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 $\frac{ 4}{ 3}|V| - 4$.
Categories: math.OC math.CO