RS. hyperplane version

main.tex

@@ -7,6 +7,8 @@
 \author{}
 \date{}
 
+\DeclareMathOperator*{\opt}{OPT}
+
 \begin{document}
 \maketitle
 % \tableofcontents
@@ -14,6 +16,8 @@
 
 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 
@@ -37,8 +41,51 @@ The last line is greater than $(\sqrt{d}/5)^d$ for large enough $d$.
 \end{proof}
 
 \begin{exercise}
-Compute clustering radius
+Compute clustering radius.
+Let $C$ and $P$ be two given set of points such that $k=|C|$ and $n=|P|$. Define the covering radius of $P$ by $C$ as $r=\max_{p\in P} \min_{c\in C} \norm{p-c}$.
+\begin{enumerate}
+\item find an $O(n+k\log n)$ expected time alg that outputs $\alpha$ such that $\alpha \leq r \leq 10\alpha$.
+\item for prescribed $\varepsilon>0$, find an $O(n+k\varepsilon^{-2}\log n)$ expected time alg that outputs $\alpha$ s.t. $\alpha<r<(1+\epsilon)\alpha$.
+\end{enumerate}
 \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 $\mathcal H$ of axis-parallel $d$ dimensional hyperplanes, find the minimum subset of $\mathcal H$ such that every rectangle is stabbed by at least one hyperplane 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 for dimension $d$. Let $K_i\subset \mathcal H$ be the set of hyperplanes that are orthogonal to the $i$th axis. For a rectangle $r\in R$, denote by $K_i^r$ the set of hyperplanes in $K_i$ that stab $r$. Consider the following LP.
+
+\begin{equation*}
+\begin{aligned}
+\min& &  \sum_{H\in \mathcal H} x_H&  &  &   \\
+s.t.& &  \sum_{i\in [d]} \sum_{H \in K_i^r} x_H&\geq 1 &    &\forall r\in R\\
+    &   &   x_H&\geq 0    &  &\forall H\in \mathcal H
+\end{aligned}
+\end{equation*}
+
+Let $\set{x^*_H: H\in \mathcal H}$ be the optimal solution to the above LP.
+For each $r$, there must be some $i\in [d]$ such that $\sum_{H \in K_i^r}x^*_H \geq 1/d$. Denote such a set for rectangle $r$ by $K_*^r$.
+Suppose that we find a subset $\mathcal H^{int}\subset \mathcal H$ and define a integral solution $\set{y_H=1}_{H\in \mathcal H^{int}}\cup \set{y_H=0}_{H\notin \mathcal H^{int}}$ such that $\sum_{H\in K_*^r}\geq 1$ for each rectangle $r$. In other words, we restrict the solution such that every rectangle $r$ is stabbed by hyperplanes in $K_*^r$.
+
+One nice property of this restriction is that now the problem becomes independent for each dimension. We assign to each rectangle $r$ a dimension $i$ such that  $\sum_{H \in K_i^r}x^*_H \geq 1/d$. This assignment indicates a partition $\set{R_i}_{i\in [d]}$ of $R$. We want to solve the following IP for dimension $i\in[d]$.
+
+\begin{equation*}
+\begin{aligned}
+IP_i=\min& &  \sum_{H\in K_i} x_H&  &  &   \\
+s.t.& &  \sum_{H \in K_i^r} x_H&\geq 1 &    &\forall r\in R_i\\
+    &   &   x_H&\in \set{0,1}    &  &\forall H\in K_i
+\end{aligned}
+\end{equation*}
+
+Another nice property is that the constraint matrix is TUM since one can sort the hyperplanes in $K_i$ by their intersection with the $i$th axis and see that element $1$'s  locate consecutively in each row in the constraint matrix. Hence, the linear relaxation of $IP_i$ (denoted by $LP_i$) is integral and we can solve $IP_i$ in polynomial time.
+
+Now we show connections between $x^*$ and solutions of $IP_i$. Let $x^*|_{K_i}$ be the optimal solution to the rectangle stabbing LP restricted to hyperplanes in $K_i$. 
+We also have $\sum_{i\in [d]} \opt(IP_i)\leq d \sum_H x^*_H$ since $d x^*|_{K_i}$ is a feasible solution to $LP_i$. Then the $d$-integrality gap follows from the fact that the union of optimal solutions to $IP_i$ is a feasible solution to the rectangle stabbing problem.
+
+
+
+\bibliographystyle{alpha}
+\bibliography{ref}
+
 \end{document}

ref.bib

@@ -0,0 +1,16 @@
+
+@article{gaur_constant_2002,
+	title = {Constant {Ratio} {Approximation} {Algorithms} for the {Rectangle} {Stabbing} {Problem} and the {Rectilinear} {Partitioning} {Problem}},
+	volume = {43},
+	issn = {0196-6774},
+	url = {https://www.sciencedirect.com/science/article/pii/S0196677402912216},
+	doi = {10.1006/jagm.2002.1221},
+	number = {1},
+	urldate = {2025-08-16},
+	journal = {Journal of Algorithms},
+	author = {Gaur, Daya Ram and Ibaraki, Toshihide and Krishnamurti, Ramesh},
+	month = apr,
+	year = {2002},
+	keywords = {approximation algorithms, combinatorial optimization, rectangle stabbing, rectilinear partitioning},
+	pages = {138--152},
+}