  The toric ideal of a $d$-dimensional flow polytope has an initial ideal
generated by square-free monomials of degree at most $d$. The toric ideal of a
flow polytope of dimension at most four has an initial ideal generated by
square-free monomials of degree at most two, with the only exception of the
four-dimensional Birkhoff polytope, whose toric ideal has an initial ideal
generated by a square-free cubic monomial. The proof is based on a method to
classify certain compressed flow polytopes, and a construction of a quadratic
pulling triangulation of them. Along the way compressed flow polytopes are
classified up to dimension four, and their Ehrhart polynomials are computed.
