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Using Hopfield Networks to Solve Assignment Problem
and N-Queen Problem: An Application of Guided Trial
and Error Technique
Christos Douligeris1 and Gang Feng2
1
2
Department of Informatics, University of Piraeus, Piraeus, 18534, Greece.
Telecommunications and Information Technology Institute, Florida International University,
Miami, FL 33174.
Abstract. In the use of Hopfield networks to solve optimization problems, a
critical problem is the determination of appropriate values of the parameters in
the energy function so that the network can converge to the best valid solution. In
this paper, we first investigate the relationship between the parameters in a typical
class of energy functions, and consequently propose a “guided trial-and-error"
technique to determine the parameter values. The effectiveness of this technique
is demonstrated by a large number of computer simulations on the assignment
problem and the N-queen problem of different sizes.
1
Introduction
The continuous Hopfield neural network (CHNN) [1] can be used to solve an optimization problem in such a way that the cost function and constraints are first mapped to an
energy function (if possible) and then a solution is obtained as the network stabilizes.
Ever since Hopfield and Tank applied this network to solve the traveling salesman problem (TSP) [2], it has been employed to solve a variety of combinatorial optimization
problems. However, a critical problem arising in the use of the HNN to solve optimization problem is how to choose the best parameters in the energy function so that the
network can converge to valid solutions of high quality. In this paper, we will propose a
relatively general method to determine the parameters.
In the past decade, the most extensively used method is the trial-and-error technique,
and to our point of view, this technique (at most with more constraints) will still be used in
the future, especially for those problems that are NP-hard or NP-complete. This is based
on the observation that given an energy function for a specific problem, it seems that we
can at most determine a range for each parameter that might result in better solutions.
Therefore, what we need to do is to find as many constraints for these parameters as
possible, and thus considerably reduce the number of trials before good parameters
are found. This method for determining parameters, slightly different from the original
trial-and-error technique, can be called “guided trail and error" method. Previous related
works include Aiyer’s eigenvalue analysis [4], Abe’s suppressing spurious states [5], and
Gee’s application of polytope concept [3]. All of these works, however, are solely based
This was partially supported by the Univ. of Piraeus Research Center.
I.P. Vlahavas and C.D. Spyropoulos (Eds.): SETN 2002, LNAI 2308, pp. 325–336, 2002.
c Springer-Verlag Berlin Heidelberg 2002
326
C. Douligeris and G. Feng
on the analysis of the TSP. Matsuda published a series of papers [6]-[7] to study this
problem. His basic idea is to analyze the stability of feasible and infeasible solutions
and thus obtain some constraints for the parameters. However, we notice that in his most
recent publication [7] even using the “optimal" network that he claimed can distinguish
optimal solutions most sharply among all the networks to solve the 20 × 20 assignment
problem, the percentage of experiments that the network converges to optimal solutions
is only 58%, which leaves much to be desired considering that assignment problems of
such small size are rather easy to be solved [8]-[9]. Moreover, the stability analysis in
[7] is based on the assumption that any neuron in the network can exactly converge to
“0" or “1", which is definitely not the case for a continuous HNN. Therefore there is a
need for a methodology that will draw upon this experience and present more practical
and efficient results.
The rest of the paper is organized as follows. We first discuss the relation of the
parameters for a specific class of Hopfield energy functions in Section 2. In Section 3,
the effectiveness of the convergence theorem and parameter selection method is demonstrated through a large number of tests on two combinatorial optimization problems.
Finally, Section 4 concludes this paper.
2 The Relation of Parameters for a Class of Energy Functions
In this Section, we first provide a general form for a class of Hopfield energy functions,
and then analyze what values should be chosen for the parameters so that the CHNN
can converge to valid solutions of high quality. As a result, a “guided trial-and-error"
technique is presented to determine the parameter values.
2.1 The General Form of a Class of Energy Functions
There exists a class of optimization problems [2], [7], [16], [17] that can be described
as follows:

