Abstract
Both the combinatorial and the circuit diameter 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 $$G=(V,E)$$G=(V,E) and prove quadratic upper bounds for both of them: the minimum of $$(|V|-1)\cdot |E|$$(|V|-1)·|E| and $$\frac{1}{6}|V|^3$$16|V|3 for the combinatorial diameter, and $$\frac{|V|\cdot (|V|-1)}{2}$$|V|·(|V|-1)2 for the circuit diameter. 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.
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