Abstract
Held and Karp have proposed, in the early 1970s, a relaxation for the Traveling Salesman Problem (TSP) as well as a branch-and-bound procedure that can solve small to modest-size instances to optimality [4, 5]. It has been shown that the Held-Karp relaxation produces very tight bounds in practice, and this relaxation is therefore applied in TSP solvers such as Concorde [1]. In this short paper we show that the Held-Karp approach can benefit from well-known techniques in Constraint Programming (CP) such as domain filtering and constraint propagation. Namely, we show that filtering algorithms developed for the weighted spanning tree constraint [3, 8] can be adapted to the context of the Held and Karp procedure. In addition to the adaptation of existing algorithms, we introduce a special-purpose filtering algorithm based on the underlying mechanisms used in Prim’s algorithm [7]. Finally, we explored two different branching schemes to close the integrality gap. Our initial experimental results indicate that the addition of the CP techniques to the Held-Karp method can be very effective.
The paper is organized as follows: section 2 describes the Held-Karp approach while section 3 gives some insights on the Constraint Programming techniques and branching scheme used. In section 4 we demonstrate, through preliminary experiments, the impact of using CP in combination with Held and Karp based branch-and-bound on small to modest-size instances from the TSPlib.
This work was partially supported by the European Community’s 7th Framework Programme (FP7/2007-2013). It was started when L.-M. Rousseau and W.-J. van Hoeve were visiting the University of Nice-Sophia Antipolis (June/July 2009).
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Benchimol, P., Régin, JC., Rousseau, LM., Rueher, M., van Hoeve, WJ. (2010). Improving the Held and Karp Approach with Constraint Programming. In: Lodi, A., Milano, M., Toth, P. (eds) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. CPAIOR 2010. Lecture Notes in Computer Science, vol 6140. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13520-0_6
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DOI: https://doi.org/10.1007/978-3-642-13520-0_6
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