In this paper, we consider the optimization problem Submodular Cover (SCP), which is to find a minimum cardinality subset of a finite universe $U$ such that the value of a submodular function $f$ is above an input threshold $\tau$. In particular, we consider several variants of SCP including the general case, the case where $f$ is additionally assumed to be monotone, and finally the case where $f$ is a regularized monotone submodular function. Our most significant contributions are that: (i) We propose a scalable algorithm for monotone SCP that achieves nearly the same approximation guarantees as the standard greedy algorithm in significantly faster time; (ii) We are the first to develop an algorithm for general SCP that achieves a solution arbitrarily close to being feasible; and finally (iii) we are the first to develop algorithms for regularized SCP. Our algorithms are then demonstrated to be effective in an extensive experimental section on data summarization and graph cut, two applications of SCP.

翻译：本文研究优化问题子模覆盖(SCP)，即寻找有限宇宙集$U$的最小基数子集，使得子模函数$f$的值超过输入阈值$\tau$。具体而言，我们考虑SCP的若干变体，包括一般情形、额外假设$f$为单调函数的情形，以及$f$为正则化单调子模函数的情形。我们的主要贡献在于：(i) 针对单调SCP提出一种可扩展算法，该算法在显著缩短运行时间的同时，能达到与标准贪心算法几乎相同的近似保证；(ii) 我们首次为一般SCP设计出能任意接近可行解的算法；(iii) 我们首次为正则化SCP开发出算法。在数据摘要和图割两个SCP应用场景的广泛实验中，我们验证了所提算法的有效性。