Assignment Problems

y

Eranda C ela

Abstract

Assignment problems arise in dierent situations where we have to nd an

optimal way to assign n ob jects to m other ob jects in an injective fashion

Dep ending on the ob jective we want to optimize we obtain dierent problems

ranging from linear assignment problems to quadratic and higher dimensional

assignment problems

The assignment problems are a well studied topic in combinatorial opti

mization These problems nd numerous application in pro duction planning

telecommunication VLSI design economics etc

We intro duce the basic problems classied into three groups linear as

signment problems three and higher dimensional assignment problems and

quadratic assignment problems and problems related to it For each group of

problems we mention some applications show some basic prop erties and de

scrib e briey some of the most successful algorithms used to solve these prob

lems

Intro duction

Assignment problems deal with the question how to assign n ob jects to m other

ob jects in an injective fashion in the b est p ossible way An assignment problem

is completely sp ecied by its two comp onents the assignments which represent

the underlying combinatorial structure and the ob jective function to b e optimized

which mo dels the b est p ossible way

In the classical assignment problem one has m n and most of the problems with

m n can b e transformed or are strongly related to analogous problems with m n

Therefore we will consider m n through the rest of this chapter unless otherwise

sp ecied

This research has b een supp orted by the Sp ezialforschun gsb erei ch F Optimierung und

Kontrolle Pro jektb ereich Diskrete Optimierung

y

Technical University Graz Institute of Mathematics Steyrergasse A Graz Austria

Email celaoptmathtugrazacat

From the mathematical p oint of view an assignment is a bijective mapping of a nite

set N f ng into itself ie a permutation assigning some j i to

each i N The set of all p ermutations assignments of n items will b e denoted by

S and has n elements Every p ermutation of the set N f ng corresp onds

n

uniquely to a permutation matrix X x with x for j i and x

ij ij ij

for j i Thus a p ermutation matrix X x can b e dened as a matrix which

ij

fullls the following conditions socalled assignment constraints

n

X

x for all j n

ij

i=1

n

X

x for all i n

ij

j =1

x f g for all i j n

ij

By replacing the conditions x f g by x in we get a doubly stochastic

ij ij

matrix The set of all doubly sto chastic matrices forms the assignment polytope

P Due to a famous result of Birkho the assignment p olytop e P is the

A A

convex hull of all assignments or equivalently every doubly sto chastic matrix can

b e written as convex combination of p ermutation matrices

The concept of an assignment is strongly related to another well known concept

in graph theory and in combinatorial optimization matching in bipartite graphs

A bipartite graph G is a triple V W E where the vertex sets V and W have no

vertices in common and the edge set E is a set of pairs i j where i V and j W

A subset M of E is called a matching if every vertex of G is incident with at most

one edge from M The cardinality of M is called cardinality of the matching The

maximum matching problem asks for a matching with as many edges as p ossible A

matching M is called a perfect matching if every vertex of G is incident with exactly

one edge from M Evidently every p erfect matching is a maximum matching A

p erfect matching in a bipartite graph G V W E with V fv v v g

1 2 n

W fw w w g can b e represented by a p ermutation of f ng such

1 2 n M

that i j if and only if v w M Hence a p erfect matching in a bipartite

M i j

graph is an assignment

p

Hop croft and Karp gave an O jE j jV jalgorithm which constructs a p erfect

p

jV jjE j algorithm for matching if it exists Even and Tarjan gave an O

the maximum ow problem on unit capacity simple networks algorithm which can

also b e applied to nd a matching of maximum cardinality in a bipartite graph

p

15

Alt et al gave an O jV j jE j log jV j implementation for the Hop croft

Karp algorithm Based on ideas similar to those in Hop croft and Karp a

fast randomized MonteCarlo algorithm is given by Mulmuley et al This

algorithm nds a p erfect matching at costs of a single matrix inversion The reader

is referred to the bibliography in Burkard and C ela for further reference

p ointers related to algorithms for cardinality matching problems

Linear Assignment Problems

The linear assignment problem LAP is one of the oldest and most studies prob

lems in combinatorial optimization Many dierent algorithms have b een develop ed

to solve this problem Also other asp ects of the problem as the asymptotic b ehav

ior or sp ecial cases have b een thoroughly investigated The reader is referred to

DellAmico and Martello for a comprehensive annotated bibliography and

to Burkard and C ela for a recent review on assignment problems

Problem denition and applications

Recall the original mo del where n items are to b e assigned to n other ob jects in

the b est p ossible way Let c b e the cost incurred by the assignment of ob ject i

ij

to ob ject j We are lo oking for an assignment which minimizes the overall cast

P

n

c Thus the linear assignment problem LAP is given as follows

i(i)

i=1

n

X

c min

i(i)

S

n

i=1

where S is the set of p ermutations of f ng Based on the description of

n

the set of all assignments see Section the LAP can also b e formulated as follows

P

min c x over all matrices X x which fulll

ij ij ij

ij

Due to Birkho s result we can relax the conditions x f g to x and

ij ij

obtain the linear programming formulation of the LAP Any basic solution of this

linear program corresp onds to a p ermutation matrix

n

X

min c x

ij ij

i=1

n

X

x j n

ij

LP

i=1

n

X

x i n

ij

j =1

x i j n

ij

As we will mention in the next section many algorithms for the LAP are based

on linear programming techniques and consider often the dual linear program

n n

X X

v u max

j i

j =1 i=1

u v c i j n

i j ij

u v IR i j n

i j

where u and v i j n are dual variables

i j

Among the numerous applications of the LAP the socalled p ersonnel assignments

are the most typical In the p ersonnel assignment we want to assign p eople to

ob jects eg jobs machines ro oms to other p eople etc Each assignment has a

cost and we want to make the assignment so as to minimize the overall sum of

the costs For example one company might want to assign graduates to vacant jobs

In this case the cost c is given by c p where p is the prociency index

ij ij ij ij

for placing candidate i to job j and the goal is to assign each candidate i to some

P

vacancy i such that the overall cost c is minimized or equivalently the

i(i)

i

P

p is maximized overall prociency

i(i)

i

There are many other applications of the linear assignment problem eg in lo cating

and tracing ob jects in space scheduling on parallel machines inventory planning

vehicle and crew scheduling wiring of typ ewriters etc The reader is referred to

Ahuja et al and Burkard and C ela for a detailed description of some

applications of the LAP and literature p ointers to other applications

Algorithms for the LAP

The LAP can b e solved eciently and the design of ecient solution metho ds for

this problem has b een an ob ject of research for many years There exists an amazing

amount of algorithms sequential and parallel for the LAP ranging from primal

dual combinatorial algorithms to simplexlike metho ds The worstcase complexity

3

of the b est sequential algorithms for the LAP is O n where n is the size of the

problem From the computational p oint of view very large scale dense assignment

6

problems with ab out no des can b e solved within a couple of minutes by sequential

algorithms see Lee and Orlin

There is a numb er of survey pap ers and b o oks on algorithms among others Derigs

