  Mixed-integer mathematical programs are among the most commonly used models
for a wide set of problems in Operations Research and related fields. However,
there is still very little known about what can be expressed by small
mixed-integer programs. In particular, prior to this work, it was open whether
some classical problems, like the minimum odd-cut problem, can be expressed by
a compact mixed-integer program with few (even constantly many) integer
variables. This is in stark contrast to linear formulations, where recent
breakthroughs in the field of extended formulations have shown that many
polytopes associated to classical combinatorial optimization problems do not
even admit approximate extended formulations of sub-exponential size.
  We provide a general framework for lifting inapproximability results of
extended formulations to the setting of mixed-integer extended formulations,
and obtain almost tight lower bounds on the number of integer variables needed
to describe a variety of classical combinatorial optimization problems. Among
the implications we obtain, we show that any mixed-integer extended formulation
of sub-exponential size for the matching polytope, cut polytope, traveling
salesman polytope or dominant of the odd-cut polytope, needs $ \Omega(n/\log n)
$ many integer variables, where $ n $ is the number of vertices of the
underlying graph. Conversely, the above-mentioned polyhedra admit
polynomial-size mixed-integer formulations with only $ O(n) $ or $ O(n \log n)
$ (for the traveling salesman polytope) many integer variables.
  Our results build upon a new decomposition technique that, for any convex set
$ C $, allows for approximating any mixed-integer description of $ C $ by the
intersection of $ C $ with the union of a small number of affine subspaces.
