Neural Information Processing  Letters and Reviews Vol.10, No.3, March 2006
51
Neural Networks for Solving Quadratic Assignment Problems
Gamil A. Azim
College of Computer, Qassim University, Saudi Arabia
Email: gazim3@hotmail.com
(Submitted on December 14, 2005; Accepted on March 31, 2006)
Abstract— In this paper the Hopfield neural networks are adopted to solve the quadratic
assignment problem, which is a generalization of the traveling salesman’s problem (TSP),
the graphpartitioning problem (GPP), and the matching problem. When the Hopfield
neural network was applied alone, a suboptimal solution was obtained. By adding the 2
exchange we obtained a solution very close to the optimum solution. The relationship
between the gain of neuron λ and the penalty coefficient α of the Hopfield model has been
experimentally analyzed.
Keywords— Neural networks, combinatorial optimization, quadratic assignment problem
1. Introduction
Neural networks can achieve high computational performance through massive parallelism. As shown in
[13], they usually provide good solutions to different combinatorial optimization problems including the NP
complete traveling salesman’s problem. The quadratic assignment problem (QAP) belongs to a class of
optimization problems best suited for neural network applications. Furthermore, the QAP serves as a generalized
extension of some other important optimization problems such as the TSP, the graph partitioning problem
(GPP), and matching problem. There exist conventional algorithms for solving these problems such as the
branch and bound algorithm. However, the computational complexity of these algorithms is non polynomial. On
other hand, the computational time of neural network is not expected to grow rapidly with problem size. Also the
technology is available to implement neural network models on a very largescale integrated (VLSI) chip. That‘s
why we are interested in neural network solutions.
This paper is organized as follows. In Section 2 and Section 3 the Hopfield neural network model and the
quadratic assignment problems are summarized, respectively. In Section 4 we show how to map the QAP into
Hopfield model, and a computational example is shown in Section 5.
2. Hopfield Neural Network
The Hopfield neural network is topologically characterized by the presence of feedback between each pair
of neurons. It is worth noting that we will focus on the continuous Hopfield model [2]. Each feedback is
associated with a weight that expresses the influence of one neuron on another, and each of them may be
supplied with external bias currents. The feedback and bias currents determine the net input of a neuron as
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−+∆+=
∑
=
n
j
old
i
ijij
old
i
new
i
y
IxTtyy
1
η
, (1)
where the subscript i denotes the index of the neuron, η is a userselected decay constant, ∆t is the time step
size, x
j
is the out put of the neuron j, I
i
is the bias current of the neuron i, and T
ij
is the weight of the connection
between the neuron i and neuron j. The output of each neuron x
j
is then obtained through an nonlinear activation
function and is given by
.)tanh1(
2
1
jj
yx λ+=
(2)
LETTER
Neural Networks for Solving Quadratic Assignment Problems Gamil A. Azim
52
The parameter λ determines the slope of the sigmoid function, and is a userselectable parameter. The updating
of neural activation can be either synchronous or asynchronous. We will assume that it is synchronous.
Hopfield and others [2] had shown that a recurrent network is stable if the weights matrix is symmetric and has
only zeros of the main diagonal. Of course this is only a sufficient condition. In the case of stable recurrent
networks it is possible to verify that the energy function (Lyapunov function) is given by
∑
∫
∑∑
=
−
==
+−−=
n
i
x
n
i
ii
n
ji
ijij
xdxgxIxxTxE
i
1
5.0
1
11,
)(
2
1
)(
λ
, (3)
where n is the number of neurons in the network. The network converges to local minima corresponding to some
vertices of the ndimensional hypercube. [1,2,3]
3. Quadratic Assignment Problem
The goal of the quadratic assignment problem (QAP) of order n is to assign n objects to n positions. It is a
classic combinatorial optimization problem and belongs to a class of NPhard problems. It is so easy to state, but
very difficult to solve.[13] The QAP has been used to develop a useful and suitable model for many different
real world situations. For an excellent review of the QAPs and their mathematical treatment, see the references
[6,7,8,9,14].
