In the online hitting set problem, sets arrive over time, and the algorithm has to maintain a subset of elements that hit all the sets seen so far. Alon, Awerbuch, Azar, Buchbinder, and Naor (SICOMP 2009) gave an algorithm with competitive ratio $O(\log n \log m)$ for the (general) online hitting set and set cover problems for $m$ sets and $n$ elements; this is known to be tight for efficient online algorithms. Given this barrier for general set systems, we ask: can we break this double-logarithmic phenomenon for online hitting set/set cover on structured and geometric set systems? We provide an $O(\log n \log\log n)$-competitive algorithm for the weighted online hitting set problem on set systems with linear shallow-cell complexity, replacing the double-logarithmic factor in the general result by effectively a single logarithmic term. As a consequence of our results we obtain the first bounds for weighted online hitting set for natural geometric set families, thereby answering open questions regarding the gap between general and geometric weighted online hitting set problems.

翻译：在线命中集问题中，集合随时间到达，算法必须维护一个元素子集以覆盖当前所有已见集合。Alon、Awerbuch、Azar、Buchbinder 与 Naor (SICOMP 2009) 针对包含 $m$ 个集合与 $n$ 个元素的（通用）在线命中集与集合覆盖问题，提出了竞争比为 $O(\log n \log m)$ 的算法；该结果对于高效在线算法已被证明是紧致的。面对通用集合系统的这一理论壁垒，我们提出：能否在结构化与几何集合系统上突破在线命中集/集合覆盖问题的双对数瓶颈？针对具有线性浅胞复杂度集合系统的加权在线命中集问题，我们提出了一种 $O(\log n \log\log n)$ 竞争比算法，将通用结果中的双对数因子有效降低为单对数项。作为本研究的推论，我们首次获得了自然几何集合族的加权在线命中集边界，从而解答了关于通用与几何加权在线命中集问题间差距的开放性问题。