The LKH algorithm based on k-opt is an extremely efficient algorithm solving the TSP. Given a non-optimal tour in a graph, the idea of k-opt is to iteratively swap k edges of this tour in order to find a shorter tour. However, the optimality of a tour cannot be proved with this method. In that case, exact solving methods such as CP can be used. The CP model is based on a graph variable with mandatory and optional edges. Through branch-and-bound and filtering algorithms, the set of mandatory edges will be modified. In this paper, we introduce a new constraint to the CP model named mandatory Hamiltonian path constraint searching for k-opt in the mandatory Hamiltonian paths. Experiments have shown that the mandatory Hamiltonian path constraint allows us to gain on average a factor of 3 on the solving time. In addition, we have been able to solve some instances that remain unsolved with the state of the art CP solver with a 1 week time out.
@InProceedings{isoart_et_al:LIPIcs.CP.2021.30, author = {Isoart, Nicolas and R\'{e}gin, Jean-Charles}, title = {{A k-Opt Based Constraint for the TSP}}, booktitle = {27th International Conference on Principles and Practice of Constraint Programming (CP 2021)}, pages = {30:1--30:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-211-2}, ISSN = {1868-8969}, year = {2021}, volume = {210}, editor = {Michel, Laurent D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2021.30}, URN = {urn:nbn:de:0030-drops-153212}, doi = {10.4230/LIPIcs.CP.2021.30}, annote = {Keywords: TSP, k-opt, 1-tree, Constraint} }
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