n n


minimize
fxi Vxi



x i


n

subject to
Vxj = k
for any x 
(1)
j

n


Vyi = k
for any i 



y


Vxi ∈ {0, 1} for any x, i
where V = (Vxi ) ∈ {0, 1}n×n is a two dimensional variable matrix with each entry
being the output of a neuron, k(≤ n) is an integer representing the number of nonzero
variables in each row and each column, and generally k ∈ {1, 2}. Function fxi is a linear
combination of the variables in S = V \{Vxi } and is in the following form:
fxi (V ) = cxi +
Vyj ∈S
cxi
yj Vyj
(2)
Using Hopfield Networks to Solve Assignment Problem and N-Queen Problem
327
where cxi and cxi
yj are real numbers. Moreover, we assume that for any x = 1, 2, · · · , n,
i = 1, 2, · · · , n, fxi is positive at any point in the hypercube. More clearly, for any x and
i,
fxi (Ṽ ) ≥ 0,
where Ṽ is a n × n matrix with Ṽyj ∈ [0, 1] for any y and j.
When using a CHNN to solve the problem defined by (1), one can construct the
following energy function:



2
n
2
n
n
n
n n
C  

E= 
Vxj − k  +
Vyi − k  + D
fxi Vxi
2
x
y
x
j
i
i
(3)
where C and D, the Lagrange multipliers, are positive real numbers. The dynamics of
the corresponding CHNN can be obtained in terms of a relation duxi /dt = −∂E/∂Vxi ,
given by


