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.