Branch-and-bound and cutting-plane methods are primarily the two types of algorithms that have been used to solve the quadratic assignment problem to optimality. Branch- and-bound algorithms have been the more successful of the two for the QAP, outperform- ing cutting-plane algorithms whose running time is simply too long [27]. In the following we will briefly discuss cutting-plane methods. We will describe in detail branch-and- bound methods for the QAP in section 2.4.1.
There are two classes of cutting-plane methods: traditional cutting-plane methods and polyhedral cutting-plane or branch-and-cut methods. Traditional cutting-plane al- gorithms have been developed by Balas and Mazzola [9], Bazaraa and Sherali [11, 12], and Kaufmann and Broeckx [69]. These algorithms make use of mixed integer linear pro- gramming formulations for the QAP which are well suited for Benders’ decomposition. They either solve the QAP to optimality or compute a lower bound. The computational experience for polyhedral cutting-plane methods is still very limited, due to lack of good understanding of the combinatorial structure of the QAP polytope [27]. Recently, en- couraging results [90, 65] in polyhedral cuts have been obtained, leading one to believe that polyhedral cutting-plane algorithms can be used to solve reasonably sized QAP instances in the future.
2.4.1
Branch-and-Bound Methods
In a branch-and-bound method for the QAP, the algorithm starts with an empty per- mutation with no facility assigned to any location and successively expands it to a full permutation in which all the facilities are assigned to the locations. The important com- ponents for a branch-and-bound algorithm for the QAP are the branching rule,selection rule, and lower bounding technique.
There are three common types of branching rules: single assignment [44, 74], pair assignment [39, 72, 88], and relative positioning [85]. In the single assignment branching, the algorithm assigns a single, not yet allocated facility i to an unoccupied location j
at each branching step. The pair assignment rule allocates a pair of facilities to a pair of locations at each branching step. In the relative positioning branching, the levels of branch-and-bound tree do not correspond to the assignments of facilities to locations. The partial permutations at each level are determined in terms of the distances between facilities, i.e., their relative positions. Numerical results show that the single assignment branching rule outperforms the pair assignment or relative positioning branching rules [19, 92]. The choice of the facility-location pair (i, j) is made according to the selection rule. Several rules have been proposed by different authors [10, 17, 79]. The appropriate rule usually depends on the bounding technique employed.
2.4.2
Lower Bounds
The performance of branch-and-bound algorithms depends critically on the quality of the lower bounds. Many efforts have been made to derive tight and yet computationally efficient lower bounds. In this section we will briefly describe five bounding techniques for the QAP: Gilmore-Lawler and related lower bounds, lower bounds based on linear programming relaxations, eigenvalue based lower bounds, lower bounds based on semidef- inite relaxations, and convex quadratic programming bounds.
Gilmore-Lawler and Related Lower Bounds. One of the first lower bounds for the QAP was derived by Gilmore [44] and Lawler [74]. For QAP(F, D) of sizeN, we define a N ×N matrix C = (cij) by cij , min π∈ΠN: π(j)=i N X k=1 fiπ(k)djk, i, j = 1, . . . , N.
It is well known that the entriescij can be easily computed by sorting vectorsFi and Dj in increasing and decreasing order respectively, where Fi denotes the i-th row of matrix
F and Dj denotes the j-th row of matrixD. It is easy to see that the following holds for
π ∈ΠN: (2.10) Z(F, D, π) = N X i=1 N X j=1 fπ(i)π(j)dij ≥ N X i=1 Fπ(i)DiT ≥ N X i=1 cπ(i)i. From (2.10) we have min π∈ΠN Z(F, D, π)≥ min π∈ΠN N X i=1 cπ(i)i ,GLB,
where GLB is the Gilmore-Lawler lower bound for QAP(F, D). After matrix C is com- puted, it takes O(n3) time to compute GLB by solving a linear assignment problem. Hence the overall complexity for computing the Gilmore-Lawler bound is O(n3).
