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，具有众多应用，包括数据摘要、影响力最大化和推荐。在这些问题中，目标通常是找到一个最大化所有用户平均效用的解，其中每个用户的效用由一个单调子模函数定义。然而，当用户群体由多个人口统计群体组成时，另一个关键问题是效用是否在不同群体之间公平分配。尽管“效用”和“公平”目标都是可取的，但它们可能相互矛盾，并且据我们所知，很少有研究关注如何联合优化它们。本文提出一个名为“双准则子模最大化”（BSM）的新问题，以在效用和公平之间取得平衡。具体而言，它要求找到一个固定规模的解以最大化效用函数，同时要求公平函数的值不低于某个阈值。由于BSM问题通常无法在任意常数因子内近似，我们将注意力转向设计实例依赖的近似方案。我们的算法方案包括两种方法，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。利用真实和合成数据集，我们在三个子模最大化问题中展示了方法的应用：最大覆盖、影响力最大化和设施选址。