In this paper we design a new primal-dual algorithm for the classic discrete optimization problem of maximizing a monotone submodular function subject to a cardinality constraint achieving the optimal approximation of $(1-1/e)$. This problem and its special case, the maximum $k$-coverage problem, have a wide range of applications in various fields including operations research, machine learning, and economics. While greedy algorithms have been known to achieve this approximation factor, our algorithms also provide a dual certificate which upper bounds the optimum value of any instance. This certificate may be used in practice to certify much stronger guarantees than the worst-case $(1-1/e)$ approximation factor.

翻译：本文针对基数约束下最大化单调子模函数的经典离散优化问题，设计了一种新的原始-对偶算法，实现了最优近似比$(1-1/e)$。该问题及其特例——最大$k$-覆盖问题，在运筹学、机器学习及经济学等多个领域具有广泛应用。尽管贪心算法已知能达到该近似因子，但我们的算法还提供了一个对偶证书，可对任意实例的最优值给出上界。在实践中，该证书可确保比最坏情况下的$(1-1/e)$近似因子强得多的性能保证。