For constrained, not necessarily monotone submodular maximization, all known approximation algorithms with ratio greater than $1/e$ require continuous ideas, such as queries to the multilinear extension of a submodular function and its gradient, which are typically expensive to simulate with the original set function. For combinatorial algorithms, the best known approximation ratios for both size and matroid constraint are obtained by a simple randomized greedy algorithm of Buchbinder et al. [9]: $1/e \approx 0.367$ for size constraint and $0.281$ for the matroid constraint in $\mathcal O (kn)$ queries, where $k$ is the rank of the matroid. In this work, we develop the first combinatorial algorithms to break the $1/e$ barrier: we obtain approximation ratio of $0.385$ in $\mathcal O (kn)$ queries to the submodular set function for size constraint, and $0.305$ for a general matroid constraint. These are achieved by guiding the randomized greedy algorithm with a fast local search algorithm. Further, we develop deterministic versions of 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：对于带约束且不要求单调性的子模最大化问题，所有已知的、近似比超过1/e的算法均需借助连续化思想，例如查询子模函数的多线性扩展及其梯度，而这类操作通常难以通过原始集函数高效模拟。在组合算法领域，目前针对基数约束和拟阵约束的最佳近似比均由Buchbinder等人[9]提出的简单随机贪心算法实现：在$\mathcal O (kn)$次查询中（其中k为拟阵秩），基数约束下的近似比为$1/e \approx 0.367$，拟阵约束下为$0.281$。本研究首次提出突破1/e界限的组合算法：通过将快速局部搜索算法与随机贪心算法相结合，我们在基数约束下以$\mathcal O (kn)$次子模集函数查询实现了$0.385$的近似比，在一般拟阵约束下实现了$0.305$的近似比。进一步地，我们构建了这些算法的确定性版本，在保持相同近似比的同时维持渐近时间复杂度不变。最后，我们提出了一种确定性、近似线性时间的算法，其近似比达到$0.377$。