On the Maximum Quadratic Assignment Problem

Quadratic assignment is a basic problem in combinatorial optimization that generalizes several other problems such as traveling salesman, linear arrangement, dense k subgraph, and clustering with given sizes. The input to the quadratic assignment problem consists of two n × n symmetric nonnegative matrices $W=(w_{i, j})$ and $D=(d_{i, j})$ . Given matrices W, D, and a permutation $\pi: [n] \rightarrow [n]$ , the objective function is $Q(\pi):= \sum_{i, j \in [n], i \ne j} w_{i, j} \cdot d_{\pi(i), \pi(j)}$ . In this paper, we study the maximum quadratic assignment problem, where the goal is to find a permutation π that maximizes $Q(\pi)$ . We give an $\tilde{O}(\sqrt{n})$ -approximation algorithm, which is the first nontrivial approximation guarantee for this problem. The above guarantee also holds when the matrices W, D are asymmetric. An indication of the hardness of maximum quadratic assignment is that it contains as a special case the dense k subgraph problem, for which the best-known approximation ratio is $\approx n^{1/3}$ (Feige et al. [Feige, U., G. Kortsarz, D. Peleg. 2001. The dense k-subgraph problem. Algorithmica29(3) 410--421]). When one of the matrices W, D satisfies triangle inequality, we obtain a $2e/(e-1) \approx 3.16$ -approximation algorithm. This improves over the previously best-known approximation guarantee of four (Arkin et al. [Arkin, E. M., R. Hassin, M. Sviridenko. 2001. Approximating the maximum quadratic assignment problem. Inform. Processing Lett.77 13--16]) for this special case of maximum quadratic assignment. The performance guarantee for maximum quadratic assignment with triangle inequality can be proved relative to an optimal solution of a natural linear programming relaxation that has been used earlier in branch-and-bound approaches (see, eg., Adams and Johnson [Adams, W. P., T. A. Johnson. 1994. Improved linear programming-based lower bounds for the quadratic assignment problem. DIMACS Ser. Discrete Math. Theoret. Comput. Sci.16 43--77]). It can also be shown that this linear program (LP) has an integrality gap of $\tilde{\Omega}(\sqrt{n})$ for general maximum quadratic assignment.