DellAmico and Toth and the b o ok on the rst DIMACS challenge

edited by Johnson and McGeo ch Among pap ers rep orting on computational

exp erience we mention Carpaneto et al Lee and Orlin DellAmico and

Toth and some of the pap ers in Johnson and McGeo ch

Most sequential algorithms for the LAP can b e classied into primaldual algorithms

and simplexbased algorithms Primaldual algorithms work with a pair consisting of

an infeasible solution x i j n of LP called primal solution and a feasible

ij

solution u v i j n of the dual called dual solution These solutions

i j

fulll the complementarity slackness conditions

x c u v for i j n

ij ij i j

These solutions are up dated iteratively until the primal solution b ecomes feasible

while keeping the complementary slackness conditions fullled and the dual solution

feasible At this p oint the primal solution would b e optimal according to duality

theory

Dierent primaldual algorithms dier on the way they obtain a starting pair of

a primal and a dual solution fullling the conditions describ ed ab ove and the

way the solutions are up dated A starting dual solution can b e obtained as in the

Hungarian metho d by setting u min fc j ng for i n and then

i ij

v minfc u i ng for j n An infeasible primal starting solution

j ij i

could b e given by a matching of maximal cardinality in the bipartite graph G

V W E where V W f ng and E fi j c c u v g

ij ij i j

Then set x if i j is an edge of the matching and x otherwise Clearly

ij ij

the pair of solutions obtained in this way fulll the complementarity slackness con

ditions One way of up dating the pair of solutions is the shortest augmenting path

metho d This metho d gives raise to a whole class of algorithms which meet the

3

b est known time complexity b ound for the LAP namely O n For a given pair

of solutions as ab ove construct a weighted directed bipartite graph G V W E

with arc set E D R with set of forward arcs D fi j i j E x g and

ij

set of backward arcs R fj i i j E x g The weights of the backward

ij

arcs are set equal to whereas the weights of the forward arcs are set equal to

the corresp onding reduced costs c Then select a no de r in V which has not b een

ij

assigned yet and solve the singlesource shortest path problem ie compute the

shortest paths from r to all no des of G The shortest among all paths from r to some

free no de in W is used to augment the current primal solution by swapping the free

and matched edges The dual solution and the reduced costs are then accordingly

up dated It can b e shown that after n augmentations an optimal primal solution

results see Derigs

There are various shortest augmenting path algorithms for the LAP Basically they

dier in the way they determine a starting pair of primal and dual solutions and

by the subroutine they use for computing the shortest paths Most of the existing

algorithms use the Dijkstra algorithm for the shortest path computations

Simplexbased algorithms are sp ecial implementations of the primal or the dual

simplex algorithm for linear programming applied to LP Simplexbased algorithms

for the LAP are sp ecic implementations of the network simplex algorithm The

latter is a sp ecialization of the simplex metho d for linear programming to network

problems The sp ecialization relies on exploiting the combinatorial structure of

network problems to p erform ecient pivots acting on trees rather than on the

co ecient matrix

It is well known that there is a onetoone corresp ondence b etween primal integer

basic solutions of the LAP and spanning trees of the bipartite graph G related to

assignment problems as describ ed in Section Moreover given a spanning tree

one can uniquely determine the values of the corresp onding dual variables so as

to fulll the complementarity slackness conditions as so on as the value of one of

those variables is xed arbitrarily Every integer primal feasible basic solution is

highly degenerate b ecause it contains n variables and n of them are equal

to Hence degeneracy p oses a problem and the rst simplexbased algorithms for

the LAP were exp onential The rst steps towards the design of p olynomialtime

simplexbased algorithms were made by intro ducing the concept of socalled strongly

feasible trees intro duced by Cunningham There are implementations of

simplexbased algorithm for the LAP which match the b est known time complexity

3

b ound of O n The reader is referred to Burkard and C ela for references

and further details

More recently Ramakrishnan et al applied an interior p oint algorithm to the

LAP and got promising results in particular for large size instances see Johnson

and McGeo ch

Since the late s a numb er of parallel algorithms for the LAP has b een prop osed

The sp eedup achieved by such algorithms is limited by the sparsity of the cost

matrices andor the decreasing load across the iterations For a go o d review on

parallel algorithms for the LAP and network ow problems in general the reader is

referred to Bertsekas et al

Asymptotic b ehavior and probabilistic analysis

When dealing with the asymptotic b ehavior of the LAP it is always assumed that

the cost co ecients c are indep endent random variables irv with a common

ij

presp ecied distribution The main question concerns the b ehavior of the exp ected

optimal value of the problem as its size tends to innity

For cost co ecients c b eing irv with a uniform distribution on it has b een

ij

shown that the optimal value of the LAP remains within constant b ounds as the

size n of the problem tends to innity The b est upp er b ound equals and is due to

Karp The b est lower b ound equals and is due to Olin Although

the gap b etween the current lower and upp er b ounds on the exp ected optimal value

of the LAP is large it is b elieved that the exp ected value is close to or more

2

exactly in the case of indep endent cost co ecients c uniformly distributed on

ij

6

For a discussion in some details and for more references see Burkard and

C ela

A more general scenario where the co ecients c are irv with a common arbitrary

ij

general distribution has b een investigated by Frenk et al and Olin

Under mild assumption on the probability distribution of the co ecients the authors

derive constant upp er and lower b ounds on the exp ected optimal value of the LAP

In the case of co ecients c b eing irv uniformly distributed on the LAP

ij

2

can b e solved in exp ected O n log n time by a randomized algorithm prop osed

by Karp Faster randomized algorithms which pro duce a solution whose

ob jective function value is within a constant factor of the optimal ob jective function

value have b een prop osed by several authors There is for instance an algorithm of

Karp et al which runs in exp ected linear time O n and provides a solution

a a

within a factor of O n of the optimum with probability O n where a

is some xed p ositive numb er The b est known linear assignment problem with an

ob jective function dierent from that of the LAP is the bottleneck linear assignment

problem BLAP

min max c

i(i)

1in

considered originally by Fulkerson et al This problem o ccurs eg in the

assignment of jobs to parallel machines so as to minimize the latest completion

time One of the rst algorithms prop osed for the BLAP is the socalled threshold

algorithm The threshold algorithm cho oses a cost element c K the threshold

ij

value and constructs a matrix C dened as follows

if c K

ij

c

ij

if c K

ij

Then the algorithm checks whether the bipartite graph with adjacency matrix C

contains a p erfect matching or not The algorithms rep eats than this pro cedure

for a new and eventually smaller value of the threshold K The smallest value

K for which the corresp onding bipartite graph contains a p erfect matching is the

optimum value of the BLAP The b est time complexity known to day amounts to

p

nm and is due to Punnen and Nair Here m is the numb er of nite O n

elements in the co ecient matrix c which would corresp ond to the numb er of

ij

edges of the bipartite graph in the graph theoretical setting

2

A randomized algorithm with quadratic exp ected running time O n has b een

given by Pferschy A computational study on the comparison of dierent

deterministic algorithms for the BLAP has b een given by Pferschy

Another linear assignment problem intro duced by Burkard and Rendl is the

socalled lexicographic bottleneck assignment problem LexBAP In the LexLAP we

want to nd a p ermutation assignment which lexicographically minimizes c

over all p ermutation where c is the vector of costs c i n sorted

i(i)

nonincreasingly

Martello et al have considered the socalled balanced assignment problem

BalAP Given a real n n matrix C c the balanced assignment problem can

ij

b e formulated as

min max c min c

i(i) i(i)

i i

4

The problem can b e solved eciently in O n time

A more general linear assignment problem which includes as sp ecial cases the linear

assignment problem and the b ottleneck assignment problem is the algebraic

assignment problem AAP intro duced by Burkard et al In the AAP the

co ecients c are elements of a totally ordered semigroup H with comp osition

ij

and order relation The AAP can then b e formulated as follows

c c c min

1(1) 2(2) n(n)

