Integer and combinatorial optimization
Integer and combinatorial optimization
Solution of large-scale symmetric travelling salesman problems
Mathematical Programming: Series A and B
Information Sciences: an International Journal
The Angular-Metric Traveling Salesman Problem
SIAM Journal on Computing
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
AAIM'06 Proceedings of the Second international conference on Algorithmic Aspects in Information and Management
Tolerance based contract-or-patch heuristic for the asymmetric TSP
CAAN'06 Proceedings of the Third international conference on Combinatorial and Algorithmic Aspects of Networking
Certification of an optimal TSP tour through 85,900 cities
Operations Research Letters
When the greedy algorithm fails
Discrete Optimization
Transforming asymmetric into symmetric traveling salesman problems
Operations Research Letters
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In this paper we introduce an extension of the Traveling Salesman Problem (TSP), which is motivated by an important application in bioinformatics. In contrast to the TSP the costs do not only depend on each pair of two nodes traversed in succession in a cycle but on each triple of nodes traversed in succession. This problem can be formulated as optimizing a quadratic objective function over the traveling salesman polytope, so we call the combinatorial optimization problem quadratic TSP (QTSP). Besides its application in bioinformatics, the QTSP is a generalization of the Angular-Metric TSP and the TSP with reload costs. Apart from the TSP with quadratic cost structure we also consider the related Cycle Cover Problem with quadratic objective function (QCCP). In this work we present three exact solution approaches and several heuristics for the QTSP. The first exact approach is based on a polynomial transformation to a TSP, which is then solved by standard software. The second one is a branch-and-bound algorithm that relies on combinatorial bounds. The best exact algorithm is a branch-and-cut approach based on an integer programming formulation with problem-specific cutting planes. All heuristical approaches are extensions of classic heuristics for the TSP. Finally, we compare all algorithms on real-world instances from bioinformatics and on randomly generated instances. In these tests, the branch-and-cut approach turned out to be superior for solving the real-world instances from bioinformatics. Instances with up to 100 nodes could be solved to optimality in about ten minutes.