Table Of ContentLectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and H. P. Künzi
184
Rai ner E. Bu rkard
Ulrich Derigs
In cooperation with
T. Bönniger G. Katzakidis
Assignment and
Matching Problems:
Solution Methods
with FORTRAN-Programs
Springer-Verlag Berlin Heidelberg GmbH
Editorial Board
H. Albach A. V. Balakrishnan M. Beckmann (Managing Editor)
P. Dhrymes J. Green W. Hildenbrand W. Krelle
H. P. Künzi (Managing Editor) K. Ritter R. Sato H. Schelbert
P. Schönfeld
Managing Editors
Prof. Dr. M. Beckmann Prol. Dr. H. P. Künzi
Brown University Universität Zürich
Providence, RI 02912/USA CH-8090 Zürich/Schweiz
Authors
Rainer E. Burkard
Mathematisches Institut
Universität zu Köln
Weyertal86
5000 Köln 41
Federal Republic of Germany
Ulrich Derigs
Industrieseminar
Universität zu Köln
Albertus-Magnus-Platz
5000 Köln 41
Federal Republic 01 Germany
AMS Subject Classilications (1980): 65K05, 90BlO, 90C08,
90C09,90C35
ISBN 978-3-540-10267-0 ISBN 978-3-642-51576-7 (eBook)
DOI 10.1007/978-3-642-51576-7
Thl$ work 15 subject 10 copynght. All rights are reserved, whether the whole or part
of the material is concerned, specifically those of translation, reprinting, re-use
01 illustrations, broadcasting, reproduction by photocopying machine orßimilar
mearos. and storage in data banks. Under § 54 01 the German COPYright Law where
copies are made for other than private use, a fee is payable 10 the publisher, the
amount 01 the fee 10 be determmed by agreement with the publisher.
9 by Sprirger-Verlaq Berhn Heidelberg 1980
Originally published by Springer-Verlag Berlin Heidelberg in 1980.
2141/3140-543210
Assignment and matching problems belang to those cornbinatorial opti
mization problems which are weIl understood in theory and have many
applications in practice. Since a research group qf the Mathematical
Institute in Cologne has worked in this field for several years, we
feIt that a publication of the developed procedures and algorithms
would be useful, both for further research and academic tu': tion as
weIl as for applications in practice.
This book covers linear assignment problems wi th SUfi and bottleneck
objective functions, cardinality matching problems, perfect matching
problems with SUffi and bottleneck objective functions, the Chinese
postman problem and optimal as weIl as heuristic algorithms for the
quadratic assignment problem. Every problem will be first described
in short and then a FORTRAN-routine is given for it, which is docu
mented in detail and illustrated by test exarnples.
Many helped us through the years to develop efficient codes for the
above rnentioned problems. Our special thanks go to T. Bönniger and
G. Katzakidis who helped us in compiling the final vers ions and by
performing extensive computational tests of the computer programs.
The development of these computer codes would not have been possible
without the excellent facilities of the Computer Center of the Uni
versity of Cologne. All programs were thoroughly tested on the CDC
CYBER 76 installation of this computer center. Independent test runs
were perfarmed on the IBM 4331 installation in the Institut for
Ökonometrie and Operations Research, University of Bonn. We are very
indepted to Prof. B. Korte and Dr. Grätschei, Bann, for their assist
ance in having these test runs perforrned. Further we thank Prof. Th.
Meis, Köln, for the permission to use his sorting routine SSORT as
a subroutine in our programs .
Various drafts of this manuscript were type.d by Miss B. Hünten and
Mrs. E. Rath who have hecome cryptologists by deciphering our hand
writings. Many thanks to thern!
The authors are grateful for any comment and report on the usage of
the \iescribed programs . May they turn out to be useful!
COlogne, August 1980 R.E.B. and U.D.
