We study the problem of maximizing a non-negative monotone submodular objective $f$ subject to the intersection of $k$ arbitrary matroid constraints. The natural greedy algorithm guarantees $(k+1)$-approximation for this problem, and the state-of-the-art algorithm only improves this approximation ratio to $k$. We give a $\frac{2k\ln2}{1+\ln2}+O(\sqrt{k})<0.819k+O(\sqrt{k})$ approximation for this problem. Our result is the first multiplicative improvement over the approximation ratio of the greedy algorithm for general $k$. We further show that our algorithm can be used to obtain roughly the same approximation ratio also for the more general problem in which the objective is not guaranteed to be monotone (the sublinear term in the approximation ratio becomes $O(k^{2/3})$ rather than $O(\sqrt{k})$ in this case). All of our results hold also when the $k$-matroid intersection constraint is replaced with a more general matroid $k$-parity constraint. Furthermore, unlike the case in many of the previous works, our algorithms run in time that is independent of $k$ and polynomial in the size of the ground set. Our algorithms are based on a hybrid greedy local search approach recently introduced by Singer and Thiery (STOC 2025) for the weighted matroid $k$-intersection problem, which is a special case of the problem we consider. Leveraging their approach in the submodular setting requires several non-trivial insights and algorithmic modifications since the marginals of a submodular function $f$, which correspond to the weights in the weighted case, are not independent of the algorithm's internal randomness. In the special weighted case studied by Singer and Thiery, our algorithms reduce to a variant of their algorithm with an improved approximation ratio of $k\ln2+1-\ln2<0.694k+0.307$, compared to an approximation ratio of $\frac{k+1}{2\ln2}\approx0.722k+0.722$ guaranteed by Singer and Thiery.

翻译：我们研究在$k$个任意拟阵约束的交集下最大化非负单调子模目标函数$f$的问题。自然贪心算法对该问题保证$(k+1)$-近似，而现有最优算法仅将近似比改进至$k$。我们给出了$\frac{2k\ln2}{1+\ln2}+O(\sqrt{k})<0.819k+O(\sqrt{k})$的近似算法。我们的结果是针对一般$k$值首次在贪心算法近似比基础上实现乘性改进。进一步证明，该算法也可用于目标函数非单调的更一般问题（此时近似比中的次线性项变为$O(k^{2/3})$而非$O(\sqrt{k})$）。所有结果在$k$-拟阵交约束替换为更一般的拟阵$k$-奇偶约束时依然成立。此外，与先前多数研究不同，我们的算法运行时间与$k$无关且关于基础集规模呈多项式级。算法基于Singer与Thiery（STOC 2025）最近针对加权拟阵$k$交问题提出的混合贪心局部搜索方法，该问题是本文所研究问题的特例。由于子模函数$f$的边际收益（对应加权情形中的权重）与算法内部随机性不独立，在子模场景中运用该方法需要若干非平凡洞见与算法修改。在Singer与Thiery研究的加权特例中，我们的算法可退化为其算法的改进变体，近似比提升至$k\ln2+1-\ln2<0.694k+0.307$，优于Singer与Thiery所保证的$\frac{k+1}{2\ln2}\approx0.722k+0.722$近似比。