The QAP can be described mathematically as follows. Given a finite set N={1,…,n} and two matrices A =
[a
ij
] and B= [b
ij
]. We have to find a permutation ρ of the set N which minimizes the quadratic functional form of
∑ ∑
= =
=
n
i
n
j
jiij
baZ
1 1
)()(
)(
ρρ
ρ
. (4a)
For every permutation ρ there is a correspondent permutation matrix X=[x
ij
] with
1
1
=
∑
=
n
j
ij
x
for i = 1 to n , (4b)
1
1
=
∑
=
n
i
ij
x
for j = 1 to n , (4c)
where x
ij
∈{0,1} for i, j = 1,…,n . Eqs.(4b) and (4c) are called as permutations constraints. Hence the QAP can
be reformulated as an Integer Program with quadratic function as
,minargminarg
1,,
xxba
ijkljl
n
ji
n
lk
ik
XX
Z
∑ ∑
=
=
(5)
with the two conditions in Eqs.(4b) and (4c), and
⎩
⎨
⎧
=
ortherwise0
positiontoassignedisobject if1 ji
x
ij
The product a
ik
b
jl
in Eq.(5) is the cost associated with assigning the object i to position j and object l to position
k. The objective function Z is defined as the total cost for the assignment. The condition in Eq.(4b) ensures that
every object will be given a single position, and the condition in Eq.(4c) ensures that each position will be
occupied by only one object.
We briefly show some examples in this section. Other applications concern the design of the write keyboards
and control panels. See for example [5]. Many exact and heuristic algorithms have been developed in the field
of operations research (OR) to solve QAPs. Interested readers should consult with Ref. [15].
3.1 Facility layout
A typical application is the facility layout problem, in which n facilities have to be assigned to n locations.
In this particular case a
ik
is the flow from facility i to facility k, b
jl
is the distance between location j and location
l. Mathematically we get the following model:
⎩
⎨
⎧
=
ortherwise0
positionin placed isfacility if1 ji
x
ij
.
3.2 Backboard wiring
This application concerns placing certain modules on a board and connect the modules by wires. The
modules should be placed such that the total wire length becomes minimal. Mathematically we get the following
model:
Neural Information Processing  Letters and Reviews Vol.10, No.3, March 2006
53
⎩
⎨
⎧
=
ortherwise0
position inplacedismodule theif1 ji
x
ij
.
where a
ik
is the number of connections between module i and k, and b
jl
is the length of connections between
location j and location l.
3.3 Graph partitioning problem (GPP)
In the graph partitioning problems one tries to make a partition of a graph G=(V,E) into two equalsized
sets with the minimum number of edges between the two sets. The GPP can be expressed as a QAP by defining
⎩
⎨
⎧
∨≤
=
otherwise0
2/,2/,if1 njinji
a
ij
f
and
⎩
⎨
⎧
∈
=
otherwise0
),(if1 Eji
b
ji
.
A partition represented by a permutation
ρ
s敦= 湥搠慳潬lo睳⸠䄠癥牴數e j belongs to the first set if
2/)( nj ≤
ρ
, and to the second set otherwise.
4. Mapping Quadratic Assignment Problem onto Hopfield Model
To map a problem onto a Hopfield neural network we need to perform the following:
(a) Choose a representation scheme which allows the outputs of neurons (states of networks) to be decoded
into a solution of the problem.
(b) Choose an energy function whose minimum value corresponds to the best solutions of the problem to be
mapped.
(c) Derive the parameters of the neural networks (the weights of the connections
T
and external bias currents
I
) from the energy function. These should appropriately represent the instance of the specific problem to
be solved.
The first step in the mapping of a combinatorial optimization problem is the representation of the objective
function as well as the problem constraints in the form of a simple energy function. There is no direct method for
mapping a constrained optimization problem onto Hopfield neural networks except through the additional terms
in the energy function E, which penalize the violation of the constraints.
