In this paper, we introduce the concept of Density-Balanced Subset in a matroid, in which independent sets can be sampled so as to guarantee that (i) each element has the same probability to be sampled, and (ii) those events are negatively correlated. These Density-Balanced Subsets are subsets in the ground set of a matroid in which the traditional notion of uniform random sampling can be extended. We then provide an application of this concept to the Matroid-Constrained Maximum Coverage problem. In this problem, given a matroid $\mathcal{M} = (V, \mathcal{I})$ of rank $k$ on a ground set $V$ and a coverage function $f$ on $V$, the goal is to find an independent set $S \in \mathcal{I}$ maximizing $f(S)$. This problem is an important special case of the much-studied submodular function maximization problem subject to a matroid constraint; this is also a generalization of the maximum $k$-cover problem in a graph. In this paper, assuming that the coverage function has a bounded frequency $\mu$ (i.e., any element of the underlying universe of the coverage function appears in at most $\mu$ sets), we design a procedure, parameterized by some integer $\rho$, to extract in polynomial time an approximate kernel of size $\rho \cdot k$ that is guaranteed to contain a $1 - (\mu - 1)/\rho$ approximation of the optimal solution. This procedure can then be used to get a Fixed-Parameter Tractable Approximation Scheme (FPT-AS) providing a $1 - \varepsilon$ approximation in time $(\mu/\varepsilon)^{O(k)} \cdot |V|^{O(1)}$. This generalizes and improves the results of [Manurangsi, 2019] and [Huang and Sellier, 2022], providing the first FPT-AS working on an arbitrary matroid. Moreover, because of its simplicity, the kernel construction can be performed in the streaming setting.

翻译：本文引入拟阵中密度平衡子集的概念，使得独立集可被采样，同时保证：(i) 每个元素被采样的概率相同，(ii) 这些事件呈负相关。密度平衡子集是拟阵基集上的子集，传统均匀随机采样的概念可在此推广。我们随后将该概念应用于拟阵约束的最大覆盖问题。该问题中，给定秩为k的拟阵$\mathcal{M} = (V, \mathcal{I})$（基集为V）与覆盖函数f，目标是在$\mathcal{I}$中寻找最大化f(S)的独立集S。该问题是受广泛研究的子模函数最大化问题在拟阵约束下的重要特例，也是图上最大k-覆盖问题的推广。本文中，假设覆盖函数具有有界频率μ（即覆盖函数底层全集中的任意元素至多出现在μ个集合中），我们设计了一个参数化为整数ρ的算法，可在多项式时间内提取大小为ρ·k的近似核，该核保证包含最优解的1 - (μ - 1)/ρ近似解。该算法进而可转化为固定参数可追踪近似方案（FPT-AS），在$(\mu/\varepsilon)^{O(k)} \cdot |V|^{O(1)}$时间内提供1 - ε近似解。本研究推广并改进了[Manurangsi, 2019]和[Huang and Sellier, 2022]的结果，首次在任意拟阵上实现了FPT-AS。此外，由于核构造的简洁性，该过程可在流式场景中执行。