Submodular optimization has become increasingly prominent in machine learning and fairness has drawn much attention. In this paper, we propose to study the fair $k$-submodular maximization problem and develop a $\frac{1}{3}$-approximation greedy algorithm with a running time of $\mathcal{O}(knB)$. To the best of our knowledge, our work is the first to incorporate fairness in the context of $k$-submodular maximization, and our theoretical guarantee matches the best-known $k$-submodular maximization results without fairness constraints. In addition, we have developed a faster threshold-based algorithm that achieves a $(\frac{1}{3} - \epsilon)$ approximation with $\mathcal{O}(\frac{kn}{\epsilon} \log \frac{B}{\epsilon})$ evaluations of the function $f$. Furthermore, for both algorithms, we provide approximation guarantees when the $k$-submodular function is not accessible but only can be approximately accessed. We have extensively validated our theoretical findings through empirical research and examined the practical implications of fairness. Specifically, we have addressed the question: ``What is the price of fairness?" through case studies on influence maximization with $k$ topics and sensor placement with $k$ types. The experimental results show that the fairness constraints do not significantly undermine the quality of solutions.

翻译：子模优化在机器学习领域日益重要，而公平性问题也备受关注。本文提出研究公平k-子模最大化问题，并开发了一种运行时间为$\mathcal{O}(knB)$的$\frac{1}{3}$近似贪心算法。据我们所知，这是首次在k-子模最大化背景下引入公平性约束的研究，且我们的理论保证与无公平性约束时最著名的k-子模最大化结果相匹配。此外，我们提出了一种更快的基于阈值的算法，该算法通过$\mathcal{O}(\frac{kn}{\epsilon} \log \frac{B}{\epsilon})$次函数$f$的评估，实现了$(\frac{1}{3} - \epsilon)$近似比。进一步地，针对两种算法，我们给出了当k-子模函数无法精确获取而只能近似访问时的近似保证。我们通过大量实证研究验证了理论发现，并探讨了公平性的实际影响。特别地，我们通过k主题影响力最大化和k类型传感器布局的案例研究，回答了"公平性的代价是什么？"这一问题。实验结果表明，公平性约束并未显著降低解的质量。