For solving the QAP by Hopfield model [3], we arrange the neurons in a matrix form, in which each
neuron is identified by a set of double indices i and j indicating its row and column, respectively. The input
output relationship of a neuron on row i and column j is given by x
ij
= g
λ
(y
ij
). There is a feedback path among
pairs of neurons, designated by T
lm,ij
and is referred to as a connection matrix. Furthermore, there is an external
bias I
ij
supplied to each neuron. The differential equation describing the dynamics of the neurons is
ij
n
m
n
l
lmlmij
ijij
IxT
y
dt
dy
++
−
=
∑ ∑
= =1 1
,
γ
,
where
γ
is the time constant of the neuron. It has been shown that the quadratic energy function may be defined
as
∑
∫
∑ ∑ ∑
−
= = =
+−−=
n
ji
x
n
ji
n
ml
n
ji
ijijlmijlmij
i
dxxgxIxxTxE
,
2/1
1
1,1,1,
,
)(
2
1
)(
λ
(6)
For the neural network its states is always in the interior of the ndimensional hypercube defined by 0 ≤x
ij
≤ 1. Furthermore, it has been proven that the local minima of this energy function occur only on the 2
n
corners of
the hypercube. We construct an energy function in the form of Eq.(6), where the local minima correspond to the
solutions of the particular QAP. Then, from the correspondence between the constructed energy function and
Eq.(6), the parameters values of the neural network are determined. We used the penalty function as
)1(11)(
1 11
2
1 1
2
11
2
1
ijij
n
i
n
j
ij
n
j
n
i
n
j
ij
n
i
ij
n
i
n
j
ij
xxnxxxx −+
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
−+
⎭
⎬
⎫
⎩
⎨
⎧
−+
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
−=Π
∑∑∑ ∑∑∑∑ ∑
= == = === =
ξγβα
. (7)
The first three terms in the right side are equal to 0 for allowed permutation matrices, but may also be zero for
other cases such as double stochastic matrices. Therefore we add the forth term to ensure that the final solution
consists of only binary values. Thus,
Neural Networks for Solving Quadratic Assignment Problems Gamil A. Azim
54
⎩
⎨
⎧
=Π
otherwise1
matrixnpermutatioaisif0
)(
x
x
Other penalty functions can be formulated using the permutation constrains in Eqs.(4b) and (4c). [11,12]
We show that by a suitable choice of
ξ
i j
the diagonal of matrix
T
become null and no local minimum
exists in the inside of hypercube [0 1]
n
. After the relaxation of the permutation constraints by using the penalty
method, the QAP can be mathematically described as
}1,0{,
2
1
minargminarg
1,1,1,
,
∈
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−−=
∑ ∑ ∑
= = =
ij
n
ji
n
ml
n
ji
ijijlmijlmij
xxIxxTZ
xx
. (8)
The relaxation of the permutation constraints means the addition of ∏(x) to the energy function, which penalizes
the violation of the constraints. By comparing Eq.(6) and Eq.(8), we find the elements of the connection matrix
and external biases as
{
}
jlikijjlikijjlikimjllmij
baT δδξδδεγδβδα −−+++−= )1(2
,
, (9a)
ijijijijij
banI
ε
ξ
γ
β
α
−
−
+
+
= 222
, (9b)
with
⎩
⎨
⎧
−
≥
=
otherwise
ba
ijij
ij
1
01
ε
.
The diagonal terms of the connection matrix become
)(2
ijijij
T
ξ
γ
β
α
−
+
+
−
=
(10)
If we choose
γ
β
α
ζ
++≥
ij
, the diagonal terms of the connection matrix become positive or zero. The matrix
T
and external bias currents
I
of the network become
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
−−
+−+−+−
−=
δδ
ξ
δδε
δδδδδδ
γβα
jlik
ij
jlikijjlik
jllmjlimlmjl
lmij
ba
T
)1(
)1()1()1(
2
,
(11)
ijijijij
banI εγβα −−++= )12(
(12)
In the following section, we apply the Hopfield model with connection matrix
T
and bias currents
I
determined by Eqs. (11) and (12).
