An instance of the quadratic assignment problem (QAP) with cost matrix Q is said to be linearizable if there exists an instance of the linear assignment problem (LAP) with cost matrix C such that for each assignment, the QAP and LAP objective function values are identical. The QAP linearization problem can be solved in O(n4) time. However, for the special cases of Koopmans-Beckmann QAP and the multiplicative assignment problem the input size is of O(n^2). We show that the QAP linearization problem for these special cases can be solved in O(n^2) time. For symmetric Koopmans-Beckmann QAP, Bookhold  gave a sufficient condition for linearizability and raised the question if the condition is necessary. We show that Bookhold’s condition is also necessary for linearizability of symmetric Koopmans- Beckmann QAP.
A.P. Punnen and S.N. Kabadi,A LINEAR TIME ALGORITHM FOR THE KOOPMANS-BECKMANN QAP LINEARIZATION AND RELATED PROBLEMS, Manuscript, Department of Mathematics, Simon Fraser University, June 27, 2011.