Recent works on cost based relaxations have improved Constraint Programming (CP) models for the Traveling Salesman Problem (TSP). We provide a short survey over solving asymmetric TSP with CP. Then, we suggest new implied propagators based on general graph properties. We experimentally show that such implied propagators bring robustness to pathological instances and highlight the fact that graph structure can significantly improve search heuristics behavior. Finally, we show that our approach outperforms current state of the art results.
An improvement of the MST relaxation is the approach of Held and Karp @cite_19 , adapted for CP by @cite_3 . It is the Lagrangian MST relaxation with a policy for updating Langrangian multipliers that provides a fast convergence. The idea of this method is to iteratively compute MST that converge towards a path by adding penalties on arc costs according to degree constraints violations. It must be noticed that since arc costs change from one iteration to another, Prim's algorithm is better than Kruskal's which requires to sort edges. Moreover, to our knowledge neither algorithm can be applied incrementally.
@cite_19 @cite_3
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The relationship between the symmetric traveling-salesman problem and the minimum spanning tree problem yields a sharp lower bound on the cost of an optimum tour. An efficient iterative method for approximating this bound closely from below is presented. A branch-and-bound procedure based upon these considerations has easily produced proven optimum solutions to all traveling-salesman problems presented to it, ranging in size up to sixty-four cities. The bounds used are so sharp that the search trees are minuscule compared to those normally encountered in combinatorial problems of this type. So far, edge-finding is the only one major filtering algorithm for unary resource constraint with time complexity O(nlog n). This paper proposes O(nlog n) versions of another two filtering algorithms: not-first not-last and propagation of detectable precedences. These two algorithms can be used together with the edge-finding to further improve the filtering. This paper also propose new O(nlog n) implementation of fail detection (overload checking).