The Gilmore-Lawler bound is one of the simplest and cheapest bounds to compute, but unfortunately this bound is not tight and, in general, the gap between the Gilmore- Lawler bound and the optimal objective value increases quickly as the problem size increases. Various reduction schemes have been proposed to improve the quality of the Gilmore-Lawler bound by transforming the coefficient matrices F and D. Li et al. [77] proposed an improvement on the bound via a variance reduction scheme. Reduction schemes based on reformulations [24, 25] and dual formulations [54, 55] have also been proposed.
Lower Bounds Based on Linear Programming Relaxations. Several authors have proposed mixed integer linear programming (MILP) formulations for the QAP. In the context of lower bound computations two MILP formulations are of importance. Frieze and Yadegar [38] replaced the products xikxjl of 0-1 variables by continuous vari- ables yijkl and introduced extra linear constraints to obtain an MILP formulation for the QAP. To obtain a lower bound, Frieze and Yadegar derived a Lagrangian relax- ation of their MILP formulation and solved it approximately by applying subgradient optimization techniques. They showed that the resulting bounds are better than the Gilmore-Lawler bounds.
Based upon the MILP formulation of Frieze and Yadegar, Adams and Johnson [1] obtained a slightly more compact MILP formulation:
minimize N X i=1 N X j=1 N X k=1 N X l=1 fijdklyijkl (2.11) subject to (1.3a)–(1.3c) N X i=1 yijkl=xjl, j, k, l= 1, . . . , N N X j=1 yijkl=xik, i, k, l= 1, . . . , N yijkl=yklij, i, j, k, l= 1, . . . , N yijkl≥0, i, j, k, l= 1, . . . , N.
They showed that the continuous relaxation of (2.11) is tighter than the continuous relaxation of the formulation of Frieze and Yadegar. Moreover, many of the previously published lower bounding techniques can be explained based upon the Lagrangian dual of this relaxation.
Eigenvalue Based Lower Bounds. These bounds are based on the relationship be- tween the objective value of the QAP in the trace formulation and the eigenvalues of the flow and distance matrices. If designed and implemented prudently, these bound- ing techniques can produce bounds of better quality than the Gilmore-Lawler bounds or bounds based on linear programming relaxations. However, the eigenvalue based bounds are expensive to compute and hence are not appropriate for use as bounding techniques in branch-and-bound algorithms. Eigenvalue based bounds for the QAP with symmetric matrices are proposed by several authors [33, 52, 53, 98].
Lower Bounds Based on Semidefinite Relaxations. Semidefinite programming (SDP) relaxations for the QAP were studied by Karisch [68], Zhao [117], and Zhao et al. [118]. In their papers, interior-point methods or cutting-plane methods are used to solve the SDP relaxations to obtain lower bounds for the QAP. These solution methods require at least O(n6) time per iteration. In terms of quality the bounds they obtained are competitive with the best existing lower bounds for the QAP. For many QAPLIB instances, such as those of Hadley et al. [53], Nugent et al. [88] and Taillard [110, 111], semidefinite relaxations provide the best existing bounds. However, the prohibitively high computation requirements makes the use of such approaches impractical.
Convex Quadratic Programming Bounds. The quadratic programming bound [5] for QAP(F, D) is defined as the optimal objective value of the following quadratic pro- gramming problem:
minimize vec(X)TQvec(X) +γ
(QPB)
subject to Xe=XTe=e
X ≥0,
whereQ= (D⊗F)−(I⊗S)−(T⊗I). The matricesSandT and constantγ are obtained from the spectral decompositions ofVTF V andVTDV, whereV is anN×(N−1) matrix
with orthonormal columns such thateTV = 0. The objective function of (QPB) is convex on the nullspace of the equality constraints, so computing the quadratic programming bound requires solving a convex quadratic programming problem. In [16] the Frank-Wolfe (FW) algorithm [36, 82] is proposed to approximately solve (QPB). Although the FW algorithm is known for its poor asymptotic performance, in this context it is attractive because the computation required at each iteration is dominated by the solution of a single, small linear assignment problem. It is worth noting that this bounding technique was used to solve several very difficult QAP instances in Table 1.3 to optimality.