Fast Approximation Algorithms for Piercing Boxes by Points

$ \newcommand{\Re}{\mathbb{R}} \newcommand{\BX}{\mathcal{B}} \newcommand{\bb}{\mathsf{b}} \newcommand{\eps}{\varepsilon} \newcommand{\polylog}{\mathrm{polylog}} $ Let $\BX={\bb1, \ldots ,\bbn}$ be a set of $n$ axis-aligned boxes in $\Re^d$ where $d\geq2$ is a constant. The piercing problem is to compute a smallest set of points $N \subset \Re^d$ that hits every box in $\BX$, i.e., $N\cap \bb_i\neq \emptyset$, for $i=1,\ldots, n$. The problem is known to be NP-Hard. Let $\psi:=\psi(\BX)$, the \emph{piercing number} be the minimum size of a piercing set of $\BX$. We first present a randomized $O(\log\log \psi)$-approximation algorithm with expected running time $O(n^{{d/2}\polylog} (n))$. Next, we show that the expected running time can be improved to near-linear using a sampling-based technique, if $\psi = O(n^{1/(d-1)})$. Specifically, in the plane, the improved running time is $O(n \log \psi)$, assuming $\psi < n/\log^{\Omega(1)} n$. Finally, we study the dynamic version of the piercing problem where boxes can be inserted or deleted. For boxes in $\Re^2$, we obtain a randomized $O(\log\log\psi)$-approximation algorithm with $O(n^{{1/2}\polylog} (n))$ amortized expected update time for insertion or deletion of boxes. For squares in $\Re^2$, the update time can be improved to $O(n^{{1/3}\polylog} (n))$. Our algorithms are based on the multiplicative weight-update (MWU) method and require the construction of a weak $\eps$-net for a point set with respect to boxes. A key idea of our work is to exploit the duality between the piercing set and independent set (for boxes) to speed up our MWU. We also present a simpler and slightly more efficient algorithm for constructing a weak $\eps$-net than in [Ezr10], which is of independent interest. Our approach also yields a simpler algorithm for constructing (regular) $\eps$-nets with respect to boxes for $d=2,3$.