The quadratic traveling salesman problem (QTSP) is an extension of the (classical) Traveling Salesman Problem (TSP) where the costs depend on each two nodes that are traversed in succession, i. e., on the edges in the symmetric (STSP) and on the arcs in the asymmetric case (ATSP). The QTSP is motivated by an application in bioinformatics. It can be used in the solution of certain Permuted Markov models that are set up for the recognition of transcription factor binding sites and of splice sites in gene regulation. Important special cases are the Angular-Metric TSP used in robotics and the TSP with Reload Costs used in the planning of telecommunication and transport networks. The SQTSP and the AQTSP can be formulated as integer optimization problems over the polytope associated with the STSP resp. ATSP together with a quadratic cost function. We study the polytopes arising from a linearization of the respective quadratic integer programming formulations. Based on the proof of the dimension of the polytopes using the so called direct method we can prove the facetness of several valid inequalities. These facets and valid inequalities can be divided into three large groups. Some are related to the Boolean quadric polytope. Furthermore we introduce the conflicting edges/arc inequalities that forbid certain configurations of edges and 2-edges resp. of arcs and 2-arcs. Finally, we strengthen valid inequalities of STSP and ATSP in order to get stronger inequalities in the quadratic case. We present two general lifting approaches. One is applicable to all inequalities with nonnegative coefficients and the second allows to strengthen clique tree inequalities. Applying these approaches to the subtour elimination constraints leads to facets in most cases, but in general facetness is not preserved. In addition, the complexity of the separation problems for some of the facet classes is studied. Finally, we present some computational results using a branch-and-cut framework, which is improved by some of the newly derived cutting planes. The tested instances from biology could be solved surprisingly well. Instances with up to 100 nodes could be solved in less than 700 seconds improving the results in the literature by several orders of magnitude. For most of the randomly generated instances using some additional separators allowed to reduce the root gaps and the numbers of nodes in the branch-and-cut tree significantly, often even the running times.

The quadratic traveling salesman problem (QTSP) is an extension of the (classical) Traveling Salesman Problem (TSP) where the costs depend on each two nodes that are traversed in succession, i. e., on the edges in the symmetric (STSP) and on the arcs in the asymmetric case (ATSP). The QTSP is motivated by an application in bioinformatics. It can be used in the solution of certain Permuted Markov models that are set up for the recognition of transcription factor binding sites and of splice sites in gene regulation. Important special cases are the Angular-Metric TSP used in robotics and the TSP with Reload Costs used in the planning of telecommunication and transport networks. The SQTSP and the AQTSP can be formulated as integer optimization problems over the polytope associated with the STSP resp. ATSP together with a quadratic cost function. We study the polytopes arising from a linearization of the respective quadratic integer programming formulations. Based on the proof of the dimension of the polytopes using the so called direct method we can prove the facetness of several valid inequalities. These facets and valid inequalities can be divided into three large groups. Some are related to the Boolean quadric polytope. Furthermore we introduce the conflicting edges/arc inequalities that forbid certain configurations of edges and 2-edges resp. of arcs and 2-arcs. Finally, we strengthen valid inequalities of STSP and ATSP in order to get stronger inequalities in the quadratic case. We present two general lifting approaches. One is applicable to all inequalities with nonnegative coefficients and the second allows to strengthen clique tree inequalities. Applying these approaches to the subtour elimination constraints leads to facets in most cases, but in general facetness is not preserved. In addition, the complexity of the separation problems for some of the facet classes is studied. Finally, we present some computational results using a branch-and-cut framework, which is improved by some of the newly derived cutting planes. The tested instances from biology could be solved surprisingly well. Instances with up to 100 nodes could be solved in less than 700 seconds improving the results in the literature by several orders of magnitude. For most of the randomly generated instances using some additional separators allowed to reduce the root gaps and the numbers of nodes in the branch-and-cut tree significantly, often even the running times.