In recent years, with the more and more researchers studying the problem of maximizing monotone (nonsubmodular) objective functions, the approximation algorithms for this problem have gotten much progress by using some interesting techniques. In this paper, we develop the approximation algorithms for maximizing a monotone function f with generic submodularity ratio \(\gamma \) subject to certain constraints. Our first result is a simple algorithm that gives a \((1-e^{-\gamma } -\epsilon )\)-approximation for a cardinality constraint using \(O(\frac{n}{\epsilon }log\frac{n}{\epsilon })\) queries to the function value oracle. The second result is a new variant of the continuous greedy algorithm for a matroid constraint. We combine the variant of continuous greedy method with the contention resolution schemes to find a solution with approximation ratio \((\gamma ^2(1-\frac{1}{e})^2-O(\epsilon ))\), and the algorithm makes \(O(rn\epsilon ^{-4}log^2\frac{n}{\epsilon })\) queries to the function value oracle.

近年来，随着越来越多的研究人员研究最大化单调（非子模）目标函数的问题，通过使用一些有趣的技术，针对该问题的逼近算法已取得了很大进展。在本文中，我们开发了一种近似算法，用于在一定约束条件下最大化具有通用子模数比\（\ gamma \）的单调函数f。我们的第一个结果是一种简单的算法给出了一个\（（1-E ^ { - \伽玛} - \小量）\） -近似用于使用基数约束\（O（\压裂{N} {\小量}日志\ frac {n} {\ epsilon}）\）查询函数值oracle。第二个结果是针对拟阵线约束的连续贪婪算法的新变体。我们将连续贪婪方法的变体与竞争解决方案相结合，以找到具有近似比率\（（\ gamma ^ 2（1- \ frac {1} {e}）^ 2-O（\ epsilon））\）的解决方案，并且该算法对函数值oracle进行\（O（rn \ epsilon ^ {-4} log ^ 2 \ frac {n} {\ epsilon}）\）查询。