TABLE OF CONTRNTS
Page
Introduction
1. The Linear Sum Assignment Problem
2. The Linear Bottleneck Assignment Problem 16
3. The Cardinality Matching Problem 25
4. The Sum Matehing Problem 37
5. The Bottleneck Matehing Problem 60
6. The Chinese Postman Problem 72
7. Quadratic Assignrnent Problems 99
8. QAP Heuristic 1: The method of increasing 120
degree of freedorn
9. QAP Heuristic 2: Cutting plane and exchange 127
method
10. General Subroutines 146
In troduction
In this book we present a number of optimization technigues and compu
ter codes which have been designed for assignment and rnatching problems.
We consider as weIl the classical SUffi objective as the bottleneck objec
tive and for the matching problem the pure cardinality case, tao.
Every problem resp. code is described in a special chapter which is
self-contained.Thus the reader will be able tci apply a special routine
without reading the whole book.
All chapters are identically structured following the scheme
(a) Formulation of the problem
(bi The algorithm
(c) References
(d) Description of the program
(e) Program listings and sample output
In (a) we introduce the specific problem and state same alternative for
mulations thereof.ln (b) the basic ideas of the implemented algorithm
are roughly described.The mathematical background is only sketched.The
reader can find detai ls in the references, lis ted in section (c) of ev
ery chapter.ln (d) we give directions for using the code.The reader
will find he re informations on the structure of the program and a de
scription of the INPUT and OUTPUT parameters with the respective dimen
sion requirements .The total storage requirement and the running time for
medium sized problems are given.The running time was determined either
by testing a great variety of randomly genera ted problems or by standard
problems from the literature on a CDC CYBER 76. In (e) the reader finds
the complete list of the FORTRAN IV subroutines . These subroutines start
.wi th a cornment where all parameters, arrays and variables are defined.
A MAIN-program is added to describe the use of the suhroutine and to
enable the solution of the given sample problem.
All computer codes in this book are wri tten in ANSI FORTR1\N and they are
therefore machine independent.The codes were developed and extensively
tested on the COC CYBER 76 of the Computer Center of the University of
Cologne.Further tests were run on an IBM 4331 of the Sonderforschungs
bereich 21 at the University of Bann.
1. The Linear Surn Assignment Problem
a) Formulation of the problem
Let the finite set N {1,2, ... ,n}, nE:JN, be given. A bijective
(one to one) mapping tp: N -+ N is called apermutation of the set N.
The set of all permutations <p of the set N is denoted by f n' Let
an integer/real (nxn) matrix C = (cij) of cast elements cij be given.
Then we can associate with every permutation tp E: fn the costs
The Linear Sum Assignment Problem (LSAP) is to find apermutation
tp E: fn with minimal costs,i.e.
(1. 1)
A graph theoretic interpretation of LSAP can be given as follows.
For <pE!n the graph M<p is a partial graph of the complete bipartite
graph Kn,n such that every node is incident to exactly one edge cf
Mtp' Such graphs are called perfeet matchings. So LSAP is equivalent
to the problem of finding a minimal weighted perfeet matching in an
edge-weighted bipartite graph. (An algorithm for solving perfect
matching problems in general (not necessarily bipartite) graphs is
presented in section 4) •
A cornmonly used formulation of L5AP is the following
min r r c .. x .. subject to
iEN JEN ~J ~J
(1.2) r x .. 1, for iEN
JEN lJ
(1.3) l: x .. 1, for JEN
iEN ~J
(1.4) xij .:: 0, for i E N, j E N.
The polyhedron PA defined by (1.2) - (1.4) is called the assignment
poZytope. It is weIl known that PA has only vertices with (0,1) compo
nents 'tlhich in turn correspond to the permutations cf fn·
b) The algorithm
The fOllowing algorithms for solving LSAP was proposed by TOMIZAWA
[3] and improved by DORHOUT [2].
During the algorithm so calledk-problems (Pk) are solved for
k == 1,2, ... ,n. Let f'n(K) denote the set of all one to one mappings
4>k: K ~ N for K = (1 ,2, ••• ,k) and kEN.
