On the Circuit Diameter of some Combinatorial Polytopes

TL;DR

This paper investigates the circuit diameter of polytopes associated with classical combinatorial optimization problems, providing insights into their geometric properties and potential implications for optimization algorithms.

Contribution

It introduces the concept of circuit diameter and analyzes it for polytopes like the Matching, Traveling Salesman, and Fractional Stable Set polytopes.

Findings

Circuit diameter differs from combinatorial diameter in certain polytopes.

Provides bounds or exact values for circuit diameters of studied polytopes.

Enhances understanding of polytope geometry in combinatorial optimization.

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 paper, we study the circuit diameter of polytopes corresponding to classical combinatorial optimization problems, such as the Matching polytope, the Traveling Salesman polytope and the Fractional Stable Set polytope.