From d9b4380c03b6c65246e929c42204f5b7f7f91539 Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Sun, 17 Aug 2025 10:15:27 +0800 Subject: [PATCH] RS. hyperplane version --- main.tex | 33 ++++++++++++++++++++------------- 1 file changed, 20 insertions(+), 13 deletions(-) diff --git a/main.tex b/main.tex index 7aac50f..8316dee 100644 --- a/main.tex +++ b/main.tex @@ -7,6 +7,8 @@ \author{} \date{} +\DeclareMathOperator*{\opt}{OPT} + \begin{document} \maketitle % \tableofcontents @@ -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. + + \bibliographystyle{alpha} \bibliography{ref}