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. To fill this gap, we propose a new problem called \emph{Bicriteria Submodular Maximization} (BSM) to balance 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, we focus on designing efficient 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 proposed methods in three submodular maximization problems: maximum coverage, influence maximization, and facility location.

翻译：子模函数最大化是一类基础组合优化问题，广泛应用于数据摘要、影响力最大化及推荐系统等领域。在许多此类问题中，目标是为所有用户找到最大化平均效用的解，其中每个用户的效用由单调子模函数定义。然而，当用户群体由若干人口统计群体组成时，另一个关键问题在于效用是否在不同群体间公平分配。尽管"效用"与"公平性"目标均具理想性，二者可能相互矛盾，且据我们所知，鲜有研究关注如何联合优化这两个目标。为填补这一空白，我们提出名为"双准则子模最大化"的新问题，旨在平衡效用与公平性。具体而言，该问题要求在公平性函数值不低于给定阈值的前提下，寻找固定规模的解以最大化效用函数。由于BSM问题无法在任意常数因子内近似求解，我们专注于设计高效的实例依赖近似方案。我们的算法方案包含两种方法，通过将BSM实例转化为其他子模优化问题实例，实现了不同的近似因子。利用真实与合成数据集，我们在最大覆盖、影响力最大化和设施选址三个子模最大化问题中展示了所提方法的应用。