Submodular optimization has been well studied in combinatorial optimization. However, there are few works considering about non-submodular optimization problems which also have many applications, such as experimental design, some optimization problems in social networks, etc. In this paper, we consider the maximization of non-submodular function under a knapsack constraint, and explore the performance of the greedy algorithm, which is characterized by the submodularity ratio \(\beta \) and curvature \(\alpha \). In particular, we prove that the greedy algorithm enjoys a tight approximation guarantee of \( (1-e^{-\alpha \beta })/{\alpha }\) for the above problem. To our knowledge, it is the first tight constant factor for this problem. We further utilize illustrative examples to demonstrate the performance of our algorithm.

在组合优化中已经对亚模优化进行了深入研究。但是，关于非次模块优化问题的研究很少，也有很多应用，例如实验设计，社交网络中的一些优化问题等。本文考虑了背包约束下非次模块函数的最大化。 ，并探索以子模数比\（\ beta \）和曲率\（\ alpha \）为特征的贪婪算法的性能。特别是，我们证明了贪心算法具有\（（1-e ^ {-\ alpha \ beta}）/ {\ alpha} \）的严格逼近保证对于上述问题。据我们所知，这是此问题的第一个严格的恒定因素。我们进一步利用说明性示例来证明我们算法的性能。