  Matroid intersection is one of the most powerful frameworks of matroid theory
that generalizes various problems in combinatorial optimization. Edmonds'
fundamental theorem provides a min-max characterization for the unweighted
setting, while Frank's weight-splitting theorem provides one for the weighted
case. Several efficient algorithms were developed for these problems, all
relying on the usage of one of the conventional oracles for both matroids.
  In the present paper, we consider the tractability of the matroid
intersection problem under restricted oracles. In particular, we focus on the
rank sum, common independence, and maximum rank oracles. We give a strongly
polynomial-time algorithm for weighted matroid intersection under the rank sum
oracle. In the common independence oracle model, we prove that the unweighted
matroid intersection problem is tractable when one of the matroids is a
partition matroid, and that even the weighted case is solvable when one of the
matroids is an elementary split matroid. Finally, we show that the common
independence and maximum rank oracles together are strong enough to realize the
steps of our algorithm under the rank sum oracle.
