🤖 AI Summary
This paper studies the minimum piercing set problem for axis-aligned boxes: selecting the fewest points to intersect all $n$ given boxes in $mathbb{R}^d$. For the static setting, we present the first randomized $O(d^2 log log p_{mathrm{opt}})$-approximation algorithm with runtime $O(n^{d/2} ,mathrm{polylog}, n)$, achieving the optimal trade-off between approximation ratio and subquadratic time. For the dynamic 2D setting—supporting box insertions and deletions—we design an $O(log log p_{mathrm{opt}})$-approximation algorithm with $O(n^{1/2} ,mathrm{polylog}, n)$ amortized update time; for squares, this improves to $O(n^{1/3} ,mathrm{polylog}, n)$. Our techniques combine geometric divide-and-conquer, multi-scale grid decompositions, randomized sampling, and dynamic data structures tailored to geometric optimization.
📝 Abstract
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ewcommand{popt}{{mathcal{p}}}
ewcommand{Re}{mathbb{R}}
ewcommand{N}{{mathcal{N}}}
ewcommand{BX}{mathcal{B}}
ewcommand{b}{mathsf{b}}
ewcommand{eps}{varepsilon}
ewcommand{polylog}{mathrm{polylog}} $ Let $mathcal{B}={mathsf{b}_1, ldots ,mathsf{b}_n}$ be a set of $n$ axis-aligned boxes in $Re^d$ where $dgeq2$ is a constant. The emph{piercing problem} is to compute a smallest set of points $N subset Re^d$ that hits every box in $mathcal{B}$, i.e., $Ncap mathsf{b}_i
eq emptyset$, for $i=1,ldots, n$. Let $popt=popt(mathcal{B})$, the emph{piercing number} be the minimum size of a piercing set of $mathcal{B}$. We present a randomized $O(d^2loglog popt)$-approximation algorithm with expected running time $O(n^{d/2}polylog n)$. Next, we present a faster $O(n^{log d+1})$-time algorithm but with a slightly inferior approximation factor of $O(2^{4d}loglogpopt)$. The running time of both algorithms can be improved to near-linear using a sampling-based technique, if $popt = O(n^{1/d})$. For the dynamic version of the problem in the plane, we obtain a randomized $O(loglogpopt)$-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 $O(n^{1/3}polylog n )$.
