  One of the most intriguing unsolved questions of matroid optimization is the
characterization of the existence of $k$ disjoint common bases of two matroids.
The significance of the problem is well-illustrated by the long list of
conjectures that can be formulated as special cases, such as Woodall's
conjecture on packing disjoint dijoins in a directed graph, or Rota's beautiful
conjecture on rearrangements of bases.
  In the present paper we prove that the problem is difficult under the rank
oracle model, i.e., we show that there is no algorithm which decides if the
common ground set of two matroids can be partitioned into $k$ common bases by
using a polynomial number of independence queries. Our complexity result holds
even for the very special case when $k=2$.
  Through a series of reductions, we also show that the abstract problem of
packing common bases in two matroids includes the NAE-SAT problem and the
Perfect Even Factor problem in directed graphs. These results in turn imply
that the problem is not only difficult in the independence oracle model but
also includes NP-complete special cases already when $k=2$, one of the matroids
is a partition matroid, while the other matroid is linear and is given by an
explicit representation.