S

n

The AAP can b e solved eciently if the order relation and the comp osition fulll

some natural algebraic prop erties For further results and reference p ointers consult

the survey on algebraic optimization by Burkard and Zimmermann

Available computer co des and test instances

FORTRAN listings of co des for the LAP and the LBAP can b e found in the b o ok

by Burkard and Derigs The co de for the LAP is a primaldual algorithm

based on shortest path computations done by a version of Dijkstras algorithm

Source co des of another primaldual algorithm for the LAP and the LBAP can

b e downloaded from httpassignmenthtml One can cho ose

among a C a PASCAL and a FORTRAN implementations of an algorithm of

Jonker and Volgenant

A compressed FORTRAN source le called Z of an implementation of the

Hungarian algorithm due to Carpaneto and Toth can b e downloaded from

ftpnetlibattcom in netlibtoms Other listings of FORTRAN co des for

the LAP can b e found in Carpaneto et al The co des are available from the

oppy disk included in the b o ok

The C co de of an ecient implementation of the scaling pushrelab el algorithm of

Goldb erg and Kennedy for the LAP can b e downloaded from Goldb ergs

network optimization library at

httpwwwnecinjneccomhomepagesavgsofthtml

Finally listings of FORTRAN co des of auction algorithms for the LAP can b e

found in Bertsekas homepage at

httpwebmitedudimitribwwwauctiontxt

Test instances of the LAP can b e downloaded as ascii les from the homepage of

the ORLibrary maintained by J Beasley at

httpmscmgamsicacukpub

Other test instances can b e obtained from the ELIB library at

ftpftpzibdepubPackagesmptestdataassignindexhtml

Clearly since the LAP can b e formulated as as a minimum cost ow problem

algorithms develop ed for the later can also b e applied to the LAP However such

algorithms are not supp osed to exploit the sp ecic features of the LAP and hence

may not b e comp etitive with algorithms develop ed esp ecially for the LAP Besides

Goldb ergs network optimization library and Bertsekas homepage other co des for

network optimization can b e found in Netlib at

httpwwwOpsResearchcomORLinksindexhtml

Furthermore C co des of implementations of the primal and the dual network simplex

algorithm due to Lob el can b e obtained through

httpwwwzibdeOptimizationindexdehtml

Multidimensional Assignment Problems

General Remarks and Applications

Multidimensional sometimes referred as multiindex assignment problems MAP

are natural extensions of the linear assignment problem They have b een considered

for the rst time by Pierskalla The most prominent representatives of this

class are axial and planar dimensional assignment problems to b e considered in the

next section The MAP asks for d p ermutations which minimize

1 2 d1

the following ob jective function

n

X

c min

i (i) (i) (i)

1 2

d1

1 2

d1

i=1

In terms of graphs a multidimensional assignment problem can b e describ ed as

follows Let a complete dpartite graph G V V V E with vertex sets V

1 2 d i

S

d

V is a jV j n i d and edge set E b e given A subset X of V

i i

i=1

clique if it meets every set V in exactly one vertex A ddimensional assignment is

i

a partition of V into n pairwise disjoint cliques If c is a real valued cost function

dened on the set of cliques of G V V V E the ddimensional assignment

1 2 d

problem asks for a ddimensional assignment of minimum cost Sp ecial cases where

the costs c of a clique are not arbitrary but given as a function of elementary costs

attached to the edges of the complete dpartite graph eg sum costs star costs tour

costs or tree costs have b een investigated and the p erformance of simple heuristics

has b een analyzed in these cases For more information see eg Burkard and C ela

and the references therein

Multidimensional assignment problems in their general form have found some ap

plications as a means to solve data asso ciation problems in in multitarget tracking

and multisensor surveillance The data asso ciation problem consists in partitioning

the observations into tracks and false alarms in real time General classes of these

problems can b e formulated as multidimensional assignment problems Other appli

cations of MAPs are related to track initiation track maintenance and multisensor

tracking Another interesting MAP arises in the context of tracking elementary par

ticles For more information on these two applications see Burkard and C ela

and the references therein

Axial Dimensional Assignment Problems

3

Consider n cost co ecients c The axial dimensional assignment problem

ij k

DAP can then b e stated as

n n n

X X X

min c x

ij k ij k

j =1 i=1

k =1

n n

X X

x i n st

ij k

j =1

k =1

n n

X X

x i n

ij k

i=1

k =1

n n

X X

x i n

ij k

j =1 i=1

x f g i j k n

ij k

We can think of c as the cost of assigning job j to b e p erformed by worker i in

ij k

machine k It follows that x if job j is assigned to worker i in machine k

ij k

and x otherwise

ij k

Equivalently a DAP can b e describ ed with the help of two p ermutations and

n

X

min c

i(i) (i)

S

n

i=1

2

Thus this problem has n feasible solutions It has b een shown that the dimension

of the axial index assignment p olytop e ie the convex hull of feasible solutions to

3

problem is n n Furthermore several classes of facet dening inequalities

have b een identied and ecient algorithms for the separation of these facets have

b een develop ed For more information on these topic see the description in Burkard

and C ela and consult the references cited therein

In contrast to the linear assignment problem LAP the DAP cannot b e solved

eciently and branch and b ound algorithms are the mostly used algorithms to solve

this problem The lower b ounds are usually computed by solving some Lagrangean

relaxation of the DAP by subgradient optimization approaches see eg Balas and

Saltzman The authors intro duce in Balas and Saltzman also a non

trivial branching strategy which exploits the structure of the problem and allows to

x several variables at each branching no de

A heuristic for solving the DAP has b een prop osed by Pierskalla

Finally there exists a numb er of eciently solvable sp ecial cases of the DAP eg

if the cost co ecients are taken from a dimensional Monge array or if the cost

co ecients are decomp osable ie c u v w and u v and w are nonnegative

ij k i j k i j k

The reader is referred to Burkard and C ela and the references therein for

more information on this topic

Planar Dimensional Assignment Problems

3

Let c i j k n b e n cost co ecients The planar dimensional assignment

ij k

problems PAP is stated as follows

n n n

X X X

c x min

ij k ij k

j =1 i=1

k =1

st

n

X

x j k n

ij k

i=1

n

X

x i k n

ij k

j =1

n

X

x i j n

ij k

k =1

x f g i j k n

ij k

The PAP has interesting applications in time tabling problems See Euler and Le

Verge for a recent study on time tables and related p olyhedra

It is easily seen that the feasible solutions of the PAP corresp ond to Latin squares

