Submodular maximization with a cardinality constraint can model various problems, and those problems are often very large in practice. For the case where objective functions are monotone, many fast approximation algorithms have been developed. The stochastic greedy algorithm (SG) is one such algorithm, which is widely used thanks to its simplicity, efficiency, and high empirical performance. However, its approximation guarantee has been proved only for monotone objective functions. When it comes to non-monotone objective functions, existing approximation algorithms are inefficient relative to the fast algorithms developed for the case of monotone objectives. In this paper, we prove that SG (with slight modification) can achieve almost $1/4$-approximation guarantees in expectation in linear time even if objective functions are non-monotone. Our result provides a constant-factor approximation algorithm with the fewest oracle queries for non-monotone submodular maximization with a cardinality constraint. Experiments validate the performance of (modified) SG.

中文翻译：

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具有基数约束的非单调子模最大化的随机贪婪算法的保证

具有基数约束的子模最大化可以对各种问题进行建模，而这些问题在实践中通常非常大。对于目标函数是单调的情况，已经开发了许多快速逼近算法。随机贪婪算法（SG）就是这样一种算法，由于其简单、高效和高经验性能而被广泛使用。然而，它的逼近保证仅在单调目标函数中被证明。当涉及非单调目标函数时，相对于为单调目标开发的快速算法，现有的近似算法效率低下。在本文中，我们证明即使目标函数是非单调的，SG（稍作修改）也可以在线性时间内实现几乎 $1/4$-近似保证。我们的结果为具有基数约束的非单调子模最大化提供了一种具有最少 oracle 查询的常数因子近似算法。实验验证了（修改后的）SG 的性能。