775 
A NEURAL-NETWORK SOLUTION TO THE CONCENTRATOR 
ASSIGNMENT PROBLEM 
Gene A. Tagliarini 
Edward W. Page 
Department of Computer Science, Clemson University, Clemson, SC 
29634-1906 
ABSTRACT 
Networks of simple analog processors having neuron-like properties have 
been employed to compute good solutions to a variety of optimization prob- 
lems. This paper presents a neural-net solution to a resource allocation prob- 
lem that arises in providing local access to the backbone of a wide-area com- 
munication network. The problem is described in terms of an energy function 
that can be mapped onto an analog computational network. Simulation results 
characterizing the performance of the neural computation are also presented. 
INTRODUCTION 
This paper presents a neural-network solution to a resource allocation 
problem that arises in providing access to the backbone of a communication 
network. In the field of operations research, this problem was first known as 
the warehouse location problem and heuristics for finding feasible, suboptimal 
solutions have been developed previously.2, a More recently it has been known 
as the multifacility location problem4 and as the concentrator assignment prob- 
lem. 
THE HOPFIELD NEURAL NETWORK MODEL 
The general structure of the Hopfield neural network models, 6, ? is illus- 
trated in Fig. 1. Neurons are modeled as amplifiers that have a sigmoid input/ 
output curve as shown in Fig. 2. Synapses are modeled by permitting the out- 
put of any neuron to be connected to the input of any other neuron. The 
strength of the synapse is modeled by a resistive connection between the output 
of a neuron and the input to another. The amplifiers provide integrative analog 
summation of the currents that result from the connections to other neurons as 
well as connection to external inputs. To model both excitatory and inhibitory 
synaptic links, each amplifier provides both a normal output V and an inverted 
output V. The normal outputs range between 0 and 1 while the inverting am- 
plifier produces corresponding values between 0 and -1. The synaptic link be- 
tween the output of one amplifier and the input of another is defined by a 
conductance Tij which connects one of the outputs of amplifier j to the input of 
amplifier i. In the Hopfield model, the connection between neurons i and j is 
made with a resistor having a value Rij= 1/Ti . To provide an excitatory synap- 
tic connection (positive Ti), the resistor is connected to the normal output of 
This research was supported by the U.S. Army Strategic Defense Command. 
American Institute of Physics 1988 
776 
I1 
I2 
I3 I4 
inputs 
T42 
! 
V 1 V 2 V 3 V 4 outputs 
Fig. 1. Schematic for a simplified 
Hopfield network with four neurons. 
1 
I 
-u 0 +u 
Fig. 2. Amplifier input/output 
relationship 
amplifier j. To provide an inhibitory connection (negative Tij), the resistor is 
connected to the inverted output of amplifier j. The connections among the 
neurons are defined by a matrix T consisting of the conductances Tij. Hop- 
field has shown that a symmetric T matrix (Ti = Ti) whose diagonal entries 
are all zeros, causes convergence to a stable state in which the output of each 
amplifier is either 0 or 1. Additionally, when the amplifiers are operated in the 
high-gain mode, the stable states of a network of n neurons correspond to the 
local minima of the quantity 
n n n 
E= (-1/2)  Z TijViV j - Z Vii i (1) 
i=l j=l i=l 
where Vi is the output of the i th neuron and Ii is the externally supplied input 
to the i th neuron. Hopfield refers to E as the computational energy of the sys- 
tem. 
THE CONCENTRATOR ASSIGNMENT PROBLEM 
Consider a collection of n sites that are to be connected to m concentra- 
tors as illustrated in Fig. 3(a). The sites are indicated by the shaded circles 
and the concentrators are indicated by squares. The problem is to find an 
assignment of sites to concentrators that minimizes the total cost of the assign- 
ment and does not exceed the capacity of any concentrator. The constraints 
that must be met can be summarized as follows: 
a) Each site i ( i = 1, 2,..., n ) is connected to exactly one concentrator; 
and 
777 
b) Each concentrator j ( j = 1, 2,..., m ) is connected to no more than kj 
sites (where k i is the capacity of concentrator j). 
Figure 3(b) illustrates a possible solution to the problem represented in Fig. 
3(a). 
[] 
O 
O 
  
[] Concentrators  Sites 
(a). Site/concentrator map (b). Possible assignment 
Fig. 3. Example concentrator assignment problem 
If the cost of assigning site i to concentrator j is cii, then the total cost of 
a particular assignment is 
n m 
total cost =   xij cij (2) 
i:l j:l 
where xij = 1 only if we actually decide to assign site i to concentrator j and is 0 
otherwise. There are m n possible assignments of sites to concentrators that 
satisfy constraint a). Exhaustive search techniques are therefore impractical 
except for relatively small values of m and n. 
