Matroid intersection is a classical optimization problem where, given two matroids over the same ground set, the goal is to find the largest common independent set. In this paper, we show that there exists a certain "sparsifer": a subset of elements, of size $O(|S^{opt}| \cdot 1/\varepsilon)$, where $S^{opt}$ denotes the optimal solution, that is guaranteed to contain a $3/2 + \varepsilon$ approximation, while guaranteeing certain robustness properties. We call such a small subset a Density Constrained Subset (DCS), which is inspired by the Edge-Degree Constrained Subgraph (EDCS) [Bernstein and Stein, 2015], originally designed for the maximum cardinality matching problem in a graph. Our proof is constructive and hinges on a greedy decomposition of matroids, which we call the density-based decomposition. We show that this sparsifier has certain robustness properties that can be used in one-way communication and random-order streaming models.

翻译：拟阵交是一个经典优化问题，给定同一基集上的两个拟阵，目标是找到最大的公共独立集。本文证明存在一种特定的“稀疏化器”：一个大小为$O(|S^{opt}| \cdot 1/\varepsilon)$的元素子集（其中$S^{opt}$表示最优解），该子集保证包含一个$3/2 + \varepsilon$近似解，同时满足特定的鲁棒性性质。我们将这种小子集称为密度约束子集（DCS），其灵感来源于最初为图中最大基数匹配问题设计的边度约束子图（EDCS）[Bernstein and Stein, 2015]。我们的证明是构造性的，并依赖于一种称为密度分解的拟阵贪心分解方法。我们证明这种稀疏化器具有特定的鲁棒性性质，可用于单向通信和随机顺序流模型。