We report on the implementation of a level-3 reformulation linearization technique (RLT-3)-based bound calculation in a branch-and-bound algorithm. The RLT-3-based bound calculation method is not guaranteed to calculate the RLT-3 lower bound exactly, but approximates it very closely and reaches it in some instances. We tested the new branch-andbound solver on six Nugent instances, 15, 18, 20, 2...

3 Why Use SDP? 5 3.1 Tractable Relaxations of Max-Cut . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1.1 Simple Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.2 Trust Region Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.3 Box Constraint Relaxation . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.4 Eigenvalue Bound . . . . . . . . . . . . ...

The quadratic assignment problem arises in a variety of practical settings. It is known to be among the hardest combinatorial problems for exact algorithms. Therefore, a large number of heuristic approaches have been proposed for its solution. In this article we introduce a new, large set of QAP instances that is intended to allow the systematic study of the performance of metaheuristics in dep...

The Quadratic Assignment Problem (QAP) can be solved by linearization, where one formulates the QAP as a mixed integer linear programming (MILP) problem. On the one hand, most of these linearization are tight, but hardly exploited within a reasonable computing time because of their size. On the other hand, Kaufman and Broeckx formulation [1] is the smallest of these linearizations, but very wea...

Few applications of ACO algorithms to multiobjective problems have been presented so far and it is not clear how to design an effective ACO algorithms for such problems. In this article, we study the performance of several ACO variants for the biobjective Quadratic Assignment Problem that are based on two fundamentally different search strategies. The first strategy is based on dominance criter...

In this paper we study two formulation reductions for the quadratic assignment problem (QAP). In particular we apply these reductions to the well known Adams and Johnson [2] integer linear programming formulation of the QAP, which we call formulation IPQAP-I. We analyze two cases: In the first case, we study the effect of constraint reduction. In the second case, we study the effect of variable...

We present a coarse-grain (outer-loop) parallel implementation of RLT1/2/3 (Level 1, 2, and 3 Reformulation and Linearization Technique—in that order) bound calculations for the QAP within a branch-and-bound procedure. For a search tree node of size S, each RLT3 and RLT2 bound calculation iteration is parallelized S ways, with each of S processors performing O(S) and O(S) linear assignment prob...

This paper studies polyhedral methods for the quadratic assignment problem. Bounds on the objective value are obtained using mixed 0–1 linear representations that result from a reformulation–linearization technique (rlt). The rlt provides different “levels” of representations that give increasing strength. Prior studies have shown that even the weakest level-1 form yields very tight bounds, whi...

Semidefinite programming (SDP) bounds for the quadratic assignment problem (QAP) were introduced in: [Q. Zhao, S.E. Karisch, F. Rendl, and H. Wolkowicz. Semidefinite Programming Relaxations for the Quadratic Assignment Problem. Journal of Combinatorial Optimization, 2, 71–109, 1998.] Empirically, these bounds are often quite good in practice, but computationally demanding, even for relatively s...

In this paper we address the Adjacent Quadratic Assignment Problem (AQAP) which is a variant of the QAP where the cost coefficient matrix has a particular structure. Motivated by strong lower bounds obtained by applying Reformulation Linearization Technique (RLT) to the classical QAP, we propose two special RLT representations for the problem. The first is based on a “flow” formulation whose li...