On the integrality Gap of Small Asymmetric Traveling Salesman Problems: A Polyhedral and Computational Approach

TL;DR

This paper explores the integrality gap of small asymmetric TSP instances by analyzing the ASEP polytope's geometry, designing heuristics to find worst-case vertices, and improving lower bounds for instances with 16 to 22 nodes.

Contribution

It introduces a geometric and computational approach to better understand the ASEP polytope, enhancing lower bounds of the integrality gap for small ATSP instances.

Findings

Improved lower bounds for integrality gap for n=16 to 22.

Identified hard-to-solve small ATSP instances.

Developed a heuristic pivoting algorithm for vertex search.

Abstract

In this paper, we investigate the integrality gap of the Asymmetric Traveling Salesman Problem (ATSP) with respect to the linear relaxation given by the Asymmetric Subtour Elimination Problem (ASEP) for instances with $n$ nodes, where $n$ is small. In particular, we focus on the geometric properties and symmetries of the ASEP polytope ($P^{n}_{ASEP}$) and its vertices. The polytope's symmetries are exploited to design a heuristic pivoting algorithm to search vertices where the integrality gap is maximized. Furthermore, a general procedure for the extension of vertices from $P^{n}_{ASEP}$ to $P^{n + 1}_{ASEP}$ is defined. The generated vertices improve the known lower bounds of the integrality gap for $ 16 \leq n \leq 22$ and, provide small hard-to-solve ATSP instances.