On The Circuit Diameters of Some Combinatorial Polytopes
Abstract
The combinatorial diameter of a polytope P is the maximum value of a shortest path between two vertices of P, where the path uses the edges of P only. In contrast to the combinatorial diameter, the circuit diameter of P is defined as the maximum value of a shortest path between two vertices of P, where the path uses potential edge directions of P i.e., all edge directions that can arise by translating some of the facets of P .
In this thesis, we study the circuit diameter of polytopes corresponding to classical combinatorial optimization problems, such as the Matching polytope, the Traveling Sales- man polytope and the Fractional Stable Set polytope. We also introduce the notion of the circuit diameter of a formulation of a polytope P. In this setting the circuits are determined from some external linear system describing P which may not be minimal with respect to its constraints. We use this notion to generalize other results of this thesis, as well as introduce new results about a formulation of the Spanning Tree polytope and a formulation of the Matroid polytope.
Collections
Cite this version of the work
Sean Kafer
(2017).
On The Circuit Diameters of Some Combinatorial Polytopes. UWSpace.
http://hdl.handle.net/10012/12413
Other formats
Related items
Showing items related by title, author, creator and subject.
-
Ehrhart Theory and Unimodular Decompositions of Lattice Polytopes
Tam, Ricci Yik Chi (University of Waterloo, 2015-01-20)Ehrhart theory studies the behaviour of lattice points contained in dilates of lattice polytopes. We provide an introduction to Ehrhart theory. In particular, we prove Ehrhart's Theorem, Stanley Non-negativity, and ... -
Entropy and Graphs
Changiz Rezaei, Seyed Saeed (University of Waterloo, 2014-01-23)The entropy of a graph is a functional depending both on the graph itself and on a probability distribution on its vertex set. This graph functional originated from the problem of source coding in information theory and ... -
Convex Algebraic Geometry Approaches to Graph Coloring and Stable Set Problems
Romero Barbosa, Julian Ariel (University of Waterloo, 2021-08-23)The objective of a combinatorial optimization problem is to find an element that maximizes a given function defined over a large and possibly high-dimensional finite set. It is often the case that the set is so large that ...