Reachability

The matroid intersection problem is a fundamental problem that has been extensively studied for half a century. In the classic version of this problem, we are given two matroids ${\cal M}_1 = (V, {\cal I}_1)$ and ${\cal M}_2 = (V, {\cal I}_2)$ on a comment ground set $V$ of $n$ elements, and then we have to find the largest common independent set $S \in {\cal I}_1 \cap {\cal I}_2$ by making independence oracle queries of the form ‘‘Is $S \in {\cal I}_1$?