Thus the numb er of feasible solutions of a PAP of size n equals the numb er of

Latin squares of order n and hence increases very fast Similarly to the DAP also

the PAP is a hard problem and cannot b e solved eciently There are not many

algorithms known for the PAP Besides two branch and b ound algorithms due to

Vlach and Magos and Miliotis there is also a tabu search algorithm

for the PAP due to Magos It uses a neighb orho o d structure based on the

relationship b etween the PAP and the latin squares See eg Burkard and C ela

for more information on algorithms for the PAP

The Quadratic Assignment Problem

The quadratic assignment problem QAP was intro duced in by Ko opmans

and Beckmann as a mo del for a plant lo cation problem Since then the QAP has

b een ob ject of intensive investigations concerning dierent asp ects of the problem

ranging from algorithms to asymptotic b ehavior and sp ecial cases In contrast to its

linear counterpart the QAP can not b e solved eciently However there are some

restricted cases of the QAP which can b e solved in p olynomial time Such sp ecial

cases are QAPs whose co ecient matrices show sp ecial combinatorial prop erties

eg Monge and Mongelike prop erties The interested reader is referred to C ela

for a detailed discussion on eciently solvable sp ecial cases of the QAP

Also from the practical p oint of view the QAP is widely considered as one of the

hardest problems in combinatorial optimization and there are instances of size

which can not yet b e solved to optimality in reasonable computational time eg the

Nugent instance of size see QAPLIB Burkard et al

The reader is referred to Burkard and C ela for a comprehensive annotated

bibliography to C ela for a recent monograph and to Burkard et al

for a recent review on quadratic assignment problems and for reference p ointers

Problem denition and applications

Among the most common and b est known applications of the QAP are those which

arise in a facility lo cation context This is one reason for cho osing the facility lo cation

terminology to intro duce the problem

Consider the problem of allo cating n facilities to n lo cations with costs dep ending

on the distance b etween the lo cations and the ow b etween the facilities plus costs

asso ciated with a facility b eing placed at a certain lo cation The ob jective is to assign

each facility to a lo cation such that the total cost is minimized More sp ecically

we are given three n n real matrices A a B b and C c where

ij k l ik

a is the ow b etween the facility i and facility j b is the distance b etween the

ij k l

lo cation k and lo cation l and c is the cost of placing facility i at lo cation k The

ik

Ko opmansBeckmann version of the QAP can b e then formulated as follows

n n n

X X X

A

min a b a

ij

(i)(j ) i(i)

S

n

j =1 i=1 i=1

where S is the set of all p ermutations of N A pro duct a b is the cost of

n ij

(i)(j )

assigning facility i to lo cation i and facility j to lo cation j

A more general version of the QAP was intro duced by Lawler in In this version

we are given a fourdimensional array D d of co ecients instead of the two

ij k l

matrices A and B and the problem can b e stated as

n n n

X X X

A

min c d

i(i) ij (i)(j )

S

n

i=1 j =1 i=1

Besides applications in facility lo cation the QAP nds applications in VLSI design

computer manufacturing scheduling pro cess communications backb oard wiring

and typ ewriter keyb ord design We describ e here a less known application on turbine

balancing Hydraulic turbine runners as used in electricity generation consist of a

cylinder around which a numb er of blades are welded at regular spacings Due to

inaccuracies in the manufacturing pro cess the weights of these blades dier slightly

and it is desirable to lo cate the blades around the cylinder in such a way that

the distance b etween the center of mass of the blades and the axis of the cylinder

is minimized This problem was intro duced by Mosevich in and has b een

formulated as a QAP by Lap orte and Mercure The places at regular spacings

on the cylinder are mo deled by the vertices v v of a regular ngon on the unit

1 n

circle in the Euclidean plane Thus

i i

v sin cos i n

i

n n

The masses of the n blades are given by the p ositive reals m m m

1 2 n

The goal is to assign each mass m to some vertex v in such a way that the center

i

(i)

of gravity

n

2i

X

sin

n

m

(i)

2i

cos

n

i=1

of the resulting mass system is as close to the origin as p ossible Minimizing the

Euclidean norm of the ab ove vector is equivalent to minimizing the expression

n n

X X

i j

m m cos

(i) (j )

n

j =1 i=1

Thus we have a quadratic assignment problem where the co ecient matrices A and

2(ij )

B have a very sp ecial structure a cos and b m m

ij ij i j

n

Finally notice that there are a numb er of other well known combinatorial optimiza

tion problems which can b e formulated as QAPs eg graph partitioning maximum

clique the traveling salesman problem the linear arrangement problem and the

minimum weight feedback arc set problem

To conclude this section notice that the QAP is a very hard problem from the

theoretical p oint of view Not only that the QAP cannot b e solved eciently but it

even cannot b e approximated eciently within some constant approximation ratio

Furthermore also nding lo cal optima is in general not a trivial task For informa

tion and references concerning the computational complexity of the QAP see C ela

Alternative formulations of the QAP

There exist several equivalent formulation of the QAP Dierent formulations stress

dierent characteristics of the problem and lead to dierent solution approaches

Recall that there is a onetoone corresp ondence b etween the p ermutations of N

f ng and the n n p ermutation matrices dened in Section Let X b e the

n

set of n n p ermutation matrices In terms of p ermutation matrices QAPABC

can b e formulated as the following quadratic integer program

n n n n n

X X X X X

a b x x c x min

ij k l ik j l ij ij

ij =1 j =1 i=1

k =1 l=1

st x X

ij n

Let us dene an inner product b etween n n matrices as follows

n n

X X

a b hA B i

ij ij

j =1 i=1

T

Clearly we have X AX a for some n n matrix A a p ermutation S

n

(i)(j )

and the asso ciated p ermutation matrix X X Thus the QAP can b e

n

formulated alternatively as

T

min hA X B X i hC X i

st X X

n

Finally consider the trace formulation of the QAP The trace trA of an n n

P

n

matrix A a is dened as sum of its diagonal elements tr A a Let

ij ii

i=1

t t

us denote B XB X where X is the p ermutation matrix corresp onding to

We get

n n

X X

a b a b trAB

ij ij

(i)(j )

j i

ij =1 ij =1

P

n

t

c the QAP in since b b for i j f ng Since tr CX

i(i) (i)(j )

i=1

ij

can b e formulated as

T T

min tr AX B C X

st X X

n

Linearizations

A rst attempt to solve the QAP would b e to eliminate the quadratic term in the

ob jective function in order to transform the problem into a mixed linear

program MILP The linearization of the ob jective function is usually achieved by

intro ducing new variables and new linear and binary constraints The very large

numb er of new variables and constraints however p oses an obstacle for eciently

solving the resulting linear integer programs The optimal value of an LP relaxation

