# Feasible bases for a polytope related to the Hamilton cycle problem

## TL;DR

This paper investigates the structure of a polytope related to the Hamilton cycle problem, revealing its independence from certain parameters, its interpretation as a network flow polytope, and providing combinatorial insights and computational evidence.

## Contribution

It advances understanding of the feasible bases of a Hamilton cycle-related polytope, including structural properties and a combinatorial interpretation, supporting a conjecture about Hamiltonian cycles.

## Key findings

- Feasible bases are independent of the parameter near 1.
- The polytope is a generalized network flow polytope.
- A combinatorial interpretation of feasible bases is provided.

## Abstract

We study a certain polytope depending on a graph $G$ and a parameter $\beta\in(0,1)$ which arises from embedding the Hamiltonian cycle problem in a discounted Markov decision process. Eshragh \emph{et al.} conjectured a lower bound on the proportion of feasible bases corresponding to Hamiltonian cycles in the set of all feasible bases. We make progress towards a proof of the conjecture by proving results about the structure of feasible bases. In particular, we prove three main results: (1) the set of feasible bases is independent of the parameter $\beta$ when the parameter is close to 1, (2) the polytope can be interpreted as a generalized network flow polytope and (3) we deduce a combinatorial interpretation of the feasible bases. We also provide a full characterization for a special class of feasible bases, and we apply this to provide some computational support for the conjecture.