Maximum quadratic assignment problem: Reduction from maximum label cover and LP-based approximation algorithm

We show that for every positive ε > 0, unless NP ∪ BPQP, it is impossible to approximate the maximum quadratic assignment problem within a factor better than 2^log_1?ε n by a reduction from the maximum label cover problem. Our result also implies that Approximate Graph Isomorphism is not robust and is, in fact, 1 ? ε versus ε hard assuming the Unique Games Conjecture. Then, we present an O(√n)- approximation algorithm for the problem based on rounding of the linear programming relaxation often used in state-of-the-art exact algorithms.