THE NEURAL NETWORK SOLUTION 
This problem is amenable to solution using the Hopfield neural network 
model. The Hopfield model is used to represent a matrix of possible assign- 
ments of sites to concentrators as illustrated in Fig. 4. Each square corresponds 
778 
CONCENTRATORS 
1 2 j rn 
2 
SITES . i ,,, ,,, i1,,, ,,,,' [[ The darkly shaded neu- 
ron corresponds to the 
hypothesis that site i 
[ [ should be assigned to 
. n 1! [llJ concentrator j. 
SLACK ' in+2 , ! g , 
Fig. 4. Concentrator assignment array 
to a neuron and a neuron in row i and column j of the upper n rows of the 
array represents the hypothesis that site i should be connected to concentrator 
j. If the neuron in row i and column j is on, then site i should be assigned to 
concentrator j; if it is off, site i should not be assigned to concentrator j. 
The neurons in the lower sub-array, indicated as "SLACK", are used to 
implement individual concentrator capacity constraints. The number of slack 
neurons in a column should equal the capacity (expressed as the number sites 
which can be accommodated) of the corresponding concentrator. While it is 
not necessary to assume that the concentrators have equal capacities, it was 
assumed here that they did and that their cumulative capacity is greater than or 
equal to the number of sites. 
To enable the neurons in the network illustrated above to compute solu- 
tions to the concentrator problem, the network must realize an energy function 
in which the lowest energy states correspond to the least cost assignments. The 
energy function must therefore favor states which satisfy constraints a) and b) 
above as well as states that correspond to a minimum cost assignment. The 
energy function is implemented in terms of connection strengths between neu- 
rons. The following section details the construction of an appropriate energy 
function. 
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THE ENERGY FUNCTION 
Consider the following energy equation: 
E: A  ( Vi j I ) 2  n+k: 
- + B (JVij - kj )2 
i:l j:l j:l i=l 
(3; 
m n+k. 
+ C   Jvij (1-Vij) 
j:l i:l 
where Vii is the output of the amplifier in row i and column j of the neuron 
matrix, m and n are the number of concentrators and the number of sites 
respectively, and kj is the capacity of concentrator j. 
The first term will be minimum when the sum of the outputs in each row 
of neurons associated with a site equals one. Notice that this term influences 
only those rows of neurons which correspond to sites; no term is used to coerce 
the rows of slack neurons into a particular state. 
The second term of the equation will be minimum when the sum of the 
outputs in each column equals the capacity kj of the corresponding concentra- 
tor. The presence of the k slack neurons in each column allows this term to 
enforce the concentrator capacity restrictions. The effect of this term upon the 
upper sub-array of neurons (those which correspond to site assignments) is 
that no more than k sites will be assigned to concentrator j. The number of 
neurons to be turned on in column j is kj; consequently, the number of neu- 
rons turned on in column j of the assignment sub-array will be less than or 
equal to kj 
The third term causes the energy function to favor the "zero" and "one" 
states of the individual neurons by being minimum when all neurons are in one 
or the other of these states. This term influences all neurons in the network. 
In summary, the first term enforces constraint a) and the second term 
enforces constraint b) above. The third term guarantees that a choice is actu- 
ally made; it assures that each neuron in the matrix will assume a final state 
near zero or one corresponding to the xij term of the cost equation (Eq. 2). 
After some algebraic re-arrangement, Eq. 3 can be written in the form of 
Eq. 1 where 
T ij,kl = { A * 8(i,k) * (1-8(j,1)) + B * 8(j,1) * (1-8(i,k)), if i<n and k<n 
C * 8(j,1) * (1-8(i,k)), if i>n or k>n. (4) 
Here quadruple subscripts are used for the entries in the matrix T. Each entry 
indicates the strength of the connection between the neuron in row i and col- 
umn j and the neuron in row k and column 1 of the neuron matrix. The func- 
tion delta is given by 
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1, ifi:j (5) 
8( i , j ) = 0, otherwise. 
The A and B terms specify inhibitions within a row or a column of the upper 
sub-array and the C term provides the column inhibitions required for the 
neurons in the sub-array of slack neurons. 
Equation 3 specifies the form of a solution but it does not include a term 
that will cause the network to favor minimum cost assignments. To complete 
the formulation, the following term is added to each Tij,n: 
D* 8(j,1) * (1-8(i,k)) 
(cost[i,j] + cost[k, 1]) 
where cost[ i, j ] is the cost of assigning site i to concentrator j. The effect of 
this term is to reduce the inhibitions among the neurons that correspond to low 
cost assignments. The sum of the costs of assigning both site i to concentrator j 
and site k to concentrator 1 was used in order to maintain the symmetry of T. 