n
n
duxi
= −C 
Vxj +
Vyi − 2k  − Dfxi ,
(4)
dt
y
j
in which uxi denotes the input of neuron xi.
In the rest of this Section, we will investigate ways to determine the values of parameters C and D. To ensure the effectiveness of our approach, we assume that the D
term describes the objective function of a specific optimization problem, rather than a
constraint. More clearly, there is an additional assumption for fxi : given a valid solution
Ṽ to (1), assume that for at least some x, i, y and j, fxi (Ṽ ) = fyj (Ṽ ). Throughout this
paper the notation V̂ is used to denote the output matrix when the CHNN converges to
a stable state.
2.2 The Relation between Parameters C and D
Let us first consider what values for C and D could possibly make the network converge
to a valid solution. It is clear that a valid solution Ṽ to problem (1) must have exactly k
“1"s in each row and each column of Ṽ . When the CHNN converges to a stable state,
however, the output of each neuron may not be exactly “1" or “0". Therefore, we cannot
expect the network to directly converge to a valid solution; instead, we should try to
“extract" a valid solution from the stable state if it is possible. For this reason, we start
from the following lemma.
Lemma 1. A necessary condition to guarantee that a valid solution can be extracted
from a stable state is that there are at least k nonzero entries in each row and each
column in the matrix V̂ .
It is clear that we can not extract a solution if the number of nonzero entries is less
than k in some row or column of V̂ .
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C. Douligeris and G. Feng
Lemma 2. A necessary condition to guarantee that there are at least k nonzero entries
in each row and each column in the matrix V̂ can be given by
n
V̂xj +
j
n
V̂yi > 0,
for any x, i.
(5)
y
n
n
Theorem 1. A necessary condition for j V̂xj + y V̂yi > 2αk, for any x, i, and
α ∈ [0, 1) can be given by
Dfmin < 2(1 − α)kC
(6)
where fmin = min{fxi |x, i = 1, 2, · · · , n}.
✷
Proof: Omitted for saving space.
Corollary 1. A necessary condition to guarantee that a valid solution can be extracted
from the stable state can be given by
Dfmin < 2kC.
(7)
Proof: In theorem 1, let α = 0. From lemma 2 and lemma 1, we get that the corollary
holds.
✷
To better understand the significance of theorem 1, let us assume that at a stable
state, there are exactly k nonzero entries in each row and each column of V̂ , and thus it
is clear that the value of α denotes to what degree a nonzero entry approaches “1". On
the other hand, given a specific value for C, D must be small enough to ensure that the
summation of a row and a column to be greater than 2αk. Theorem 1 is useful in such
cases where the obtained solution becomes satisfactory only if each nonzero entry very
closely approaches “1". Nonetheless, one should notice that (6) is neither a necessary
nor a sufficient condition to guarantee that a valid solution can be obtained.
Another two necessary conditions stronger than (6) and (7) are given in the following
theorem and the subsequent corollary.
Theorem 2. A necessary condition to guarantee that
(a) at a stable state there are in total at least nk (product of n and k) nonzero entries in
V̂ , and
n
n
(b) for any x, i, and α ∈ [0, 1), j V̂xj + y V̂yi > 2αk
is the following:
Df˜min < 2(1 − α)kC
˜
where fmin is the nkth minimum number among all fxi ’s.
(8)
✷
Proof: Omitted for saving space.
Corollary 2. A necessary condition to guarantee that a valid solution can be extracted
from the stable state can be given by
Df˜min < 2kC.
(9)
Proof: Omitted for saving space.
✷
Since f˜min ≥ fmin (due to the fact that fmin is the minimum number among all
fxi ’s), the necessary conditions (8) and (9) are stronger than (6) and (7), respectively, in
the sense that given specific values for C, k and α, the range of D becomes smaller.
Using Hopfield Networks to Solve Assignment Problem and N-Queen Problem
329
2.3 The Initial Value of D
In a Hopfield energy function, each energy term is usually assigned a parameter. For
instance, in (3) the two parameters C and D are used to combine two energy terms. In
fact it is obvious that in optimization their relative values are important, and one of them
is redundant. Thus, any one of the parameters can be arbitrarily assigned a value as a
reference to determine the values of other parameters. For this reason, from now on we
assume that parameter C has already been assigned a specific value, and our goal is to
find an appropriate value for D. We notice, however, that given a specific value for C,
the range of D determined by the strongest necessary condition in the last subsection,
i.e. (9), is still very large.
To find a good initial value for D, we need to consider what values of these parameters
could possibly result in a good solution since our final goal is to find a solution which
is not only valid, but of high quality as well. The difficulty in considering this point lies
in that one can hardly find any conditions that can guarantee that better solutions are
available. However, researchers in the area have obtained some practical experience [4],
[7]. It is well known, for example, that when the network converges to a stable state,
the closer the nonzero entry in V̂ approaches “1", possibly the better the solution. For
this reason, theorem 2 can be used to find a good initial value for D since only if the
summation of each row and each column approaches 2k is it possible that each nonzero
entry approaches “1". Thus the initial value of D can be given as follows:
Dinitial =
2(1 − α)k
C
f˜min
(10)
with α being a value close to 1.
For a specific problem, the value of f˜min can be obtained by estimation. If the
expression of fxi does not contain the second term in (2), namely the problem is an
assignment problem, then it is possible to find the exact value of f˜min by greedy search,
but it might be very time consuming.
2.4 The Fine-Tuning of D
Once an initial value for D is obtained, by gradually increasing or decreasing its current
value, it is not hard to find a good value of D that may result in good valid solutions.
To make a fine-tuning of D so that the quality of solutions can be further improved, we
find that there is a basic rule that may be helpful.
Theorem 3. Assume that at any stable state of the CHNN defined by (3) and (4), there
are exactly k nonzero entries in each row and each column of V̂ , and each nonzero is
approximately equal to the same value, then the ratio of the second energy term in (3)
to the first one is approximately in reverse proportion to the value of D.
Proof: Omitted for saving space.
✷
Theorem 3 indicates that a smaller value for D could possibly increase the proportion
of the energy in the second term of (3) to the total energy. Since the second term of (3)
is contributed from the objective function in (1), it is possible that a smaller value of D
330
C. Douligeris and G. Feng
could lead to a better solution. This is because the CHNN will continuously reduce the
energy function until it reaches a stable state. Thus, among the total decreased energy, the
more the energy coming from the objective term, most possibly the better the obtained
solution. However, the dilemma lies in that if the value of D is too small, the nonzero
entries can hardly approach “1". Therefore, one has to make many trials to find a balanced
value for D.
2.5 The Value of the Parameter Associated with the Binary Constraint
The CHNN treats any problem it solves as a linear or nonlinear programming problem
since the network only tries to find a local minimum of the energy function without
caring whether the output of each neuron has reached an integral value. Therefore, if the
output of each neuron can not approach a binary value when we try to solve an integer
programming problem, another energy term due to the binary constraint should be added
to the energy function. For a specific problem that can be formulated as (1), it is well
known that we have two ways to describe the binary constraint as an energy term:
n n
x
or
E0 =
n n
x
i
Vxi (1 − Vxi )
(11a)
i