of some MILP formulation is a lower b ound for the QAP In this context the tight

ness of the continuous relaxation of the linear integer program mentioned ab ove is

a desirable prop erty Several linearizations of the QAP have b een prop osed in the

literature eg by Lawler by Kaufmann and Bro eckx this lineariza

tion has the smallest numb er of variables and constraints by Frieze and Yadegar

and by Adams and Johnson The last linearization unies most of the

previous linearizations and is imp ortant for getting lower b ounds The QAP with

array of co ecients C d is proved to b e equivalent to the following mixed

ij k l

linear program

n n

X X

d y min

ij k l ij k l

ij =1

k l=1

st x X

ij n

n

X

y x j k l n

ij k l j l

i=1

n

X

y x i j l n

ij k l j l

k =1

y y i j k l n

ij k l j ilk

y i j k l n

ij k l

where each y represents the pro duct x x Although as noted by Adams and

ij k l ik j l

Johnson a signicant smaller formulation in terms of b oth the variables and

constraints could b e obtained the structure of the continuous relaxation of the ab ove

formulation is favorable for solving it approximately by Lagrangean dual metho ds

The theoretical strength of the linearization relies on the fact that the con

straints of the continuous relaxations of previous linearizations can b e expressed as

linear combinations of the constraints of the continuous relaxation of More

over many of the previously published lowerb ounding techniques can b e explained

based on the Lagrangean dual of this relaxation For more details on this topic we

refer to Section

Lower b ounds

Lower b ounding techniques are used within implicit enumeration algorithms such

as branch and b ound to p erform a limited search of the set of feasible solution until

an optimal solution is found Numerous b ounding techniques have b een develop ed

for the QAP so far The most successful b ounding techniques for the QAP can

b e classied in groups GilmoreLawler typ e lower b ounds b ounds based on LP

relaxation eigenvalue related b ound

GilmoreLawler typ e lower b ounds

The basic idea of these typ e of lower b ounds go es back to the early s when

Gilmore and Lawler develop ed the socalled GilmoreLawler b ound

GLB for the QAP Nowadays the GilmoreLawler typ e lower b ounds and esp ecially

the GLB are the most frequently used b ounds within branch and b ound algorithms

for the QAP The most advantageous prop erty of these b ounds is that they can b e

computed eciently The main drawback is the fast deterioration of their quality

with increasing problem size

2 2

To compute the GLB for a given QAP of size n one has to solve n LAPs n of

them of size n and the last one of size n as describ ed b elow

Consider an instance of the Lawler QAP with co ecients D d For each

ij k l

ordered pair of indices i k i k n solve an LAP with co ecient matrix

(ik )

D d under the additional constraint that i k Let us denoted by

ij k l

l the optimal solutions of the ab ove mentioned LAP Finally solve an LAP with

ik

co ecient matrix l c its optimal value is the GLB Since the LAP can b e

ij ij

solved eciently also the GLB can b e computed eciently

Several GilmoreLawler typ e lower b ounds have b een develop ed aiming at improving

the quality of the GLB One of the ideas on which such b ounds are based are the so

called reduction methods These metho ds decomp ose each quadratic cost co ecient

into several terms so as to guarantee that some of them end up in b eing linear cost

co ecients and can b e moved to the linear term of the ob jective function This

would yield a tighter lower b ound b ecause the LAP can b e solved exactly

More recently another b ounding pro cedure which shares the basic idea of the GLB

has b een prop osed by Hahn and Grant This pro cedure combines GLB ideas

with reduction steps in a general framework The resulting b ound HGB shows a

go o d trade o b etween computation time and b ound quality when tested in instances

from QAPLIB see Burkard et al

Bounds based on linear programming relaxations

Consider a mixed integer linear programming MILP formulations of the QAP

Clearly the optimal solution of the continuous relaxation of an MILP formulation

is a lower b ound for the optimal value of the corresp onding QAP Moreover each

feasible solution of the dual of this relaxation is also a lower b ound Even for QAPs

of mo derate size it is practically imp ossible to solve to optimality the LP relaxations

of any MILP formulation Generally such formulations are highly degenerated and

already for QAPs of mo derate size the memory requirements b ecome prohibitive

as shown by Resende et al The LP relaxations of the MILPs are there

fore approximately solved by applying subgradient optimization based techniques

or Lagrangean relaxations Adams et al have shown that a particular Lagrangean

relaxation of the MILP can b e solved eciently for each xed set of Lagrangean

multipliers see Adams and Johnson The Lagrangean multipliers are then

up dated iteratively in the fashion of a dual ascent pro cedure The strength of the

metho d of Adams and Johnson relies on the fact that it can pro duce all Gilmore

Lawlerlike b ounds describ ed in Section and others but the HGB for dierent

settings of the Lagrangean multipliers

Recently Karisch et al have shown that although HGB cannot b e obtained

by applying the algorithm of Adams and Johnson b oth AJB and HGB can b e

obtained as feasible solutions of the dual of the continuous relaxation of the MILP

formulation Karisch et al prop ose an iterative algorithm to approximately

solve this dual and show that AJB HGB and all other GilmoreLawlerlike b ounds

can b e obtained by applying this algorithm with sp ecic settings for the control

parameters

Eigenvalue based lower b ounds

These b ounds were intro duced by Finke et al and can b e applied to the

Ko opmansBeckmann QAP in They are based on the relationship b etween the

ob jective function value of the QAP in the trace formulation and the eigen

values of its co ecient matrices When designed and implemented carefully these

techniques pro duce b ounds of go o d quality in comparison with GilmoreLawlerlike

b ounds or more generally with b ounds based on linear relaxations However these

b ounds are quite exp ensive in terms of computation time requirements and are

therefore not appropriate for use within branch and b ound algorithms

Exact solution metho ds

Since QAP is a hard problem from the theoretical and also from the practical p oint

of view and no ecient algorithms are known for this problem All existing exact

algorithms are in principle enumeration metho ds like branch and b ound cutting

plane algorithms and branch and cut algorithms

Branch and b ound algorithms

Nowadays branch and b ound algorithms app ear to b e the most ecient exact algo

rithms for solving the QAP

The most ecient branch and b ound algorithms for the QAP employ the Gilmore

Lawler b ound GLB The reason is that other b ounds which outp erform GLB in

terms of b ound quality are simply to o exp ensive in terms of computation time More

recently some eorts have b een made to employ other GilmoreLawlerlike b ounds

The b ound of HGB has b een used in a branch and b ound algorithm by Hahn et al

and the results are promising

Three typ es of branching strategies are mostly used for the QAP single assignment

branching pair assignment branching and branching based on relative positioning

The most ecient strategy is the single assignment branching It assigns a facility to

a lo cation in each branching step ie each problem is divided into subproblems by

xing the lo cation of one of the facilities which are not assigned yet Several rules for

the choice of the facilitylo cation pair to determine the subproblems of a new level

