A linear time algorithm for the Koopmans–Beckmann QAP linearization and related problems

• A linear time algorithm for QAPKB linearization problem is presented.
• As a by-product, we have answer a question raised in 1990 by Bookhold.

An instance of the quadratic assignment problem (QAP) with cost matrix QQ is said to be linearizable if there exists an instance of the linear assignment problem (LAP) with cost matrix CC such that for each assignment, the QAP and LAP objective function values are identical. The QAP linearization problem can be solved in O(n4)O(n4) time. However, for the special cases of Koopmans–Beckmann QAP and the multiplicative assignment problem the input size is of Ω(n2)Ω(n2). We show that the QAP linearization problem for these special cases can be solved in O(n2)O(n2) time. For symmetric Koopmans–Beckmann QAP, Bookhold [I. Bookhold, A contribution to quadratic assignment problems, Optimization 21 (1990) 933–943.] 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.