Low rank quadratic assignment problem: Formulations and experimental analysis

In this thesis, we study the quadratic assignment problem (QAP) with a special emphasis on the case where the associated cost matrix is of rank r (QAP(r)), for small values of r. We first consider different representations of the cost matrix Q which were shown to be beneficial for the quadratic set covering problem (QSCP). Unlike QSCP, these representations were unable to solve QAP of size n >= 20 and had a behaviour different from that of QSCP. To reconfirm this, additional experiments were carried out using the quadratic knapsack problem (QKP). We did notice statistically significant preferred representations for QKP and QAP, but were different from what was observed and known for QSCP. Next we consider four different mixed integer linear programming (MILP) formulations of QAP(r), extending the known case of r=1. Extensive experimental results are provided for r=2,3,4. One of our new formulations was shown to be very effective in solving large size QAP(r) for r=2,3,4. The performance of the model is observed to deteriorate as the rank is increased. Finally, we present theoretical and experimental comparisons of the linear programming relaxations of our MILP formulations of QAP(r). Our MILP formulations for QAP(r) could be used as a heuristic for QAP by computing a low-rank approximation of the data matrix Q.