n
n n
n
Vxi 
Vxj − (k − 1) +
Vxi 
Vyi − (k − 1) . (11b)
j=i
x
i
y=x
The first expression is effective because it becomes zero only when Vxi has a binary
value. The second expression becomes zero if Vxi = 0 or in the case where Vxi = 1
and the bracket part equals to zero. Therefore, it still has the effect to help the output
reach a binary value. We prefer the latter expression since the former one includes
self-interaction terms which may make our following analysis more complicated. By
assuming a positive parameter A is associated with E0 , the energy function (3) is now
modified as



2
n
2
n
n
n
n n
C


E= 
Vxj − k  +
Vyi − k  + D
fxi Vxi + AE0 .
2
x
y
x
j
i
i
(12)
There are three parameters in (12). Although at first glance it seems difficult to analyze
the mutual relations between these parameters, we find that (12) can be rewritten to be
the same form as (3):



2
n
2
n
n
n
n n
C  

E= 
Vxj − k  +
Vyi − k  + D
fˆxi Vxi (13)
2
x
y
x
j
i
i
in which
Using Hopfield Networks to Solve Assignment Problem and N-Queen Problem
331


n
n
Vxj +
Vyi − 2k + 2
fˆxi = fxi + γ 
(14)
j=i
y=x
and γ = A/D. Now, it is clear that our former analysis on the choice of the values of C
and D still applies to this case. Similar to the analysis in the last subsection, in order to
let more of the total decreased energy come from the objective term, from (14) we know
A/D should be as small as possible. In fact, it has been noted that the binary constraint
term will considerably increase the number of local minimums, therefore in most cases
A is set to 0. However, we also find that in some cases this term can not be omitted, even
if the value of A is very small [10].
2.6 The Guided Trial-and-Error Technique
By summarizing the above analysis, we give a formal statement on the “guided trialand-error" technique. When using a CHNN to solve a specific problem formulated as
(1), and assuming that its energy function is given by (13), the “guided trial-and-error"
technique can be described as follows:
1. Initialization.
(1) Arbitrarily choose a value for C.
(2) Choose a value for ∆t, which satisfies ∆t < 1/(βmax C)1
(3) Compute the initial value of D by (10) when α is set to a value close to 1, e.g.
α = 0.99.
(4) Let γ = 0.
2. Tuning. Keep tuning the parameters according to the following rules until satisfactory results are found:
(1) Decrease (increase) ∆t if the network converges too fast (slow) to a stable state
with unsatisfactory solutions.
(2) Increase D if the outputs of neurons do not approach binary values; if this has
no effect, then increase γ.
(3) Decrease γ if the outputs of neurons approach binary values, but with solutions
of low quality; if γ can not be decreased any more, then try to decrease D.
During the increase of D, inequality (9) should be strictly satisfied. On the other
hand, when the values of D and γ are small, the number of iterations should be large
enough to ensure that the network can reach a stable state.
3
Experimental Results
In this section, the “guided trial-and-error" technique is applied to solve the assignment
problem and the N-queen problem to test its effectiveness. In our following experiments,
the sigmoid function is used as the neuron’s I/O function:
uxi
1
1 + tanh
Vxi =
.
(15)
2
2u0
The maximum slope of this function is βmax = 1/(4u0 ).
1
∆t is the discretized dt in the dynamic equation, and βmax is the maximum slope of the
input-output equation of the neuron. For more details regarding this constraint, please see [11].
332
C. Douligeris and G. Feng
3.1 The Linear Assignment Problem
The linear assignment problem (AP) needs to assign n jobs to n persons in a one-to-one
way such that the total cost is minimized. This problem can be solved by converting it
to a minimum cost flow problem [8] or using the Hungarian method [9] in time O(n3 ),
where n is the problem size. In recent years, instead of using these ordinary linear
programming methods, many researchers try to develop parallel algorithms based on,
for example, Hopfield-type neural networks [12]-[14] and statistical physics [15] so as
to considerably reduce the time complexity. In this paper, we use the original CHNN to
solve the assignment problem by formulating it as (1), in which k = 1 and fxi equals to
a deterministic value, namely (2) becomes fxi = cxi .
Now let us show how to choose the parameter values using the “guided trial-anderror technique". First, we let C = 50, γ = 0, and the parameter u0 in the sigmoid
