Dynamic Matrix Factorization Methods for Using Formulations Derived From Higher Order Lifting Techniques in the Solution of the Quadratic Assignment Problem

This paper concerns the use of linear programming based methods for the exact solution of the Quadratic Assignment Problem (QAP). The primary obstacles facing such an approach are the large size of the formulation resulting from the linearization of the quadratic objective function and the poor quality of the lower bounds. Special purpose linear programming methods using dynamic matrix factorization provide a promising avenue for solving these large scale linear programs (LP). This enables a large portion of the LP basis to be represented implicitly and generated from the remaining explicit part. Computational results demonstrating the strength of this approach are also presented. For this approach to be effective in the solution of QAPs, dynamic matrix factorization should be combined with formulations that yield superior lower bounds. Lifting techniques have been theoretically proven to improve bound strength at the cost of a dramatic increase in formulation size. A formulation including third order interactions is derived using this methodology. However, degeneracy poses a significant problem in the solution of these linear programs. Incorporation of third order interaction costs in the objective function is proposed as a possible way to mitigate problems due to stalling. Computational results indicate that this formulation yields much stronger lower bounds than the currently best known lower bounds. Unifying these various observations, it is suggested that the development of specialized dual simplex algorithms using dynamic matrix factorization can provide a promising approach to overcome these barriers.