The external input currents were derived from the energy equation (Eq.3) 
and are given by 
2*kj,ifi<n 
I ij = 2 * kj- 1, otherwise. (6) 
This exemplifies a technique for combining external input currents which arise 
from combinations of certain basic types of constraints. 
AN EXAMPLE 
The neural network solution for a concentrator assignment problem con- 
sisting of twelve sites and five concentrators was simulated. All sites and con- 
centrators were located within the unit square on a randomly generated map. 
For this problem, it was assumed that no more than three sites could be 
assigned to a concentrator. The assignment cost matrix and a typical assign- 
ment resulting from the simulation are shown in Fig. 5. It is interesting to 
notice that the network proposed an assignment which made no use of concen- 
trator 2. 
Because the capacity of each concentrator kj was assumed to be three 
sites, the external input current for each neuron in the upper sub-array was 
Iij= 6 
while in the sub-array of slack neurons it was 
Iij= 5. 
The other parameter values used in the simulation were 
A=B=C=-2 
and 
D=0.1. 
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SITES 
A 
B 
C 
D 
E 
F 
G 
H 
J 
K 
L 
.47 
.72 
.95 
.88 
.31 
.25 
.17 
CONCENTRATORS 
2 3 4 
28 .55 
75 
71 
78 
62 
.81 
39 
38 
56 
51 76 46 
39 77 41 
81 
67 
84 
33 
60 1 
54 52 
76 .66 
05 .71 
5 
.46 
.63 
.92 
.82 
. 
. 
 
.56 
.51 
.48 
.38 
.18 
Fig. 5. The concentrator assignment cost matrix with choices circled. 
Since this choice of parameters results in a T matrix that is symmetric 
and whose diagonal entries are all zeros, the network will converge to the 
minima of Eq. 3. Furthermore, inclusion of the term which is weighted by the 
parameter D causes the network to favor minimum cost assignments. 
To evaluate the performance of the simulated network, an exhaustive 
search of all solutions to the problem was conducted using a backtracking algo- 
rithm. A frequency distribution of the solution costs associated with the assign- 
ments generated by the exhaustive search is shown in Fig. 6. For comparison, 
a histogram of the results of one hundred consecutive runs of the neural-net 
simulation is shown in Fig. 7. Although the neural-net simulation did not find 
a global minimum, ninety-two of the one hundred assignments which it did 
find were among the best 0.01% of all solutions and the remaining eight were 
among the best 0.3%. 
CONCLUSION 
Neural networks can be used to find good, though not necessarily opti- 
mal, solutions to combinatorial optimization problems like the concentrator 
782 
Frequency 
4000000 
3500000 
3000000 
2500000 
2000000 
1500000 
1000000 
500000 
0 
3.2 4.2 
Cost 
5.2 6.2 7.2 8.2 
Frequency 
25 
20 
15 
10 
5 
Cost 
3.2 3.4 3.6 3.8 4.0 4.2 
Fig. 6. Distribution of assignment 
costs resulting from an exhaustive 
search of all possible solutions. 
Fig. 7. Distribution of assignment 
costs resulting from 100 consecu- 
tive executions of the neural net 
simulation. 
assignment problem. In order to use a neural network to solve such problems, 
it is necessary to be able to represent a solution to the problem as a state of the 
network. Here the concentrator assignment problem was successfully mapped 
onto a Hopfield network by associating each neuron with the hypothesis that a 
given site should be assigned to a particular concentrator. An energy function 
was constructed to determine the connections that were needed and the result- 
ing neural network was simulated. 
While the neural network solution to the concentrator assignment prob- 
lem did not find a globally minimum cost assignment, it very effectively re- 
jected poor solutions. The network was even able to suggest assignments which 
would allow concentrators to be removed from the communication network. 
REFERENCES 
1. A. S. Tanenbaum, Computer Networks (Prentice-Hall: Englewood Cliffs, 
New Jersey, 1981), p. 83. 
2. E. Feldman, F. A. Lehner and T. L. Ray, Manag. Sci. V12, 670 (1966). 
3. A. Kuehn and M. Hamburger, Manag. Sci. V9, 643 (1966). 
4. T. Aykin and A. J. G. Babu, J. of the Oper. Res. Soc. V38, N3, 241 (1987). 
5. J. J. Hopfield, Proc. Natl. Acad. Sci. U.S. A., V79, 2554 (1982). 
6. J. J. Hopfield and D. W. Tank, Bio. Cyber. V52, 141 (1985). 
7. D. W. Tank and J. J. Hopfield, IEEE Trans. on Cir. and Sys. CAS-33, N5, 
533 (1986). 
