  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.
