Gilmore-Lawler bound of quadratic assignment problem

Abstract

The Gilmore-Lawler bound (GLB) is one of the well-known lower bound of quadratic assignment problem (QAP). Checking whether GLB is tight is an NP-complete problem. In this article, based on Xia and Yuan linearization technique, we provide an upper bound of the complexity of this problem, which makes it pseudo-polynomial solvable. We also pseudopolynomially solve a class of QAP whose GLB is equal to the optimal objective function value, which was shown to remain NP-hard.Keywords Quadratic assignment problem (QAP), Gilmore-Lawler bound, computational complexity, NP-hard MSC 68Q17, 90C11, 90C27

Solving the quadratic assignment problem by means of general purpose mixed integer linear programming solvers

“…Xia and Yuan [19,20,21] tightened Kaufman and Broeckx formulation, basically, by introducing new constraints based on the Gilmore-Lawler constants l ij :…”

“…Although this is the smallest QAP linearization, its LP relaxation is known to be usually weak. Recently, Xia and Yuan [19,20,21] tightened Kaufman-Broeckx formulation, basically, by introducing new constraints based on the Gilmore-Lawler constants (see Section 2.3.). In this paper we will concentrate on linerizations derived from the Kaufman-Broeckx formulation, which we will call the Kaufman-Broeckx (KB) family of formulations.…”

Effective formulation reductions for the quadratic assignment problem

“…Different QAP bounds have been proposed: Gilmore-Lawler bound, eigenvalue bounds, quadratic programming bounds, LP bounds, polyhedral bounds, semidefinite bounds, among others. More details about QAP lower bounds can be found in [38,1,28,6,23,39].…”

The quadratic assignment problem: the linearization of Xia and Yuan is weaker than the linearization of Adams and Johnson and a family of cuts to narrow the gap

“…We furthermore develop a Branch and Cut approach using the insights obtained in the proof that enhances the linearization of Xia and Yuan via a new family of cuts called ab-cuts. Keywords quadratic assignment problem • linearization • lift and project • linear relaxation • cutting planes Mathematics Subject Classification (2010) MSC 52-B05 • MSC 05-C38 1 The quadratic assignment problemThe quadratic assignment problem is a well-known optimization problem which has a long history, numerous applications and has obtained broad coverage in the literature (see, e. g., [10] for a survey, or [1,3,14,15,5,11,13,12,2] for recent approaches) since its introduction in [8]. Its aim is to find an optimal…”

“…The quadratic assignment problem is a well-known optimization problem which has a long history, numerous applications and has obtained broad coverage in the literature (see, e. g., [10] for a survey, or [1,3,14,15,5,11,13,12,2] for recent approaches) since its introduction in [8]. Its aim is to find an optimal…”