of the search tree have b een prop osed by dierent authors The appropriate rule

usually dep ends on the b ounding technique If the GLB is employed the branching

rule is frequently formulated in terms of the reduced costs of the last assignment

problem solved to b ound the subproblem which is currently b eing branched

As for the selection rule there seems to b e no clear winner among dierent strate

gies tested for the QAP ranging from problemindep endent depth or breadth rst

search to instance dep endent criteria related to the maximization of lower b ounds

or reduced costs

More recently a numb er of parallel branch and b ound algorithms have b een devel

op ed for the QAP eg by Pardalos and Crouse Bruengger et al

and Clausen and Perregaard

Cutting plane metho ds

The cutting plane metho ds in general can b e classied into traditional cutting plane

methods and polyhedral cutting plane methods Traditional cutting plane algorithms

for the QAP have b een develop ed by dierent authors eg Bazaraa and Sherali

Balas and Mazzola a Balas and Mazzola b and Kaufmann and

Bro eckx These algorithms make use of mixed integer linear programming

MILP formulations for the QAP which are suitable for Benders decomp osition

Generally the time needed for these metho ds to converge is to o large and hence

these metho ds may solve to optimality only very small QAPs However heuristics

derived from cutting plane approaches pro duce go o d sub optimal solutions in early

stages of the search

Also p olyhedral cutting planes or branch and cut algorithms make use of MILP

formulations of the QAP Additionally p olyhedral cutting plane metho ds make use

of a class of nontrivial valid or facet dening inequalities known to b e fullled by

al l feasible solutions of the original problem Some prop erties and few facet dening

inequalities of the QAP p olytop e are already known but still p olyhedral cutting

plane metho ds for the QAP are not yet backed by a strong theory Some eorts to

design branch and cut algorithms for the QAP have b een made by Padb erg and

Rijal and Kaib el The numerical results are encouraging although

the develop ed software is of preliminary nature as claimed by the authors

Heuristics

The QAP is a very dicult problem from the practical p oint of view instances of

dimension n b eing still not practical to solve b ecause of very high computa

tion time requirements This is probably the reason why the literature ab ounds in

heuristics which are the only available algorithms to provide go o d quality solutions

for the QAP in a reasonable computational time The reader is referred to Burkard

et al for numerous literature p ointers to numerous heuristics applied to the

QAP

The numerous heuristic approaches develop ed for the QAP can b e classied in con

struction metho ds limited enumeration metho ds lo cal search algorithms including

improvement metho ds tabu search and simulated annealing genetic algorithms

greedy randomized adaptive search pro cedures GRASP and ant systems In the

case of the QAP there is no widely accepted winner among these strategies The con

struction metho ds the limited enumeration metho ds and the improvement metho ds

seem however to b e outp erformed by the other heuristic approaches

We refer to other chapters of this handb o ok for a detailed description of the general

metaheuristic approaches mentioned ab ove In the following we briey mention the

most frequently used neighb orho o d structures included in lo cal search algorithms

for the QAP

Frequently used neighb orho o ds for the QAP are the pairexchange neighb orho o d

and the cyclic tripleexchange neighb orho o d In the case of pairexchanges the

neighb orho o d of a given solution p ermutation consists of all p ermutations which

can b e obtained from the given one by applying a transp osition to it The size of this

n

neighb orho o d is O In the case of cyclic tripleexchanges the neighb orho o d of

2

a solution p ermutation consists of all p ermutations obtained from by a cyclic

n

exchange of some triple of indices The size of this neighb orho o d is O In

3

general cyclic tripleexchanges do not lead to b etter results when compared with

pairexchanges

Available computer co des for the QAP

Burkard et al have compiled a library of QAP instances QAPLIB which is

widely used to test b ounds exact algorithms and heuristics for the QAP Many of

these instances have not b een solved to optimality yet the most celebrated among

them b eing the instances of Nugent et al of size larger than QAPLIB

can b e found at

httpwwwoptmathtugrazacat ekarischqaplib

Two co des for computing lower b ounds are also available from the QAPLIB web

page a FORTRAN co de due to Burkard and Derigs to compute the GLB for

instances of size up to and another FORTRAN co de to compute the elimination

b ound ELI for symmetric QAP instances of size up to Finally a FORTRAN

co de of the branch and b ound algorithm develop ed by Burkard and Derigs

can also b e downloaded from the QAPLIB web page

Recently Esp ersen et al have develop ed QAPpack which is a JAVA package

containing a branch and b ound algorithm to solve the QAP In QAPpack a numb er

of b ounds based on linearization are implemented the GilmoreLawler b ound the

b ound of Carraresi and Malucelli the b ound of Adams and Johnson

the b ound of Hahn and Grant and the b ound of Karisch et al The

implementation is based on the dual framework provided by Karisch et al

QAPpack can b e found at httpwwwimmdtudk eteQAPpack

There are also some co des of heuristics available The compressed FORTRAN

source le Z of a heuristic due to West can b e downloaded at

ftpnetlibattcom in netlibtoms

The source les compressed tarles of two FORTRAN implementations of GRASP

for dense QAPs by Resende et al and sparse QAPs by Pardalos et al

can b e downloaded from Resendes web page at httpwwwresearchattcom emgcrsrcindexhtml

The source le of a FORTRAN implementation of the simulated annealing algorithm

of Burkard and Rendl can b e downloaded from the QAPLIB web page

The source le of a C implementation of the simulated annealing algorithm of

Connolly due to Taillard can b e downloaded from Taillards web page at

httpwwwidsiach eericcodesdirsa qapc Also a source le of a PAS

CAL implementation of the robust tabu search algorithm by Taillard can b e

found at Taillards web page

Asymptotic b ehavior

While b eing a very dicult problem b oth from the theoretical and from the practical

p oint of view the QAP shows an interesting asymptotic b ehavior which suggests

that under certain probabilistic conditions on the problem data QAPs which are

large enough are trivial to solve Namely it can b e shown that the ratio b etween the

b est and worst values of the ob jective function approaches as the size of the

QAP approaches innity Thus the relative error of every heuristic metho d vanishes

as the size of the problem tends to innity ie every heuristic nds almost always

an almost optimal solution when applied to QAP instances which are large enough

A numb er of authors have investigated the asymptotic b ehavior of the QAP Burkard

and Fincke Burkard and Fincke and Frenk et al have shown the

convergence of the ab ove mentioned ratio to in probability Later Szpankowski

improved the convergence to almost surely Summarizing we get the following

theorem

Theorem Consider a sequence of QAPs P for n IN with n n coecient

n

(n) (n) (n) (n)

(n)

matrices A a and B b n IN and b Assume that a

ij ij ij ij

i j n are independently distributed random variables on M where M is a

(n)

positive constant Moreover assume that entries a n IN i j n have a

ij

(n)

n IN i j n have also a common common distribution and entries b

ij

(n)

distribution which does not necessarily coincide with that of a Furthermore

ij

assume that these random variables have nite expected values variances and third

moments

(n)

