Finding diverse solutions to optimization problems has been of practical interest for several decades, and recently enjoyed increasing attention in research. While submodular optimization has been rigorously studied in many fields, its diverse solutions extension has not. In this study, we consider the most basic variants of submodular optimization, and propose two simple greedy algorithms, which are known to be effective at maximizing monotone submodular functions. These are equipped with parameters that control the trade-off between objective and diversity. Our theoretical contribution shows their approximation guarantees in both objective value and diversity, as functions of their respective parameters. Our experimental investigation with maximum vertex coverage instances demonstrates their empirical differences in terms of objective-diversity trade-offs.

翻译：寻找优化问题的多样性解已在实践领域受到数十年的关注，并近年来在研究中获得越来越多的重视。尽管子模优化已在多个领域得到严格研究，但其多样性解的扩展尚未被充分探索。本研究考虑子模优化的最基本变体，提出两种简单的贪心算法，这些算法已知对于最大化单调子模函数是有效的。这些算法配备了控制目标值与多样性之间权衡的参数。我们的理论贡献展示了它们在目标值和多样性方面的近似保证，这些保证是各自参数的函数。针对最大顶点覆盖实例的实验研究揭示了它们在目标-多样性权衡方面的经验差异。