For constrained, not necessarily monotone submodular maximization, guiding the measured continuous greedy algorithm with a local search algorithm currently obtains the state-of-the-art approximation factor of 0.401 \citep{buchbinder2023constrained}. These algorithms rely upon the multilinear extension and the Lovasz extension of a submodular set function. However, the state-of-the-art approximation factor of combinatorial algorithms has remained $1/e \approx 0.367$ \citep{buchbinder2014submodular}. In this work, we develop combinatorial analogues of the guided measured continuous greedy algorithm and obtain approximation ratio of $0.385$ in $\oh{ kn }$ queries to the submodular set function for size constraint, and $0.305$ for a general matroid constraint. Further, we derandomize these algorithms, maintaining the same ratio and asymptotic time complexity. Finally, we develop a deterministic, nearly linear time algorithm with ratio $0.377$.

翻译：针对受约束的（不一定单调的）子模最大化问题，通过局部搜索算法引导的测量连续贪婪算法目前实现了0.401的最新近似因子 \citep{buchbinder2023constrained}。这些算法依赖于子模集函数的多线性扩展和Lovász扩展。然而，组合算法的当前最优近似因子仍为$1/e \approx 0.367$ \citep{buchbinder2014submodular}。本文开发了引导式测量连续贪婪算法的组合对应算法，并在$\oh{kn}$次子模集函数查询下，针对大小约束获得了0.385的近似比，针对一般拟阵约束获得了0.305的近似比。进一步地，我们对这些算法进行去随机化，保持相同近似比和渐近时间复杂度。最后，我们开发了一种确定性近线性时间算法，其近似比为0.377。