On Fair Epsilon Net and Geometric Hitting Set

2025-07-11

AI Summary

This paper introduces group fairness into classical geometric approximation problems—specifically, ε-nets and geometric hitting sets—for the first time. It formalizes two new concepts: fair ε-nets and fair geometric hitting sets, supporting both demographic parity (preserving group proportions) and customizable proportional fairness (achieving arbitrary target ratios). Method: Leveraging bounded VC-dimension, the authors design approximation algorithms combining random sampling with discrepancy theory, ensuring theoretical guarantees while maintaining practicality. Contribution/Results: The fair ε-net algorithm achieves zero unfairness with asymptotically unchanged size; the fair hitting set algorithm attains an $O(log ext{OPT} cdot log k)$ approximation ratio, where OPT is the optimal solution size and $k$ is the number of groups. Empirical evaluation demonstrates an effective trade-off between accuracy and fairness. This work establishes the first systematic framework for fairness-aware geometric approximation algorithms.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Machine Learning: Ethics, Bias, and Fairness
• Computer Vision: Bias, Fairness & Privacy

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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 $O(log mathsf{OPT} imes log k)$ 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.

Problem

• Incorporating fairness into geometric approximation problems like epsilon-nets.
• Addressing demographic parity and custom-ratios fairness in data summaries.
• Developing algorithms for fair geometric hitting sets and epsilon-nets.

Innovation

• Fairness added to geometric approximation problems
• Sampling-based algorithm for fair ε-net
• Discrepancy-based algorithm for smaller ε-net