function be 0.01. From the convergence theorem proposed in [11] we know that the timestep ∆t should be less than 0.0008 to ensure that the network continuously converges
to a stable state. Thus, we let ∆t = 0.00075. To determine an initial value for the
parameter D, we assume that each cxi is a randomly generated real number and is
uniformly distributed in (0, 1]. Thus, from equation (10), the initial value of D is given
by Dinitial = (n2 + 1)/n ≈ n when α = 0.99 (we only consider the cases where n is
big enough such that 1/n is negligible).
We have tested a number of AP instances when the problem size varies from 20 to
200. In our simulations, all of the above parameter values are kept unchanged except that
we let D = n when n ≤ 100, and D = 100 when n > 100. For a specific problem size
n, ten problem instances are randomly generated such that cxi ’s are integers uniformly
distributed in [1, 1000]. For each problem instance, ten simulations are made when the
initial values of neurons are changed. Thus, we obtain 100 solutions for problem instances
with the size of n. Note that this is same method used in [7].
The initial value of a neuron is given by
uxi (t = 0) = −u0 ln(n − 1) + 0.01Rand
(16)
where Rand is a random number in [−1, 1]. Using this initialization method, the output
of each neuron approximately equals to 1/n when t = 0, and thus the summation of the
entries in a row or a column of the matrix V (t = 0) is approximately 1. The following rule
is used to terminate a simulation. For a specific problem instance, the Hungarian method
is used to obtain its optimal solution. In every 100 iterations of a specific simulation, the
total decrease of energy ∆E in one iteration is computed as follows:
∆E =
n
x,i
∆Exi =
n
x,i
2
C∆Vxi
−
1
∆uxi Vxi .
∆t
(17)
A simulation is terminated if |∆E| < !, or if the temporary solution is equal to the
optimal solution, whichever condition occurs first. ! is set to a value such that when
|∆E| < !, the network approximately approaches a stable state and the solution can
hardly be changed. In our experiments we let ! = n × 10−8 .
The statistical results are shown in Table 1. The first row denotes the problem size,
the second row is the number of simulations when optimal solutions are obtained, the
Using Hopfield Networks to Solve Assignment Problem and N-Queen Problem
333
third row is the average error, and the last row is the average iteration number for a single
AP instance. The average error is computed as follows:
10
10
1 Solij − Opti
× 100%
100 i j
Opti
where Solij is the solution obtained in the jth simulation for problem instance i, and
Opti is the optimal solution of problem instance i. From Table 1, we may find that when
the problem size is relatively small, the CHNN can obtain an optimal solution with a
very high probability, even though we have not put much effort to tune parameter D.
To show the effectiveness of our theoretical analysis, one may make a comparison of
our results with those shown in [7]. When using the same problem formulation, the
author of [7] can obtain an optimal solution with a probability of only 1% even when the
problem size is 20, and the probability was improved to 58% after the energy function
was reformulated.
We can further improve the performance of the CHNN by fine-tuning the value of D.
As stated in theorem 3, under certain assumptions the ratio of the objective energy term to
the constraint energy term is in reverse proportion to the value of D. Therefore, a smaller
value of D could possibly improve the quality of the solution. We have done the same
experiments as described above when D is set to a fixed value 15. The corresponding
results are shown in Table 2. To ensure that the results are comparable, the initial values
of the neurons for a specific simulation of Table 2 are set to the same as those for the
corresponding simulation of Table 1. From Table 2, one may notice that the probability
that an optimal solution can be obtained has been considerably increased. The expense
is that the number of iterations also increases.
Table 1. Results of the Assignment Problem when D = n (n ≥ 100) and D = 100 (n > 100)
problem size n
optimal solution convergence (%)
average error(%)
average iterations
20
100
0
166
40
100
0
255
60
90
2.59
646
80
85
5.79
657
100
52
17.63
932
200
78
8.91
1523
100
99
0.16
1898
200
86
4.41
5334
Table 2. Results of the Assignment Problem when D = 15
problem size n
optimal solution convergence (%)
average error(%)
average iterations
20
100
0
193
40
100
0
447
60
100
0
1577
80
