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<sup>log<sup>1-ϵ</sup> <i>n</i></sup> 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 <i>O</i>(√n)-approximation algorithm for the problem based on rounding of the linear programming relaxation often used in state-of-the-art exact algorithms.

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