RS. hyperplane version

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+\DeclareMathOperator*{\opt}{OPT}
+
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@@ -50,33 +52,38 @@ Let $C$ and $P$ be two given set of points such that $k=|C|$ and $n=|P|$. Define
 
 \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.
+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. Let $K_i$ be the subset of $d$th-axis parallel lines in $L$. For a rectangle $r\in R$, denote by $K_i^r$ the set of lines in $K_i$ that stab $r$. Consider the following LP.
+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_{\ell\in L} 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
+\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 $\set{x^*_\ell: \ell\in L}$ be the optimal solution to the above LP.
-For each $r$, there must be some $i\in [d]$ such that $\sum_{\ell \in K_i^r}x^*_\ell \geq 1/d$. Denote such a set for rectangle $r$ by $K_*^r$.
-Suppose that we find a subset $L^{int}\subset L$ and define a integral solution $\set{y_\ell=1}_{\ell\in L^{int}}\cup \set{y_\ell=0}_{\ell\notin L^{int}}$ such that $\sum_{\ell\in K_*^r}\geq 1$ for each rectangle $r$. In other words, we restrict the solution to be a subset of lines that stabs every rectangle $r$ with lines in $K_*^r$.
+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  $\sum_{\ell \in K_i^r}x^*_\ell \geq 1/d$. Let $R_i\subset R$ be the set of rectangles assigned dimension $i$. We want to solve the following IP for dimension $i\in[d]$.
+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}
-\min& &  \sum_{\ell\in K_i} x_\ell&  &  &   \\
-s.t.& &  \sum_{\ell \in K_i^r} x_\ell&\geq 1 &    &\forall r\in R_i\\
-    &   &   x_\ell&\in \set{0,1}    &  &\forall \ell\in K_i
+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 and therefore the its LP relaxation is integral. wait... the definition for d-dimensional rectangle stabbing is different. they do not use lines, they use hyperplanes... I cannot show that the constraint matrix is TUM under my settings.
+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.
+
+
 
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