The Asymmetric Quadratic Traveling Salesman Problem

The quadratic traveling salesman problem asks for a tour of min-imal costs where the costs are associated with each two arcs that are traversedin succession. This structure arises, e. g., if the succession of two arcs representsthe costs of loading processes in transport networks or a switch between dif-ferent technologies in communication networks. Based on a quadratic integerprogram we present a linearized integer programming formulation and studythe corresponding polyhedral structure of the asymmetric quadratic travelingsalesman problem (AQTSP), where the costs may depend on the direction oftraversal. The constructive approach that is used to establish the dimensionof the underlying polyhedron allows to prove the facetness of several classes ofvalid inequalities. Some of them are related to the Boolean quadric polytope.Two new classes are presented that exclude conflicting configurations. Amongthese the first one is separable in polynomial time, the separation problem forthe second class is NP-complete under certain conditions. We provide a generalstrengthening approach that allows to lift valid inequalities for the asymmetrictraveling salesman problem (ATSP) to stronger valid inequalities for AQTSP.Applying this approach to the subtour elimination constraints gives rise tofacet defining inequalities, but finding a maximally violated inequality amongthese is NP-complete. For the (D3)-, (D−4 )-, (D +4 )-inequalities of the ATSPthe strengthening approach is not sufficient to obtain a facet. First computa-tional results are presented to illustrate the importance of the new inequalities.In particular, with these inequalities the running times can be improved forsome real-world instances from biology by orders of magnitude in comparisonto other state of the art methods in the literature.