Scope and Purpose-This paper shows how a minor modilication to the Hungarian algorithm for the linear assignment problem can drastically reduce execution ...
Computers Opns Rrs. Vol. 17. No. I. pp. 95-96. 1990 Prmted 10 Great Brttam All rights reserved
SPEEDING
UP THE HUNGARIAN M. B.
Department
of Operational
0305-0548 90 s3 00 + 0 00 t. 1990 Pcrgamon Press plc
Cop>rtpht
Research and Operations
(Received
ALGORITHM
WRIGHT*
Management, England
University
April 1988; rerised January
of Lancaster.
Lancaster
LA1 4YX.
19891
Scope and Purpose-This paper shows how a minor modilication to the Hungarian algorithm for the linear assignment problem can drastically reduce execution times. This may be useful not only to academic studies which compare the performances of different algorithms. but also for the many implementations which use the Hungarian algorithm. Abstract-A simple modification execution time by up to 90”/0.
to the “Hungarian”
algorithm
for the linear assignment problem reduces
INTRODUCTION
The linear assignment problem is a standard OR problem. Given an N by N cost matrix. N entries must be chosen, exactly one from each row and one from each column, in such a way as to minimize the total
cost of the selection.
The algorithm Hungarian
most
algorithm
widely [I],
used for the solution
although
other
of this problem
algorithms
is generally
have since been devised
known
which
as the
are claimed
to be faster [2, 3-J. However.
execution
time is detcrmincd
code used. For example,
Jonker
codes used for the Hungarian assesses the effect of another In practical anyway, result
not only by the algorithm
and Volgenant algorithm
minor
can cause large reductions that all the algorithms
so that there is no need for further
assignments
very quickly,
as in an interactive
THE
Let c[i, j] whole total
improvement.
be the initial successive
cost matrix.
alterations
amount
subtracted
from
to the original time. This paper
The standard
to this matrix,
column
and computer
codes used are quick
there are occasions
or where
a long
to use as quick
where a
series of successive
a method
as possible.
MODIFICATION
codes for the Hungarian
by means
So, if u[i] is the total
rows and columns.
modifications in execution
However,
system,
needs to be made [S], when it is important
by making
itself but also by the computer
that minor
modification.
terms, it may be argued
is required
[4] showed
amount
j, the altered
of subtractions
subtracted
matrix
is given
algorithm from
from
row
proceed
or additions i, and
c[j]
to
is the
by:
a[i, j] = c[i, j] - u[i] - r[j]. For
most
linear
assignment
by far the most time-consuming The modification statements
that
the matrix
is therefore
problems,
the arrays
tested
would
does not
have used a[i, j] now saved, but the other
Two codes for the Hungarian modification, and the second
cm
altered.
frequently
These alterations
during
the
are generally
part of the algorithm.
in this paper
matrix
at all. So all computer
use (c[i. j] - u[i] - cfj]).
The time spent altering
algorithm (“NEW”)
use an altered
parts of the process take slightly
EXPERIMENTAL
‘M.
u and c are changed
and so a[i, j] must also be frequently
course of the algorithm,
longer.
TESTS
were compared, the lirst (“OLD”) without the above with it. Both codes were otherwise fairly eflicient,
B. Wright is a Lecturer in operational research at the University of Lancaster, England. Before that he was with the operational research group at the Headquarters of British Rail. His main research interests are the practical uses of heuristics and the application of OR to transport. health services and sport. 17:1-G
95
96
M. 8. WRIGHT Tdblr I
Computer r~r~utmn times txcl
for “OLD”
OLD r
n
Mean
and “NEW”
NEW SD
Mean
NEW OLD SD
Mean
SD
50
1.88
0.31
0.82
0.16
043
0.04
100
100
8.52
0.87
3.53
0.53
0.41
0.04
100
150
18.49
2.61
7.88
1.05
0.43
0.04
100
200
30.78
3.96
14.77
I.85
0.48
0.05
100
250
46.02
4.51
27.96
4.78
0.61
0.08
IOIl
300
63.37
7.51
47.42
4.75
0.76
0.09
50
3.18
0.85
I.00
0.23
0.32
0.03
1.000
100
21.03
3.41
4.66
0.84
0.22
0.02
l.ooo
I50
59.09
8.47
2.0X
0.19
0.02
l.OOG l.CQO
ml
I27 32
15.4.i
II.15 ,? 1') __.__
3.01
0.18
0.02
250
215.02
21.24
34.58
4.93
0.16
0.01
1.000
300
32963
25.43
51.68
6.65
0.16
0.02
loo
Ian0
10.000
50
4.25
0.83
1.23
0.23
0.29
0.02
5.03
5.78
0.85
O.IR
0.01
I596
14.36
2.17
0.14
001
235 27
28.73
26.99
3.88
0.11
0.01
250
432.56
46.42
44.43
5.93
0.10
0.01
300
742.03
50.x1
67.44
5.51
0.09
0.01
lO.OOfI
loo
3190
lO.OOG
150
IO5 76
IO.OMl
200
10.000 IO.MXI
incorporating pointers and the first two of the improvements described by Jonker and Volkenant [4]. Six values of n, the number of rows in the cost matrix, and three values of r, the range of values in the matrix, were used, giving I8 combinations. For each combination 20 runs of OLD and NEW were made. For each of the 360 runs, a cost matrix was created by choosing a random integer between 0 and r for each matrix element c[i, j]. OLD and NEW were then run successively on this matrix and the execution times were recorded. The computer used was an Amstrad PC1512, and the language used was Pascal, using Borland’s TURBO system, version 4. The results are summarized in Table I. COMMENTS
These results show that NEW is substantially quicker than OLD, especially for large values of n and r, where the improvement is as high as 90%. Regression analysis shows that the execution time for NEW is O(n2~22ro~“), compared with O(n2.48r0.38) for OLD. In addition, when each value of r is considered separately, the power of n for NEW is almost constant (between 2.20 and 2.23), whereas for OLD it ranges from 1.95 (for r = 100, when virtually no matrix alteration is ever necessary) to 2.88 (for r = 10,000). So NEW is not only quicker than OLD, it is also less sensitive to variation in the size of the matrix and the range of costs within it. REFERENCES I. H. W. Kuhn, The Hungarian method for the assignment problem. Naval Rcs. Loyisr. Q. 2, 83-97 (1955). 2. M. S. Hung and W. 0. Rom, Solving the assignment problem by relaxation. Opns Res. 28, 969-982 (1980). 3. R. Jonker and A. Volgenant. A shortest augmenting path algorithm (or dense and sparse-linear assignment problems. Compuring 38.325340 (1987). 4. R. Jonker and A. Volgenant. Improving the Hungarian assignment algorithm. Opl Res. Lett. 5. 171-175 (1986). 5. M. B. Wright. Applying stochasticalgorithms to a locomotivescheduling problem. J. Opl Res. Soc.40.187-192 (1989).