5. Computation Tests
We have treated several examples of different sizes taken from the literature where matrices
A
and
B
are
symmetric with zero diagonal elements. The parameter
λ
=桥慩渠潦h湥畲潮⤠楳湣r敡獥搠慴敧畬慲n瑥牶慬猠慳†
1,)()1( >
=
+
amam
λ
λ
, (13)
where m is the interval size. It should be noted that we adopt the idea of simulated annealing where
λ
=
捯牲敳灯湤猠瑯p瑨攠楮癥牳≥映=emp 敲慴畲e.=坥W癥h潳敮⁴桥o灥湡汴礠捯pf晩捩敮琠慳f景汬潷献†f
nji
ij
,....1,0,,
=
∀
=
+
=
=
γ
β
α
ζ
β
α
.
The modification of the penalty coefficients is made in the following way:
1,
1
>
=
+
dd
kk
α
α
.
Here
α
0
is a positive arbitrary real number. To improve the neural solution, we have applied the 2exchange
improvement method, which consists of choosing the best value of the objective function among all
permutations obtained by exchanging two elements of the neural solution [4].
For the examples of Nugent [10] we have tested several examples with different sizes for n=12, 20, and 30,
and show only the results of the simulation for n=30 in the Tables 1 and 2. The optimal solution has the cost
function of 6124.
The model parameters are defined as follows:
0
λ
㨠楮楴楡氠ia汵攠潦e
λ
⸠†
= MI吺a硩xum=mb敲e潦≥敲慴i潮o⸠.
Neural Information Processing  Letters and Reviews Vol.10, No.3, March 2006
55
Table 1. Experiment results for (n=
3
0),
0
λ
=0.5, MIT=10000, m=100,
0
α
=600, and d=1.1
CPUIIMPITPFon
4.32
632
8
1867094
4.32
6328
197
7160
4.55
6672
179
7290
4.35
6480
178
7514
4.34
6334
139
7322
Table 2. Experiment results for (n =30),
0
λ
=0.5, MIT=10000, m=300,
0
α
=450, and d=1.1
CPUI
IMP
ITP
Fon
4.34
6328
2417094
4.33
6328
190
7094
4.34
6372
212
7140
4.39
6256
170
7318
4.34
6328
241
7094
m: interval size of
λ
桡湧n⸠.
=
0
α
›=楴楡氠ia汵攠of⁴=攠pena汴礠捯敦晩捩敮琮†
=
f
α
: final penalty coefficient.
FON: value of the cost function.
ITP: number of iteration used to obtain the permutation (feasible solution).
IMP: cost function after improvement.
CPUI: computation time for improvement.
We observed that the computational times are similar with and without the improvement method. It may
show that the obtained solution belongs to the neighborhood of the optimal solutions. As shown in Figure 1 we
also found the relationship between
λ
湤=
α
景爠晥慳楢汥⁰e牭u≥慴楯n=汵瑩≥ns⸠
=
=
䙩Fu牥‱⸠周牥攠獯汵瑩潮= 牥ri潮猠楮=
λ
⡴桥h条gnf= 畲on)湤n
α
⡴桥⁰= 湡汴礠捯n晦楣楥湴⤠獰慣f⸠债P獥琠潦s
晥慳楢汥l汵瑩潮=⡰敲eu瑡瑩≥ns⤬⁂㨠獥琠潦obi湡特n琠 no≥=晥慳楢汥llu瑩≥湳Ⱐ䌺整映捯n≥in畯u猠獯汵瑩潮猠
=
Neural Networks for Solving Quadratic Assignment Problems Gamil A. Azim
56
6. Conclusion
We have developed a neural mapping solution method for finding a good solution to quadratic assignment
problems. We have shown that recurrent neural networks provide an interesting
alternative to classical
algorithms for the optimization problems.
However the
neural approach does not ensure convergence
deterministically. It has to be studded by more effective means such as the
characterization of the problem and
its surrounding conditions.
