4.5 Summary
6.1.2 Linearizations
Analogous to the previous chapters 4 and 5, the here considered linearization for the QAP is the standard linearization, which we apply to (QAP). As usual, denote with Q the set of index tuples (i, j, k, l) with nonzero costs qijkl 6= 0, and replace each product term xijxkl
6.1. PROPERTIES AND ALGORITHMS 83 The equivalent ILP formulation then reads
(LQPQAP) min X i,j∈N cijxij + X (i,j,k,l)∈Q qijklyijkl s. t. X i∈N xij = 1 ∀ j ∈ N (6.4) X j∈N xij = 1 ∀ i ∈ N (6.5) yijkl≤ xij, xkl ∀ (i, j, k, l) ∈ Q (6.6) yijkl≥ xij+ xkl− 1 ∀ (i, j, k, l) ∈ Q (6.7) x ∈ {0, 1}n2 y ∈ {0, 1}|Q|.
Besides the standard linearization and the alternatives discussed in Section 3.1 there exist several different custom-built approaches to get rid of the quadratic terms in the objective function. Lawler modeled the quadratic dependency yijkl= xijxkl by 1 + n4 additional linear constraints
X
i,j,k,l∈N
yijkl= n2 and xij + xkl≥ 2yijkl ∀ i, j, k, l ∈ N (6.8)
for n4additional binary variables yijkl[121]. The set of inequalities guarantees yijkl= 0 whenever
one or both of the corresponding linear variables are zero. Combined with the linear degree constraints (6.4) and (6.5), which force xij = 1 for exactly n linear variables, this yields the fact
that at most n2 y-variables can obtain a value of one. The equality of (6.8) guarantees that this cardinality of n2 is indeed obtained and thus y
ijkl = 1 when xij = xkl= 1. Thus, yijkl= xijxkl
for all i, j, k, l ∈ N .
Assuming without loss of generality nonnegative coefficients qijkl, Kaufmann and Broeckx rear-
ranged the quadratic part of the objective function X i,j,k,l∈N qijklxijxkl= X i,j∈N xij X k,l∈N qijklxkl
and replaced the latter part of the right-hand side by a set of n2 new real and nonnegative variables wij := xijPk,l∈Nqijklxkl for all i, j ∈ N . Then, they minimize the mixed integer
linear minimization problem with objective functionPi,j∈Ncijxij+Pi,j∈Nwij, the degree con-
straints (6.4) and (6.5) and the n2 additional constraints aijxij +
X
k,l∈N
qijklxkl− wij ≤ aij ∀ i, j ∈ N
for constants aij :=Pk,l∈Nqijkl [113]. The additional constraints yield
wij ≥ X k,l∈N (xij+ xkl− 1) = 0 if xij = 0 X k,l∈N qijklxkl if xij = 1
for all i, j ∈ N and equality emanates from the fact that the objective function minimizes the nonnegative w-variables.
84 CHAPTER 6. QUADRATIC ASSIGNMENTS
An extensive linearization in terms of variables and constraints is the one of Frieze and Yadegar. They consider n4 continuous variables y
ijkl∈ [0, 1] and add 4n3+ n2 constraints
X i∈N yijkl= xkl ∀ j, k, l ∈ N, X j∈N yijkl= xkl ∀ i, k, l ∈ N, (6.9) X k∈N yijkl = xij ∀ i, j, l ∈ N, X l∈N yijkl= xij ∀ i, j, k ∈ N, (6.10) and yijij = xij ∀ i, j ∈ N, (6.11)
obtaining a mixed integer linear programming formulation being equivalent to (QAP) [73]. The equivalence follows since the constraints model yijkl = xijxkl for all i, j, k, l ∈ N : if on the one
hand for one linear variable xij = 0, the equalities (6.10) and the nonnegativity of y yield yijkl= 0
for all k, l ∈ N , analogously for xkl = 0 and yijkl = 0 for all i, j ∈ N . On the other hand, if
two linear variables xij = xkl = 1, constraints (6.9) yield Pt∈Nytjkl = 1. Assume that there
exist two variables yt1jkl, yt2jkl> 0. Then, by the previous arguments, the corresponding linear
variables xt1j = 1 and xt2j = 1, leading to t1 = t2 = i due to the degree constraints (6.5),
analogously for the other indices j, k, l.
