We study the non-submodular maximization problem, whose objective function can be expressed as the Difference between two Set (DS) functions or the Ratio between two Set (RS) functions. For the cardinality-constrained and unconstrained DS maximization problems, we present several deterministic algorithms and our analysis shows that the algorithms can provide provable approximation guarantees. As an application, we manage to derive an improved approximation bound for the DS minimization problem under certain conditions compared with existing results. As for the RS maximization problem, we show that there exists a polynomial-time reduction from the approximation of RS maximization to the approximation of DS maximization. Based on this reduction, we derive the first approximation bound for the cardinality-constrained RS maximization problem. We also devise algorithms for the unconstrained problem and analyze their approximation guarantees. By applying our results to the problem of optimizing the ratio between two supermodular functions, we give an answer to the question posed by Bai et al. (in Proceedings of The 33rd international conference on machine learning (ICML), 2016). Moreover, we give an example to illustrate that whether the set function is normalized has an effect on the approximability of the RS optimization problem.

我们研究了非子模最大化问题，其目标函数可以表示为两个集合 （DS） 函数之间的差值或两个集合 （RS） 函数之间的比率。对于基数约束和无约束的 DS 最大化问题，我们提出了几种确定性算法，我们的分析表明这些算法可以提供可证明的近似保证。作为一个应用程序，与现有结果相比，我们设法在特定条件下推导出 DS 最小化问题的改进近似边界。至于 RS 最大化问题，我们表明从 RS 最大化的近似到 DS 最大化的近似存在多项式时间减少。基于这个约简，我们推导出了基数约束的 RS 最大化问题的第一个近似边界。我们还为无约束问题设计了算法并分析了它们的近似保证。通过将我们的结果应用于优化两个超模函数之间比率的问题，我们回答了 Bai 等人提出的问题（在第 33 届机器学习国际会议 （ICML） 的会议记录中，2016 年）。此外，我们举了一个例子来说明集合函数是否归一化对 RS 优化问题的近似性有影响。