The problem of maximizing a given set function with a cardinality constraint has widespread applications. A number of algorithms have been provided to solve the maximization problem when the set function is monotone and submodular. However, reality-based set functions may not be submodular and may involve large-scale and noisy data sets. In this paper, we present the Stochastic-Lazier-Greedy Algorithm (SLG) to solve the corresponding non-submodular maximization problem and offer a performance guarantee of the algorithm. The guarantee is related to a submodularity ratio, which characterizes the closeness of to submodularity. Our algorithm also can be viewed as an extension of several previous greedy algorithms.

中文翻译：

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单调非子模最大化的随机懒惰贪婪算法

最大化具有基数约束的给定集合函数的问题具有广泛的应用。已经提供了许多算法来解决当集合函数是单调和子模时的最大化问题。然而，基于现实的集合函数可能不是子模块的，并且可能涉及大规模和嘈杂的数据集。在本文中，我们提出了 Stochastic-Lazier-Greedy Algorithm (SLG) 来解决相应的非子模最大化问题，并为算法提供性能保证。保证与子模块性比率有关，它表征与子模块性的接近程度。我们的算法也可以看作是之前几个贪心算法的扩展。