In many submodular optimization applications, datasets are naturally partitioned into disjoint subsets. These scenarios give rise to submodular optimization problems with partition-based constraints, where the desired solution set should be in some sense balanced, fair, or resource-constrained across these partitions. While existing work on submodular cover largely overlooks this structure, we initiate a comprehensive study of the problem of Submodular Cover with Partition Constraints (SCP) and its key variants. Our main contributions are the development and analysis of scalable bicriteria approximation algorithms for these NP-hard optimization problems for both monotone and nonmonotone objectives. Notably, the algorithms proposed for the monotone case achieve optimal approximation guarantees while significantly reducing query complexity compared to existing methods. Finally, empirical evaluations on real-world and synthetic datasets further validate the efficiency and effectiveness of the proposed algorithms.

翻译：在许多子模优化应用中，数据集天然被划分为互不相交的子集。这类场景催生了具有基于划分约束的子模优化问题，其中期望的解集应在某种意义下在这些划分间保持平衡、公平或受资源限制。尽管现有关于子模覆盖的研究大多忽视了这一结构，我们首次对具有划分约束的子模覆盖问题及其关键变体展开了系统性研究。我们的主要贡献是为这些NP难优化问题（包括单调与非单调目标函数）开发并分析了可扩展的双准则近似算法。值得注意的是，针对单调情形所提出的算法在实现最优近似保证的同时，显著降低了与现有方法相比的查询复杂度。最后，在真实世界与合成数据集上的实证评估进一步验证了所提出算法的高效性与有效性。