(n)

Let and denote an optimal and a worst solution of P with their objective

opt

n

wor

(n)

(n)

function values z and z respectively Then the fol lowing equality holds

opt

wor

(n)

(n)

almost surely lim z z

opt

wor

n

The asymptotic b ehavior of the QAP has b een exploited by Dyer et al to

analyze the p erformance of branch and b ound algorithms for QAPs with co ecients

generated randomly as describ ed ab ove Dyer et al have shown that any branch

and b ound algorithm that uses single assignment branching and employs a b ound

obtained by solving the continuous relaxation of the linearization of Frieze and Yade

(1o(1))n4

gar would branch on at least n no des with probability tending to as the

size n of the problem tends to innity

The biquadratic assignment problem

A generalization of the QAP arises if we consider ob jective functions of higher degree

and obtain in this way cubic biquadratic and generally N adic assignment problems

as intro duced by Lawler In this way we get among other also the biquadratic

assignment problem denoted by BiQAP and stated as follows

n n n n

X X X X

a b x x x x min

ij k l mpst im j p k s lt

st=1 mp=1

ij =1

k l=1

st X x X

ij n

4 4

where A a and B b are two n n arrays

ij k l mpst

An application of the BiQAP arises in Very Large Scale Integrated VLSI circuit

design A detailed description of the mathematical mo deling of the VLSI problem as

a BiQAP is given by Burkard et al Similarly to the QAP also the BiQAP is

a hard problem and cannot b e solved eciently GilmoreLawlerlike lower b ounds

branch and b ound algorithms and some lo cal search heuristics as well as a GRASP

implementation for the BiQAP can b e found in the literature See Burkard and

C ela or C ela for more information and reference p ointers

The Bottleneck QAP

Another problem related to the QAP is the bottleneck quadratic assignment problem

BQAP obtained by substituting the sum by a max op eration in the ob jective

function of the QAP

maxfa b i j ng min

ij

(i)(j )

S

n

The rst o ccurrence of the BQAP in the literature is due to Steinb erg and

arises as an application in backb oard wiring while trying to minimize the maximum

length of the involved wires A well studied problem in graph theory which can b e

mo deled as a BQAP is the bandwidth problem In the bandwidth problem we are

given an undirected graph G V E with vertex set V and edge set E and seek a

lab eling of the vertices of G by the numb ers n where jV j n such that the

maximum distance of entries of the resulting adjacency matrix from the diagonal

is minimized ie the bandwidth of the adjacency matrix is minimized

Analogously to the QAP also the BQAP is a hard problem which cannot b e solved

eciently Some enumeration algorithms to solve BQAP to optimality have b een

prop osed by Burkard Those algorithms employ a GilmoreLawlerlike b ound

for the BQAP

Burkard and Fincke investigated the asymptotic b ehavior of the BQAP and

proved results analogous to those obtained for the QAP Under certain probabilistic

