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 $(k+1)\ln2<0.694k+0.694$, 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，而当前最优算法仅将近似比改进至$k$。我们给出$\frac{2k\ln2}{1+\ln2}+O(\sqrt{k})<0.819k+O(\sqrt{k})$的近似比。这是首个对一般$k$在贪心算法近似比上实现乘法改进的结果。我们进一步证明，该算法可应用于目标函数不保证单调性的更一般问题，并获得大致相同的近似比（此时近似比中的次线性项为$O(k^{2/3})$而非$O(\sqrt{k})$）。所有结果在$k$-拟阵交集约束替换为更一般的拟阵$k$-对偶约束时仍然成立。此外，与先前多项工作不同，我们的算法运行时间与$k$无关，且关于基集规模呈多项式级。我们的算法基于Singer和Thiery（STOC 2025）针对加权拟阵$k$-交问题（我们所研究问题的特例）最新提出的混合贪心局部搜索方法。由于子模函数$f$的边际值（对应加权情形中的权重）与算法内部随机性相关，在子模场景中运用该方法需要多项非平凡洞见与算法改进。在Singer和Thiery研究的加权特例中，我们的算法可简化为其算法变体，近似比提升至$(k+1)\ln2<0.694k+0.694，而Singer和Thiery保证的近似比为$\frac{k+1}{2\ln2}\approx0.722k+0.722$。