在线命中集问题：$d$ 维胖物体
Online hitting set of $d$-dimensional fat objects
AI总结：
本文研究几何最小命中集的在线版本，提出竞争比为 $((4\alpha+1)^{2d}\log N)$ 的确定性算法，适用于 $\alpha$-胖物体，并证明该比率在固定 $\alpha$ 和 $d$ 时最优，解决了二维圆盘和 $d$ 维立方体的开放问题。
摘要：
我们考虑几何最小命中集问题的在线版本，该问题可以描述为对手与算法之间的博弈。对于某些整数 $d$ 和 $N$，设 $P$ 为 $(0, N)^d$ 中具有整数坐标的点的集合，并设 $\mathcal{O}$ 为 $P$ 的子集族，称为物体。$P$ 和 $\mathcal{O}$ 均由算法和对手事先知晓。然后，对手逐个给出一些物体，算法必须使用 $P$ 中的点来维护这些物体的有效命中集，且决策是即时且不可撤销的。我们通过算法的竞争比来衡量其性能，即算法使用的点数与对手选择的物体子序列的离线最小命中集之间的比率。\n我们提出一个简单的确定性在线算法，当物体对应于 $\alpha$-胖物体族时，其竞争比为 $((4\alpha+1)^{2d}\log N)$。非正式地，$\alpha$-胖度衡量物体与立方体的相似程度。我们证明当 $\alpha$ 和 $d$ 为固定常数时，没有算法能达到更好的比率。特别是，我们的算法适用于二维圆盘和 $d$ 维立方体，这回答了先前相关论文中在点集对应于固定 $d$ 维立方体的所有整数坐标点这一特殊情况下的两个开放问题。
英文摘要：
We consider an online version of the geometric minimum hitting set problem that can be described as a game between an adversary and an algorithm. For some integers $d$ and $N$, let $P$ be the set of points in $(0, N)^d$ with integral coordinates, and let $\mathcal{O}$ be a family of subsets of $P$, called objects. Both $P$ and $\mathcal{O}$ are known in advance by the algorithm and by the adversary. Then, the adversary gives some objects one by one, and the algorithm has to maintain a valid hitting set for these objects using points from $P$, with an immediate and irrevocable decision. We measure the performance of the algorithm by its competitive ratio, that is the ratio between the number of points used by the algorithm and the offline minimum hitting set for the sub-sequence of objects chosen by the adversary. We present a simple deterministic online algorithm with competitive ratio $((4α+1)^{2d}\log N)$ when objects correspond to a family of $α$-fat objects. Informally, $α$-fatness measures how cube-like is an object. We show that no algorithm can achieve a better ratio when $α$ and $d$ are fixed constants. In particular, our algorithm works for two-dimensional disks and $d$-cubes which answers two open questions from related previous papers in the special case where the set of points corresponds to all the points of integral coordinates with a fixed $d$-cube.
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