An O(n4) Algorithm for the QAP Linearization Problem

Authors:
Santosh N. Kabadi;Abraham P. Punnen
Affiliations:
Formerly at the University of New Brunswick, Fredericton, New Brunswick, Canada;Department of Mathematics, Simon Fraser University Surrey, Central City, Surrey, British Columbia V3T 0A3, Canada
Venue:
Mathematics of Operations Research
Year:
2011

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Abstract

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. Several sufficiency conditions are known that guarantee linearizability of a QAP. However, no polynomial time algorithm is known to test if a given instance of QAP is linearizable. In this paper, we give a necessary and sufficient condition for an instance of a QAP to be linearizable and develop an O(n4) algorithm to solve the corresponding linearization problem, where n is the size of the QAP.