The formulation of Frieze and Yadegard is larger but also stronger than the others when re- garding the lower bounds obtained by Lagrangean relaxation, which leads to the bounds FY1 and FY2 which are better than all bounds obtained by the reduction techniques on the Gilmore- Lawler bound, see Section 6.1.4.
The linearization of Adams and Johnson resembles the one of Frieze and Yadegar but here the inequalities (6.10) and (6.11) are replaced by the constraints
yijkl = yklij ∀ i, j, k, l ∈ N.
Note that this formulation with n4new variables and n4+2n3new constraints can be obtained by applying the Sherali-Adams linearization on the IP formulation (QAP). Furthermore, relaxing the integrality constraints in all mentioned linearizations, all constraints of the other relaxations can be expressed as linear combinations of the constraints of the relaxed Adams and Johnson linearization, and, moreover, are less tight [3, 105].
6.1.3 Complexity
As already mentioned, Sahni and Gonzales proved that QAP is NP-hard. Moreover, they showed that even the problem of finding an ǫ-approximation, i. e. a solution ¯x with
z(¯x) − z(x∗) z(x∗) ≤ ǫ
for any fixed ǫ > 0 and an optimal solution x∗ with respect to a quadratic objective function z, is not polynomially solvable unless P=NP [156]. Both statements have been shown by a reduc- tion from the NP-complete Hamiltonian cycle problem. Furthermore the quadratic assignment problem is related to the NP-hard traveling salesman problem (TSP). Namely, the TSP can be modeled as a QAP where a facility (position number in the tour) needs to be assigned to each location (city), regarding the distances (costs for traveling between the cities) and with zero construction costs. Other special cases of the QAP such as the linear ordering problem are surveyed in [40]. Further results on the hardness of the quadratic assignment problem con- cerning different types of approximations can be found in [108]. Some easy special cases of the QAP based on anti-Monge and benevolent Toeplitz matrices are analyzed by Burkard and C¸ ela et al. [33, 41].
6.1. PROPERTIES AND ALGORITHMS 85
6.1.4 Lower bounds
To reduce the set of feasible solutions and therefore the search domain of optimal QAP solutions, such as in branch-and-bound algorithms or related methods, good lower bounds are necessary to cut off as many non-optimal solutions as possible. In the past 15 years, great effort was put into bound improvements. In fact, Loiola et al. state that there exist more than a hundred papers about the development and improvement of heuristics and bounds since 1999 [128].
A natural way of obtaining lower bounds are relaxations of (mixed) integer linear programming formulations. Especially the formulation of Frieze and Yadegar and the one of Adams and Johnson, c. f. Section 6.1.2, both result in fairly good results improving the gaps of the Gilmore- Lawler bound [3, 73].
Reformulations using other coefficients c′ and q′, which provide the original objective value for each feasible QAP solution, are also popular methods for the calculation of lower bounds. An iterative process, where an appropriate reformulation rule models the next reformulation by adapting the last, yields different Gilmore-Lawler bounds, out of which the best is chosen [8,38]. Another approach, called reduction method, reduces the contribution of the quadratic term by decomposing and moving quadratic costs to the linear term [46, 73]. A combination of the reduction method and the GLB yield the Hahn-Grant bound (HGB) with promising results for B&B approaches [95, 96].
Further methods are eigenvalue bound functions, which are based on the relationship between the objective function value and the eigenvalues of the flow and distance matrices F and D [67,93,155], semidefinite relaxations [109,172,173] and an approach based on a continuous convex quadratic optimization problem [6]. The current best solutions with respect to instances of the QAPLIB are based on SDP relaxations, such as matrix-splitting [147, 148] or the exploitation of group symmetry [54].
All these approaches yield lower bounds which are different in their complexity and efficiency in the branch-and-bound tree and in the gap quality. Therefore a comparison turns out to be difficult, but concerning the asymptotic behavior of the QAP, a small gap seems to be the most important factor for a succesful solution of larger QAP instances. Loiola et al. give more details on the different lower bounds and discuss their qualities [128].