On the Maximum Quadratic Assignment Problem - Nagarajan, Viswanath; Sviridenko, Maxim (Mathematics of operations research, 2009)
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 x n symmetric nonnegative matrices and . Given matrices W, D , and a permutation , the objective function is . In this paper, we study the maximum quadratic assignment problem , where the goal is to find a permutation that maximizes . We give an -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 (Feige et al. [Feige, U., G. Kortsarz, D. Peleg. 2001. The dense k -subgraph problem. Algorithmica 29 (3) 410–421]). When one of the matrices W, D satisfies triangle inequality , we obtain a -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 for general maximum quadratic assignment.