论公平的 ε 网与几何击中集
On Fair Epsilon Net and Geometric Hitting Set
AI总结：
本文将公平性引入ε网、ε样本和几何击中集等经典几何近似问题，提出基于抽样和差异理论的两种算法，在保证零不公平性的同时仅适度增加输出规模。
摘要：
公平性已成为数据驱动决策中的一项艰巨挑战。许多数据问题，例如为近似查询处理创建紧凑的数据摘要，可以通过使用计算几何中的概念（如 ε 网）来有效解决。然而，这些强大的工具尚未从公平性的角度进行审视。为了填补这一研究空白，我们将公平性引入到 ε 网、ε 样本和几何击中集等经典几何近似问题中。我们引入并探讨了两种群体公平性概念：人口统计平权（要求保留输入分布中的群体比例）和自定义比例公平性（要求满足任意目标比例）。我们开发了两种算法来强制实现公平性：一种基于抽样，另一种基于差异理论。基于抽样的算法速度更快，计算出的公平 ε 网的规模仅比标准（不公平）ε 网大一个 log(k) 因子，其中 k 是人口统计群体的数量。基于差异的算法稍慢（对于有界 VC 维），但它能计算出更小的公平 ε 网。值得注意的是，我们将公平几何击中集问题简化为寻找公平 ε 网。这产生了公平几何击中集的 O(log OPT × log k) 近似。此外，我们展示了在某些输入分布下，构建公平 ε 样本是不可行的，凸显了公平抽样的局限性。除了理论保证外，我们的实验结果还验证了所提算法的实际有效性。具体而言，与不公平设置相比，我们实现了零不公平性，且输出规模仅有适度增加。
英文摘要：
Fairness has emerged as a formidable challenge in data-driven decisions. Many of the data problems, such as creating compact data summaries for approximate query processing, can be effectively tackled using concepts from computational geometry, such as $\varepsilon$-nets. However, these powerful tools have yet to be examined from the perspective of fairness. To fill this research gap, we add fairness to classical geometric approximation problems of $\varepsilon$-net, $\varepsilon$-sample, and geometric hitting set. We introduce and address two notions of group fairness: demographic parity, which requires preserving group proportions from the input distribution, and custom-ratios fairness, which demands satisfying arbitrary target ratios. We develop two algorithms to enforce fairness: one based on sampling and another on discrepancy theory. The sampling-based algorithm is faster and computes a fair $\varepsilon$-net of size which is only larger by a $\log(k)$ factor compared to the standard (unfair) $\varepsilon$-net, where $k$ is the number of demographic groups. The discrepancy-based algorithm is slightly slower (for bounded VC dimension), but it computes a smaller fair $\varepsilon$-net. Notably, we reduce the fair geometric hitting set problem to finding fair $\varepsilon$-nets. This results in a $O(\log \mathsf{OPT} \times \log k)$ approximation of a fair geometric hitting set. Additionally, we show that under certain input distributions, constructing fair $\varepsilon$-samples can be infeasible, highlighting limitations in fair sampling. Beyond the theoretical guarantees, our experimental results validate the practical effectiveness of the proposed algorithms. In particular, we achieve zero unfairness with only a modest increase in output size compared to the unfair setting.
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