Proximate Solution of Dense Quadratic Assignment Problems Using Grasp, Lower Bounds Based on Linear Programming for the Quadratic

bounds for the quadratic assignment problem with an interior point algorithm for linear programming, tech. A greedy randomized adaptive search procedure for the quadratic assignment problem, in Quadratic assignment and related problems, P.tation of a variance reduction based lower bound in a branch and bound algorithm for the quadratic assignment problem, tech. 16 nodes. As the size of the QAP grew the CPU time ratio of the time taken by the new code to the time taken by the GLB based code decreased dramatically. However, for problems of the dimensions considered, it is still faster to use the GLB-based branch and bound algorithm. In a forthcoming paper, we expect to produce examples for which the new approach, besides scanning much fewer nodes, is also much faster. For this, we plan several extensions to our implementation. Since linear programs on diier-ent branches of the search tree are essentially independent of each other, it is possible to solve them in parallel. We are implementing a distributed algorithm that solves diierent linear programs on diierent processors, broadcasting the value of a new upper bound whenever one is found. In the version of the ADP code used in this paper, the algorithm does not have the capability to do warm starts, i.e. start from an advanced solution. The linear programs are always solved from the start, even when one LP varies from the other by a single column in the dual program. We are implementing a version of ADP that can start from a warm solution. With this, we expect to speed up the solution process signiicantly. 15 Table 5: CPU time ratios for QAPs of consecutive sizes n t n t n =t n?1 2 0.93 where t n is the average time taken to solve a problem of dimension n. As can be observed, for problems in this experiment, it takes slighltly over 50% longer to solve a problem of dimension n as it takes to solve a problem of dimension n ? 1. 6 Concluding Remarks In this paper, we presented implementation details and computational results of a new branch and bound algorithm for solving the quadratic assignment problem. The branch and bound algorithm uses a branching scheme discussed in 28] with a lower bounding scheme that uses linear programming described in 31]. An eecient implementation 15] of an interior point algorithm is used to compute the lower bounds. The use …