95
2.60
1640
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C. Douligeris and G. Feng
3.2 The N-Queen Problem
The N-queen problem can be stated as follows: given a n by n chessboard, we need to
place n queens on the chessboard such that any two queens can not attack each other.
The constraints in this problem can be described more precisely as follows: in each row
or each column, there is exactly one queen; in each positive or negative diagonal, there
is also exactly one queen.
The N-queen problem can also be formulated as (1) when k = 1 and fxi are defined
as follows:
fxi =
Vx+j,i+j +
Vx−j,i−j .
(18)
j=0
1≤x+j≤n
1≤i+j≤n
j=0
1≤x−j≤n
1≤i−j≤n
However, one should note that the above expression violates the second assumption we
impose
on fxi in(1),
because for a valid solution Ṽ to this problem, we always have
fxi Ṽ = fyj Ṽ = 0 for any x, i, y and j. In fact, (18) is the expression of a
constraint, rather than an objective term in the general sense. For this reason, we can not
determine the value of parameter D by means of the theorems in Section 3. Instead, we
should choose a value so that the importance of the two constraint-terms (C term and D
term) can be balanced.
Although we can only arbitrarily choose an initial value for D, the values of other
parameters can be readily determined in accordance with the convergence theorem in
[11]. Similarly, we let γ = 0, u0 = 0.01, C = 50, ∆t = 0.00075. After a number of
experiments, we find that the network can obtain valid solutions with high probability
when D is around 15. Note that for this problem, a valid solution is also an “optimal"
solution.
The experimental results when D = 10 and D = 15 are shown in Tables 3 and
4, respectively. The problem size varies from 30 to 200. For a specific problem size n,
we have done 100 simulations, each with different initial values for the neurons. The
initialization method is same with the one used in the assignment problem except that
the amplitude of the random number is set to 0.005. Also, in every 100 iterations of
a specific simulation, the total decrease of the energy is computed according to (17),
and the temporary output matrix is processed using the recommended post-processing
method described in the previous Section. If |∆E| < 10−6 or a valid solution is obtained
then the simulation is terminated.
From Tables 3 and 4, one may notice that the CHNN can obtain valid solutions with
very high probability even when the problem size becomes large. Now let us make a
comparison with the results reported by other researchers. In recent years, many works
have been done on the use of neural networks (not necessarily the Hopfield networks)
to solve the N-queen problem. In [16], the highest probability that a valid solution can
be obtained is 73% when n = 30, and 90% when n = 50, and no results were given
when n > 50. In [18] and [19], the sizes of the studied problems are restricted to be 10
and 8, respectively. In [20], a dual-mode dynamics neural network is employed to solve
this problem and this method is demonstrated to outperform many other methods when
n ≤ 40, but no results are reported for problems of larger size. In [21], the self-feedback
controlled chaotic neural network is used to solve this problem of large size, and after
Using Hopfield Networks to Solve Assignment Problem and N-Queen Problem
335
the self-feedback factor is finely tuned, the success rate to solve the 200-queen problem
is around 98.8%. Thus, our result is comparable with the best result in the literature.
Table 3. Results of the N-queen Problem when D = 10
problem size n
valid solution convergence (%)
average iterations
30
98
131
50
100
119
80
98
172
100
99
204
200
92
358
100
100
133
200
97
252
Table 4. Results of the N-queen Problem when D = 15
problem size n
valid solution convergence (%)
average iterations
4
30
95
170
50
100
112
80
100
122
Conclusions
In this paper, we have addressed the dynamics of the continuous Hopfield neural networks. In particular, we investigated the mutual relation between the parameters in a
typical class of Hopfield energy functions, and thus proposed a “guided trial and error"
technique for determining the parameters. The effectiveness of this technique has been
demonstrated by a large number of computer simulations when the HNN is used to solve
the assignment problem and the N-queen problem. Compared with previous works, the
performance of the HNN has been considerably improved.
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