Facets for the Cardinality Constrained Quadratic Knapsack Problem and the Quadratic Selective Travelling Salesman Problem

This paper considers the Cardinality Constrained Quadratic Knapsack Problem (QKP) and the Quadratic Selective Travelling Salesman Problem (QSTSP). The QKP is a generalization of the Knapsack Problem and the QSTSP is a generalization of the Travelling Salesman Problem. Thus, both problems are NP hard. The QSTSP and the QKP can be solved using branch-and-cut methods, and in doing so, good bounds can be obtained if strong constraints are used. Hence it is important to identify strong or even facet-defining constraints. This paper presents the polyhedral combinatorics of QSTSP and QKP, i.e. amongst others identify facet-defining constraints for the QSTSP and the QKP, and provide mathematical proofs that they do indeed define facets.