34 research outputs found
Quadratic diameter bounds for dual network flow polyhedra
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 -flows
on directed graphs and prove quadratic upper bounds for both of them:
the minimum of and for the combinatorial
diameter, and 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
Edges vs Circuits: a Hierarchy of Diameters in Polyhedra
The study of the graph diameter of polytopes is a classical open problem in
polyhedral geometry and the theory of linear optimization. In this paper we
continue the investigation initiated in [4] by introducing a vast hierarchy of
generalizations to the notion of graph diameter. This hierarchy provides some
interesting lower bounds for the usual graph diameter. After explaining the
structure of the hierarchy and discussing these bounds, we focus on clearly
explaining the differences and similarities among the many diameter notions of
our hierarchy. Finally, we fully characterize the hierarchy in dimension two.
It collapses into fewer categories, for which we exhibit the ranges of values
that can be realized as diameters