Quadratic diameter bounds for dual network flow polyhedra

TL;DR

This paper establishes quadratic upper bounds for the combinatorial and circuit diameters of dual network flow polyhedra, extending previous results beyond bipartite graphs and providing new counterexamples.

Contribution

It introduces quadratic bounds for diameters of dual network flow polyhedra on general graphs, surpassing prior bipartite-only results, and constructs counterexamples for circuit diameter bounds.

Findings

Quadratic upper bounds for combinatorial diameter: min((|V|-1)*|E|, |V|^3/6)

Quadratic upper bound for circuit diameter: |V|*(|V|-1)/2

Counterexamples violating bipartite circuit diameter bounds

Abstract

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 $\frac{4}{3}|V| - 4$.