On Fair Epsilon Net and Geometric Hitting Set

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.