The Traveling Salesman Problem: A Computational Study (Princeton Series in Applied Mathematics)
The Traveling Salesman Problem: A Computational Study (Princeton Series in Applied Mathematics)
The "Not-Too-Heavy Spanning Tree" Constraint
CPAIOR '07 Proceedings of the 4th international conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems
CPAIOR'08 Proceedings of the 5th international conference on Integration of AI and OR techniques in constraint programming for combinatorial optimization problems
A constraint programming approach for the traveling purchaser problem
CP'12 Proceedings of the 18th international conference on Principles and Practice of Constraint Programming
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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.