Then the k-problem is defined by:
(Pk) min L ci,w (i)
4>Efn (K) iEK
The solution of (P1) is obvious. Starting with a solution of (Pk)
for 1 2 k~n-l the solution ~ of (Pk+1) is constructed by means of
a shortest augmenting path. Here an augmenting path is a sequence
ß~ of mutually distinct matrix entries
with j1 = 4>(i1) for 1 == 1, ... , r
jr+1 t <j>(K).
Then we find
\)J(k+1)= j1
for 1 = 1,2, ... Ir
for iEK'{ i1 I 1=1, ... ,r} .
Figure 1 shows such an augmenting path.
Here the circled matrix entries represent the mapping Wk E f n (K) .
e k+1
We define c(4) 44» := L C
i=l i,~di) .
•
Lr
•
•
•
•
k+l
Figure 1: Augmenting path
Let Dk be the set of all augmenting paths with respect to <j) E :f n (Kl •
Then .6.\jJ EDk is called shartest auqmenting path if c (tP ® tHO) 1s minimal.
Such a shartest augrnenting path can be faund using a modification
of the shortest path algorithm of DIJKSTRA [1]. With an optimal solu
tion ~k for (Pk) and a shartest augmenting path .6.\jJ we obtain
l\J = <j) G) LI<j) as optimal solution for (Pk+11.
After the augmentation the cast coefficients of the matrix C =(Cij)
have to be transformed. This is done by rneans of a vector
y E m2n whose elements can be interpreted as dual variables. After
termination of the algorithm the vector y represents the associated
dual solution i.e.
for i, JEN
Computational experiments show that the running time of the algorithm
can be decreased by means of astart heuristic in which an initial
set K ~ N and a mapping l.Ok E::fn (K) are determined in the following way:
Par t;!"lery i € N we calculate
Si := min C ••
JEN 1)
Then ~e calculate
t). := min (ci)·-si) for every JEN
iEN
Every entry (i, j) wi th cij -si -tj==O is called admissible. Inspecting
the rows first and the columns thereafter we deterrnine a maximal set
in
K ::: N and an associated mapping <llk E (k) such that (i, <llk (i»
is ad.missible for i E K.
Then the shortest augmenting path rnethod is star ted with respect
to <llk'
c) References
[1] Dijkstra, E.W.: A note on two problems in connection with
graphs. Numerische Mathematik.l (1959),
269-271.
[2] Dorhout, B.: Het lineaire toewij zingsproblem, vergelijking
van algoritrnen. Report BN 21, Stichting Mathe
matisches Centrum, Amsterdam, 1973.
(3} Tomizawa, N.: On some techniques useful for solution of
transportation network problems.
Networks (1972), 179-194.
d) Description of the program
The program for solving the linear sum assignment problem consists
of a MAIN-prograrn for INPUT and OUTPUT operations and a single sub
routine LSAP.
In MAIN all arrays and vectors are specified in DIMENSION statements
for the test problem given below. They rnay be altered for the parti-
cular problems.
INPUT: As input data the dimension n of the problem, the cost
matrix C and a parameter SUP are required. The cast coeffi
eients cij should be nonnegative integers. The matrix
C = (Cij) has to be read row by row, i.e.
and is then stored on a vector C(I) of length n2•
The parameter SUP has to be a sufficiently large number,
i.e.
SUP ~ 1: 1: ciJ. •
iEN JEN
OUTPUT: The MAIN program prints out the name of the problem and
the cost matrix C = (cij). The optimal permutation <jl is
stored on the vector SPALTE(I) and the optimal assignment
is pr int ed out in the form
<jl(i) , i = 1,2, ... ,n.
Then the value 1: of the optimal permutation
iEN
is given.
The associated dual solution is stored on the two vectors
YS(I) and YT(J) of length n, but is not printed out here.
The program requires a storage capacity of n2 + Sn.
Computational tests with randomly generated problems on a
CDC CYBER 76 showed that the expected (= mean)
CPU-time is
for n 100 Rf 0.1 sec
and
for n 200 .. 0.5 sec.
