The hitting set problem is a well-known NP-hard optimization problem in which, given a set of elements and a collection of subsets, the goal is to find the smallest selection of elements, such that each subset contains at least one element in the selection. Many geometric set systems enjoy improved approximation ratios, which have recently been shown to be tight with respect to the shallow cell complexity of the set system. The algorithms that exploit the cell complexity, however, tend to be involved and computationally intensive. This paper shows that comparable approximation ratios for the hitting set problem can be attained using a much simpler algorithm: solve the linear programming relaxation, take one initial random sample from the set of elements with probabilities proportional to the LP-solution, and, while there is an unhit set, take an additional sample from it proportional to the LP-solution. Our algorithm is based on a generalization of the elegant net-finder algorithm of Nabil Mustafa. To analyze this algorithm for the hitting set problem, we generalize the classic Packing Lemma, and the more recent Shallow-Packing Lemma, to the setting of weighted epsilon nets.

翻译：击中集问题是一个著名的NP难优化问题：给定一个元素集合及其子集族，目标是最小化所选元素数量，使得每个子集至少包含一个被选元素。许多几何集合系统具有改进的近似比，这些近似比最近被证明与集合系统的浅层单元复杂度紧密相关。然而，利用单元复杂度的算法往往结构复杂且计算量大。本文证明，采用一种更简单的算法即可获得与击中集问题相当的近似比：求解线性规划松弛，根据与LP解成比例的概率从元素集合中随机抽取初始样本，然后当存在未被击中的子集时，从中额外抽取与LP解成比例的样本。该算法基于Nabil Mustafa提出的优雅网查找算法的推广。为分析该算法在击中集问题上的表现，我们将经典的打包引理及近年提出的浅层打包引理推广至加权ε-网场景。