problem}
+This problem is NP-hard even for the 2D case. There is a LP rounding method which gives a $d$-approximation. Let $K_i$ be the subset of $d$th-axis parallel lines in $L$. For a rectangle $r$, denote by $K_i[r]$ the set of lines in $K_i$ that stab $r$. Consider the following LP.
+
+\begin{equation*}
+\begin{aligned}
+\min& &  \sum_{i\in [d]} \sum_{\ell \in K_i} x_\ell&  &  &   \\
+s.t.& &  \sum_{i\in [d]} \sum_{\ell \in K_i[r]} x_\ell&\geq 1 &    &\forall r\in R\\
+    &   &   x_\ell&\geq 0    &  &\forall \ell\in L
+\end{aligned}
+\end{equation*}
+
+Let $\set{x^*_\ell: \ell\in L}$ be the optimal solution to the above LP. For each $r$, there must be some $i\in [d]$ such that $\sum_{\ell \in K_i[r]}x^*_\ell \geq 1/d$.
+
+\bibliographystyle{alpha}
+\bibliography{ref}
+
 \end{document}