constraints on the problem data the relative dierence b etween the worst and the

b est value of the ob jective function approaches with probability tending to as

the size of the problem approaches innity

References

Adams W P and Johnson T A Improved linear programmingbased

lower b ounds for the quadratic assignment problem In Pardalos P M and

Wolkowicz H editors Quadratic Assignment and Related Problems volume

of DIMACS Series on Discrete Mathematics and Theoretical Computer Science

pages AMS Providence RI

Ahuja R K Magnanti T L Orlin J B and Reddy M R Applications

of network optimization In Ball M O Magnanti T L Monma C L and

Nemhauser G L editors Network Models Handbooks of Operations Research

and Management Science pages Elsevier Amsterdam

Alt H Blum N Mehlhorn K and Paul M Computing maximum cardi

p

15

nality matching in time on m log n Information Process Letters

Balas E and Mazzola J B a Nonlinear programming I linearization

techniques Mathematical Programming pages

Balas E and Mazzola J B b Nonlinear programming Ii dominance rela

tions and algorithms Mathematical Programming

Balas E and Saltzman M J An algorithm for the threeindex assignment

problem Operations Research

Bazaraa M S and Sherali H D On the use of exact and heuristic cutting

plane metho ds for the quadratic assignment problem Journal of Operations

Research Society

Bertsekas D P non D A C Eckstein J and Zenios S Parallel comput

ing in network optimization In Ball M O Magnanti T L Monma C L

and Nemhauser G L editors Network Models Handbooks in Operations Re

search and Management Science pages Elsevier Amsterdam The

Netherlands

Birkho G Tres observaciones sobre el algebra lineal Rev univ nac

Tucuman A

Bruengger A Clausen J Marzetta A and Perregaard M Joining forces

in solving largescale quadratic assignment problems in parallel In Proceedings

of the th IEEE International Paral lel Processing Symposium IPPS pages

Burkard R E Quadratische b ottleneckprobleme Operations Research

Verfahren

Burkard R E and C ela E Quadratic and threedimensional assignments

In DellAmico M Maoli F and Martello S editors Annotated Bibliogra

phies in combinatorial Optimization pages Wiley Chichester

Burkard R E and C ela E Linear assignment problems and extensions In

Pardalos P and Du DZ editors Handbook of Combinatorial Optimization

Kluwer Academic Publishers to app ear

Burkard R E C ela E and Klinz B On the biquadratic assignment

problem In Pardalos P M and Wolkowicz H editors Quadratic Assignment

and Related Problems volume of DIMACS Series on Discrete Mathematics

and Theoretical Computer Science pages AMS Providence RI

Burkard R E C ela E Pardalos P and Pitsoulis L The quadratic

assignment problem In Pardalos P and Du DZ editors Handbook of Com

binatorial Optimization pages Kluwer Academic Publishers

Burkard R E and Derigs U Assignment and Matching Problems Solution

Methods with FORTRAN Programs Springer Berlin

Burkard R E and Fincke U On random quadratic b ottleneck assignment

problems Mathematical Programming

Burkard R E and Fincke U The asymptotic probabilistic b ehavior of

the quadratic sum assignment problem Zeitschrift fur Operations Research

Burkard R E and Fincke U Probabilistic asymptotic prop erties of some

combinatorial optimization problems Discrete Applied Mathematics

Burkard R E Hahn W and Zimmermann U An algebraic approach to

assignment problems Mathematical Programming

Burkard R E Karisch S E and Rendl F Qaplib a quadratic

assignment problem library Journal of Global Optimization

An online version is available via World Wide Web at the following URL

httpwwwoptmathtugrazacatekarischqaplib

Burkard R E and Rendl F A thermo dynamically motivated simulation

pro cedure for combinatorial optimization problems European Journal of Op

erational Research

Burkard R E and Rendl F Lexicographic b ottleneck problems Operations

Research Letters

Burkard R E and Zimmermann U Combinatorial optimization in lin

early ordered semimo dules a survey In Korte B editor Modern Applied

Mathematics pages North Holland Amsterdam

Carpaneto G Martello S and Toth P Algorithms and co des for the

assignment problem Annals of Operations Research

Carpaneto G and Toth P Solution of the assignment problem ACM

Transactions on Mathematical Software

Carraresi P and Malucelli F A reformulation scheme and new lower b ounds

for the qap In Pardalos P and Wolkowicz H editors Quadratic Assignment

and Related Problems volume of DIMACS Series in Discrete Mathematics

and Theoretical Computer Science pages AMS Providence RI

C ela E The Quadratic Assignment Problem Theory and Algorithms

Kluwer Academic Publishers Dordrecht

Clausen J and Perregaard M Solving large quadratic assignment problems

in parallel Computational Optimization and Applications

Connolly D T An improved annealing scheme for the qap European

Journal of Operational Research

Cunningham W H A network simplex metho d Mathematical Program

ming

DellAmico M and Martello S Linear assignment In DellAmico M Maf

oli F and Martello S editors Annotated Bibliographies in Combinatorial

Optimization pages Wiley Chichester

DellAmico M and Toth P Algorithms and co des for dense assignment

problems To app ear

Derigs U The shortest augmenting path for solving assignment problems

motivation and computational exp erience

Dyer M E Frieze A M and McDiarmid C J H On linear programs

with random costs Mathematical Programming

Esp ersen T Karisch S E C ela E and Clausen J QAPPACK a java package for

solving quadratic assignment problems working pap er Department of Math

ematical Mo delling Technical University of Denmark Denmark and Institute

of Mathematics Technical University Graz Austria

Euler R and Le Verge H Timetables p olyhedra and the greedy algorithm

Discrete Applied Mathematics

Even S and Tarjan R E Network ow and testing graph connectivity

SIAM Journal on Computing

Finke G Burkard R E and Rendl F Quadratic assignment problems

Annals of Discrete Mathematics

Frenk J B G van Houweninge M and Kan A H G R Asymptotic

prop erties of the quadratic assignment problem Mathematics of Operations

Research

Frenk J B G van Houweninge M and Rinno oy Kan A H G Order

statistics and the linear assignment problem Computing bf

Frieze A M and Yadegar J On the quadratic assignment problem Discrete

Applied Mathematics

Fulkerson R Glicksb erg I and Gross O A pro duction line assignment

problem Technical Rep ort RM The Rand Corp oration Sta Monica CA

Gilmore P C Optimal and sub optimal algorithms for the quadratic assign

ment problem SIAM Journal on Applied Mathematics

Goldb erg A V and Kennedy R An ecient cost scaling algorithm for the

assignment problem Mathematical Programming

Hahn P and Grant T Lower b ounds for the quadratic assignment problem

based up on a dual formulation Operations Research

Hahn P Grant T and Hall N Solution of the quadratic assignment prob

lem using the hungarian metho d European Journal of Operational Research

5

2

Hop croft J E and Karp R M An n algorithm for maximum matchings

in bipartite graphs SIAM Journal on Computing

Johnson D S and McGeo ch C C editors Network Flows and Matching

First DIMACS Implementation Chal lenge volume of DIMACS Series in

Discrete Mathematics and Theoretical Computer Science AMS Providence

RI

Jonker R and Volgenant A Improving the hungarian assignment algo

rithm Operations Research Letters

Kaib el V Polyhedral Combinatorics of the Quadratic Assignment Problem

PhD thesis Universitat zu Koln Germany

Karisch S E C ela E Clausen J and Esp ersen T A dual framework

for lower b ounds of the quadratic assignment problem based on linearization

Computing

Karp R M An algorithm to solve the m n assignment problem in exp ected

time omn log n Networks

Karp R M An upp er b ound on the exp ected cost of an optimal assignment

In Discrete Algorithms and Complexity pages Academic Press Boston

Karp R M Kan A H G R and Vohra R V Average case analysis of

a heuristic for the assignment problem Mathematics of Operations Research

Kaufmann L and Bro eckx F An algorithm for the quadratic assignment

problem using b enders decomp osition European Journal of Operational Re

search

Lap orte G and Mercure H Balancing hydraulic turbine runners A

quadratic assignment problem European Journal of Operational Research

Lawler E L The quadratic assignment problem Management Science

Lee Y and Orlin J B On very large scale assignment problems In Hager

W W Hearn D W and Pardalos P M editors Large Scale Optimization

State of the Art pages Kluwer Academic Publishers Dordrecht The

Netherlands

Magos D Tabu search for the planar threeindex assignment problem

Journal of Global Optimization

Magos D and Miliotis P An algorithm for the planar threeindex assign

ment problem European Journal of Operational Research

Martello S Pulleyblank W R Toth P and de Werra D Balanced

optimization problems Operations Research Letters

Mulmuley K Vazirani U V and Vazirani V V Matching is as easy as

matrix inversion Combinatorica

Nugent C E Vollmann T E and Ruml J An exp erimental comparison

of techniques for the assignment of facilities to lo cations Journal of Operations

Research

Olin B Asymptotic Properties of Random Assignment Problems PhD the

sis Division of Optimization and Systems Theory Department of Mathematics

Royal Institute of Technology Sto ckholm

Padb erg M W and Rijal M P Location Scheduling Design and Integer

Programming Kluwer Academic Publishers Boston

Pardalos P and Crouse J A parallel algorithm for the quadratic assignment

problem In Proceedings of the Supercomputing Conference pages

ACM Press

Pardalos P M Pitsoulis L S and Resende M G C Fortran subroutines

for approximate solution of sparse quadratic assignment problems using grasp

ACM Transcations on Mathematical Software

Pferschy U The random linear b ottleneck assignment problem RAIRO

Operations Research

Pferschy U Solution metho ds and computational investigations for the

linear b ottleneck assignment problem Computing

Pierskalla W P The trisubstitution metho d for the threemultidimensional

assignment problem Canadian ORS Journal

Punnen A P and Nair K P K Improved complexity b ound for the

maximum cardinality b ottleneck bipartite matching problem Discrete Applied

Mathematics

Ramakrishnan K G Karmarkar N K and Kamath A P An approximate

dual pro jective algorithm for solving assignment problems In Johnson D S

and McGeo ch C C editors Network ows and matching First DIMACS Im

plementation Chal lenge volume of DIMACS Series in Discrete Mathematics

and Theoretical Computer Science pages AMS Providence RI

Resende M G C Pardalos P M and Li Y Fortran subroutines for

approximate solution of dense quadratic assignment problems using grasp ACM

Transcations on Mathematical Software

Resende M G C Ramakrishnan K G and Drezner Z Computing lower

b ounds for the quadratic assignment problem with an interior p oint algorithm

for linear programming Operations Research

Steinb erg L The backb oard wiring problem A placement algorithm SIAM

Review

Szpankowski W Combinatorial optimization problems for which almost

every algorithm is asymptotically optimal Optimization

Taillard E Robust tabu search for the quadratic assignment problem

Paral lel Computing

Vlach M Branch and b ound metho d for the threeindex assignment prob

lem EkonomickoMatematicky Obzor

West D H Algorithm Approximate solution of the quadratic assign

ment problem ACM Transactions on Mathematical Software