We have noted the difficulty of finding an effective definition and weighting for the numerous coefficients
associated with the
Hopfield network which are essential to achieve good behavior. It may be the reason why so
far only limited applications are investigated to the optimization problems.
This works has shown that the solutions found by using only the Hopfield method are not satisfactory.
However, if we consider them as initial solutions for our simple improvement method, then the obtained results
are much closer to the optimal solutions. We also observed that the additional computational times for the
improvement method is negligible, This observation shows that the neural solution belongs to the neighborhood
of the optimal solution. We also experimentally found the relationship between
λ
慮搠
α
潲摭i獳楢汥s
灥牭畴慴i潮潬u≥i潮献†
=
References
[1] J.J. Hopfield, “Neuronal networks and physical systems with emergent collective computational abilities,”
Proc. Nat. Acad. Sc. U.S.A., vol. 79, pp. 2554 2558, 1982.
[2] J.J. Hopfield, “Neurons with granded response have collective computational properties like those of two
state neurons,” Proc. Nat. Acad. Sc. U.S.A., vol. 81, pp. 3088 –3092, 1984.
[3] J.J. Hopfield and D.W. Tank, “Neuronal computation of decisions in optimization problems,” Biol. Cybern,
vol. 52, pp 141 152, 1985.
[4] G.C. Armour and E.S. Buffa, “A heuristic logarithm and simulative approach to relative location of
facilities,” Management Science, Vol. 9, pp. 294 309, Jan. 1963.
[5] R. E. Burkard and J. Offerman, “Entwurf von Schreibmaschinentastaturen mittels quadratischer
zuordnungsprobleme,” Z. Oper. Res., vol. 21, pp. B121B132, 1977.
[6] R.E. Burkard and U. Derigs, “Assignment and matching problems: solution methods with FORTAN
program,” Lecture Notes in Economics and Mathematical Systems, pp. 184, Springer, Berlin, 1980.
[7] R. E. Burkard and T. Bonniger, “A heuristic for quadratic Boolean programs with applications to quadratic
assignment problems,” European J. Operational Res., Vol. 13(4), pp. 374386, 1983.
[8] R.E. Burkard and F. Rendl, “A Thermodynamically motivated simulation procedure for combinatorial
optimization problems,” European J. Operational Res., Vol. 17, pp. 169174, 1984.
[9] R. E. Burkard, “Quadratic assignment problems,” European J. Operational Research., Vol. 15 (3), pp. 283
289, 1984.
[10] C. E. Nugent, T.E. Vollman, and J. Ruml, “An experimental comparison of Techniques for the assignment
of facilities to locations,” Operations Research, Vol. 16, pp. 150173, 1968.
[11] G.A. Azim, “Applications of neural networks to combinatorial optimization,” Ph.D., Thesis, Paris Dauphine
University, May 1992, France.
[12] G.A. Azim, “Mapping the bipartition graph and quadratic assignment problems into Hopfield neural
networks,” Far East Journa of applied Mathematics, Vol.2., 1998.
[13] T.C. Koopmans and M.J. Beckmann, “Assignment problem and the location of economic activities,”
Econometrica, vol. 25, pp. 5376, 1957.
[14] R. E. Burkard, S. Karisch, and F. Rendl, “QAPLIB A quadratic assignment problem library,” Eur. J. Oper.,
pp. 115119, 1991.
[15] G. Finke, E. Rainer, R. E. Burkard, and F. Rendl, “Quadratic assignment problem,” Annals of Discrete
Mathematics, Vol. 31, pp. 6182, 1987.
Neural Information Processing  Letters and Reviews Vol.10, No.3, March 2006
57
Gamil A. Azim
received M.S. and the Ph.D. degrees from Paris Dauphine
University, France, in 1988 and 1992, respectively. He is currently an Assistant
Professor in the Department of Computer Sciences, College of Computer, Qassim
University, K.S.A. His Current research interests include Neural Networks,
Combinatorial Optimization, Metaheuristic Algorithms, Pattern Recognition and
Evolutionary Computation (Genetic algorithms and Genetic Programming).
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