# On the maximum quadratic assignment problem

## Abstract

Quadratic Assignment is a basic problem in combinatorial optimization, which 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 non-negative matrices W = (wi,j) and D = (di,j). Given matrices W, D, and a permutation π: [n] → [n], the objective function is Q(π) = ∑i,jε[n],i≠j wi,j·d π(i),π(j). In this paper, we study the Maximum Quadratic Assignment Problem, where the goal is to find a permutation π that maximizes Q(π). We give an Õ(√n) approximation algorithm, which is the first non-trivial 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 ≈ n 1/3 (Feige et al. [8]). When one of the matrices W, D satisfies triangle inequality, we obtain a 2e/e-1 ≈ 3.16 approximation algorithm. This improves over the previously best-known approximation guarantee of 4 (Arkin et al. [4]) 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 [1]). It can also be shown that this LP has an integrality gap of Ω̄(√n) for general Maximum Quadratic Assignment. Copyright © by SIAM.