Submodular function maximization is a fundamental combinatorial optimization problem with plenty of applications -- including data summarization, influence maximization, and recommendation. In many of these problems, the goal is to find a solution that maximizes the average utility over all users, for each of whom the utility is defined by a monotone submodular function. However, when the population of users is composed of several demographic groups, another critical problem is whether the utility is fairly distributed across different groups. Although the \emph{utility} and \emph{fairness} objectives are both desirable, they might contradict each other, and, to the best of our knowledge, little attention has been paid to optimizing them jointly. In this paper, we propose a new problem called \emph{Bicriteria Submodular Maximization} (BSM) to strike a balance between utility and fairness. Specifically, it requires finding a fixed-size solution to maximize the utility function, subject to the value of the fairness function not being below a threshold. Since BSM is inapproximable within any constant factor in general, we turn our attention to designing instance-dependent approximation schemes. Our algorithmic proposal comprises two methods, with different approximation factors, obtained by converting a BSM instance into other submodular optimization problem instances. Using real-world and synthetic datasets, we showcase applications of our methods in three submodular maximization problems: maximum coverage, influence maximization, and facility location.

翻译：子模函数最大化是一个基础的组合优化问题，广泛应用于数据摘要、影响力最大化和推荐等场景。在许多此类问题中，目标是找到一个能最大化所有用户平均效用的解决方案，其中每个用户的效用由一个单调子模函数定义。然而，当用户群体由多个不同的人口统计组构成时，另一个关键问题是效用是否在不同组之间公平分配。尽管"效用"和"公平性"目标都是可取的，但它们可能相互矛盾，而据我们所知，鲜有研究关注如何同时优化这两个目标。本文提出一个新问题——双准则子模最大化（BSM），旨在平衡效用与公平性。具体而言，它要求找到一个固定大小的解，在满足公平函数值不低于某个阈值的前提下最大化效用函数。由于BSM在一般情况下无法在任意常数因子内近似，我们转而设计依赖于实例的近似方案。我们的算法策略包括两种方法，具有不同的近似因子，通过将BSM实例转化为其他子模优化问题实例获得。利用现实世界和合成数据集，我们展示了这些方法在三个子模最大化问题中的应用：最大覆盖、影响力最大化和设施选址。