In many data-mining applications, including recommender systems, influence maximization, and team formation, the goal is to pick a subset of elements (e.g., items, nodes in a network, experts to perform a task) to maximize a monotone submodular utility function while simultaneously minimizing a cost function. Classical formulations model this tradeoff via cardinality or knapsack constraints, or by combining utility and cost into a single weighted objective. However, such approaches require committing to a specific tradeoff in advance and return only a single solution, offering limited insight into the space of viable utility-cost tradeoffs. In this paper, we depart from the single-solution paradigm and examine the problem of computing representative sets of high-quality solutions that expose different tradeoffs between submodular utility and cost. For this, we introduce $(α_1,α_2)$-approximate Pareto frontiers that provably approximate the achievable tradeoffs between submodular utility and cost. Specifically, we formalize the Pareto-$\langle f,c \rangle$ problem and develop efficient algorithms for multiple instantiations arising from different combinations of submodular utility $f$ and cost functions $c$. Our results offer a principled and practical framework for understanding and exploiting utility-cost tradeoffs in submodular optimization. Experiments on datasets from diverse application domains demonstrate that our algorithms efficiently compute approximate Pareto frontiers in practice.

翻译：在许多数据挖掘应用中，包括推荐系统、影响力最大化和团队组建，目标是从元素集合（例如物品、网络节点、执行任务的专家）中选择一个子集，以最大化单调子模效用函数，同时最小化成本函数。经典建模方法通过基数或背包约束，或将效用与成本合并为单一加权目标来刻画这种权衡。然而，此类方法需要预先确定具体的权衡方案，且仅返回单一解，难以揭示可行的效用-成本权衡空间。本文突破单一解范式，研究计算高质量解的代表性集合的问题，这些解展现了子模效用与成本之间的不同权衡关系。为此，我们引入$(α_1,α_2)$-近似帕累托前沿，其可证明地逼近子模效用与成本之间的可实现权衡。具体而言，我们形式化定义了Pareto-$\langle f,c \rangle$问题，并针对子模效用函数$f$与成本函数$c$不同组合产生的多种实例开发了高效算法。我们的研究结果为理解和利用子模优化中的效用-成本权衡提供了原则性且实用的框架。在多个应用领域数据集上的实验表明，我们的算法在实践中能高效计算近似帕累托前沿。