  Both the combinatorial and the circuit diameters of polyhedra are of interest
to the theory of linear programming for their intimate connection to a
best-case performance of linear programming algorithms.
  We study the diameters of dual network flow polyhedra associated to $b$-flows
on directed graphs $G=(V,E)$ and prove quadratic upper bounds for both of them:
the minimum of $(|V|-1)\cdot |E|$ and $\frac{1}{6}|V|^3$ for the combinatorial
diameter, and $\frac{|V|\cdot (|V|-1)}{2}$ for the circuit diameter. The latter
strengthens the cubic bound implied by a result in [De Loera, Hemmecke, Lee;
2014].
  Previously, bounds on these diameters have only been known for bipartite
graphs. The situation is much more involved for general graphs. In particular,
we construct a family of dual network flow polyhedra with members that violate
the circuit diameter bound for bipartite graphs by an arbitrary additive
constant. Further, it provides examples of circuit diameter $\frac{4}{3}|V| -
4$.
