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 (β£Vβ£β1)β β£Eβ£ and 61ββ£Vβ£3 for the combinatorial
diameter, and 2β£Vβ£β (β£Vβ£β1)β 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 34ββ£Vβ£β4