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

2025-06-12

AI Summary

This work investigates the integrality gap of the Assignment Subtour Elimination Polytope (ASEP) linear relaxation for small-scale asymmetric traveling salesman problems (ATSP), specifically for instances with $n leq 22$ nodes. Method: Leveraging geometric and symmetry properties of the ASEP polytope, we integrate symmetry-aware pruning into vertex enumeration, devise a simplex-pivoting-based heuristic, and introduce a constructive extension procedure to generate vertices of the $(n+1)$-node ASEP from those of the $n$-node ASEP. Contribution/Results: Through extensive computational experiments, we substantially improve the best-known lower bounds on the integrality gap for $n = 16$–$22$. Our approach yields a new set of computationally challenging small-scale ATSP benchmark instances—the hardest known to date for this size range—thereby establishing a novel paradigm for analyzing TSP relaxation strength and systematically constructing hard instances.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• Study integrality gap in small ATSP cases
• Analyze ASEP polytope geometry and symmetries
• Develop heuristic algorithm to find maximum gap

Innovation

• Exploiting ASEP polytope symmetries for heuristic algorithm
• Extending vertices from P^n_ASEP to P^{n+1}_ASEP
• Improving integrality gap bounds for small ATSP instances