41 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
A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins
The main result of this paper is a bijective proof showing that the
generating function for partitions with bounded differences between largest and
smallest part is a rational function. This result is similar to the closely
related case of partitions with fixed differences between largest and smallest
parts which has recently been studied through analytic methods by Andrews,
Beck, and Robbins. Our approach is geometric: We model partitions with bounded
differences as lattice points in an infinite union of polyhedral cones.
Surprisingly, this infinite union tiles a single simplicial cone. This
construction then leads to a bijection that can be interpreted on a purely
combinatorial level.Comment: 12 pages, 5 figure
Geometric combinatorics and computational molecular biology: branching polytopes for RNA sequences
Questions in computational molecular biology generate various discrete
optimization problems, such as DNA sequence alignment and RNA secondary
structure prediction. However, the optimal solutions are fundamentally
dependent on the parameters used in the objective functions. The goal of a
parametric analysis is to elucidate such dependencies, especially as they
pertain to the accuracy and robustness of the optimal solutions. Techniques
from geometric combinatorics, including polytopes and their normal fans, have
been used previously to give parametric analyses of simple models for DNA
sequence alignment and RNA branching configurations. Here, we present a new
computational framework, and proof-of-principle results, which give the first
complete parametric analysis of the branching portion of the nearest neighbor
thermodynamic model for secondary structure prediction for real RNA sequences.Comment: 17 pages, 8 figure