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 a slightly improved asymptotic approximation ratio 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 a simple generalization of the elegant net-finder algorithm by 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：Hitting set 问题是一个著名的 NP-难优化问题，给定一个元素集合和一个子集族，目标是找到元素的最小子集，使得每个子集至少包含所选元素中的一个。许多几何集合系统具有改进的近似比，最近的研究表明这些近似比相对于集合系统的浅层胞复杂度是紧的。然而，利用浅层胞复杂度的算法通常较为复杂且计算密集。本文表明，使用一种更简单的算法即可实现 hitting set 问题近似比的略微改进的渐近上界：求解线性规划松弛，以与 LP 解成比例的概率从元素集合中取初始随机样本，并在存在未被击中的集合时，从中以与 LP 解成比例的概率额外采样。我们的算法是 Nabil Mustafa 优雅的搜网算法的一种简单推广。为了分析该算法在 hitting set 问题上的表现，我们将经典的 Packing 引理和更新近的 Shallow Packing 引理推广到加权 epsilon-网的场景中。