Tolerance-based Branch and Bound algorithms for the ATSP
The selection of entries to be included/excluded in Branch and Bound algorithms is usually done on the basis of cost values. We consider the class of Depth First Search algorithms, and we propose to use upper tolerances to guide the search for optimal solutions. In spite of the fact that it needs time to calculate tolerances, our computational experiments for Asymmetric Traveling Salesman Problems show that in most situations tolerance-based algorithms outperform cost-based algorithms. The solution time reductions are mainly caused by the fact that the branching process becomes much more effective, so that optimal solutions are found in an earlier stage of the branching process. The use of tolerances also reveals why the widely used choice for branching on a smallest cycle in assignment solutions is on average the most effective one. Moreover, it turns out that tolerance-based DFS algorithms are better in solving difficult instances than the Best First Search algorithm from Carpaneto et al. [Carpaneto, G., Dell'Amico, M., Toth, P., 1995. Exact solution of large-scale asymmetric traveling salesman problems. ACM Transactions on Mathematical Software 21 (4), 394-409].
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- Lawrence Hubert & Phipps Arabie, 1985. "Comparing partitions," Journal of Classification, Springer;The Classification Society, vol. 2(1), pages 193-218, December.
- Giorgio Carpaneto & Paolo Toth, 1980. "Some New Branching and Bounding Criteria for the Asymmetric Travelling Salesman Problem," Management Science, INFORMS, vol. 26(7), pages 736-743, July.
- Gessner, Guy & Malhotra, Naresh K. & Kamakura, Wagner A. & Zmijewski, Mark E., 1988. "Estimating models with binary dependent variables: Some theoretical and empirical observations," Journal of Business Research, Elsevier, vol. 16(1), pages 49-65, January.
- Helsgaun, Keld, 2000. "An effective implementation of the Lin-Kernighan traveling salesman heuristic," European Journal of Operational Research, Elsevier, vol. 126(1), pages 106-130, October.
- Lin, Chi-Jen & Wen, Ue-Pyng, 2003. "Sensitivity analysis of the optimal assignment," European Journal of Operational Research, Elsevier, vol. 149(1), pages 35-46, August.
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