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Learning Lagrangian Multipliers for the Travelling Salesman Problem

Authors Augustin Parjadis, Quentin Cappart , Bistra Dilkina , Aaron Ferber , Louis-Martin Rousseau

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Author Details

Augustin Parjadis
  • Polytechnique Montréal, Canada
Quentin Cappart
  • Polytechnique Montréal, Canada
Bistra Dilkina
  • Center for Artificial Intelligence in Society, University of Southern California, Los Angeles, CA, USA
Aaron Ferber
  • Center for Artificial Intelligence in Society, University of Southern California, Los Angeles, CA, USA
Louis-Martin Rousseau
  • Polytechnique Montréal, Canada

Acknowledgements

We sincerely thank the anonymous reviewers for their constructive feedback. Their comments helped us better position our contribution within the field. Furthermore, their insights have provided guidance for our future research directions.

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Augustin Parjadis, Quentin Cappart, Bistra Dilkina, Aaron Ferber, and Louis-Martin Rousseau. Learning Lagrangian Multipliers for the Travelling Salesman Problem. In 30th International Conference on Principles and Practice of Constraint Programming (CP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 307, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CP.2024.22

Abstract

Lagrangian relaxation is a versatile mathematical technique employed to relax constraints in an optimization problem, enabling the generation of dual bounds to prove the optimality of feasible solutions and the design of efficient propagators in constraint programming (such as the weighted circuit constraint). However, the conventional process of deriving Lagrangian multipliers (e.g., using subgradient methods) is often computationally intensive, limiting its practicality for large-scale or time-sensitive problems. To address this challenge, we propose an innovative unsupervised learning approach that harnesses the capabilities of graph neural networks to exploit the problem structure, aiming to generate accurate Lagrangian multipliers efficiently. We apply this technique to the well-known Held-Karp Lagrangian relaxation for the traveling salesman problem. The core idea is to predict accurate Lagrangian multipliers and to employ them as a warm start for generating Held-Karp relaxation bounds. These bounds are subsequently utilized to enhance the filtering process carried out by branch-and-bound algorithms. In contrast to much of the existing literature, which primarily focuses on finding feasible solutions, our approach operates on the dual side, demonstrating that learning can also accelerate the proof of optimality. We conduct experiments across various distributions of the metric traveling salesman problem, considering instances with up to 200 cities. The results illustrate that our approach can improve the filtering level of the weighted circuit global constraint, reduce the optimality gap by a factor two for unsolved instances up to a timeout, and reduce the execution time for solved instances by 10%.

Subject Classification

ACM Subject Classification
  • Computing methodologies → Artificial intelligence
  • Theory of computation → Constraint and logic programming
  • Computing methodologies → Machine learning
Keywords
  • Lagrangian relaxation
  • unsupervised learning
  • graph neural network

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