Classes of quadratic assignment problem instances: isomorphism and
difficulty measure in a statistical approach
Nair Maria Maia de Abreu
Tania Maia Querido
1
1, 2
Paulo Oswaldo Boaventura Netto
Elisabeth Ferreira Gouvea
1
1
1 -Introduction: The Quadratic Assignment Problem (QAP) is still one of the most
difficult challenge in the combinatorial optimization area. For this reason, considerable
attention is given to the complexity theory focusing the features of a particular QAP
instance. A complete survey and applications of the QAP can be found in [3].
In this work we define a novel class of QAP instances that leads to the same linearization.
Besides a discussion about isomorphism, we will present a difficulty measure in a
statistical basis, which approach was partially suggested by Graves and Whinston in the
early 70’s and remained an open question so far.
2 - QAP partition into classes of relative instances: The QAP can be stated as:
z = Min
  n
 f
i
ij
d  ( i ) ( j ) for 1 i < j  n,
(2.1)
j
where  n is the set of permutations in {1, 2, …, n} and the coefficients of the objective
function are defined by the vectors F= [ f ij ] and D = [ d kl ], with N = Cn,2 elements. An
instance, represented by the pair of vectors (F,D) will be denoted by QAP(F,D) and Q =
F D t will be the matrix consisting of the coefficients in 2.1. Ranking F and D elements
in non-increasing and non-decreasing order, respectively, we obtain ( F  , D  ) with
coefficient matrix Q * = ( F  )( D  ) T . We will designate Relclass( Q * ) the family of
relative instances {QAP(F,D) / ( F  )( D  ) T = Q * }. A particular feature of this set lies on
the fact that instances in Relclass( Q * ) induce the same pair ( F  , D  ). Thus, we can
think of the QAP as:
QAP =

Relclass (QF* , D ) .
F ,D
3 - Costs average and variance for QAP(F,D): The large space of local optimums for
the QAP instances suggests the use of the average and variance as parameters to establish
the particular difficulty an instance reveals. Many researchers have been ranking the
problem hardness or introduced algorithms based on a variance approach, such as in [5]
and [6]. Graves and Whinston [4] indicated how the variance of QAP solutions could be
determined. However, up to this date, a polynomial expression for this variance was not
developed.
1
2
Programa de Engenharia de Produção, COPPE/UFRJ
CEFET-RJ
e-mails: {boaventu, nair, bgouvea}@pep.ufrj.br;
querido@openlink.com.br
N
Consider S   f r
r 1
N
d
s 1
s
. The average of the costs z for QAP(F,D) is  = S / N ,
over   n [4], [1], [5]. Seeking a cost variance parameter, we first decompose
 z   f

2
 n
1i  j  n
1 r  s  n
ij
d  (i ) ( j ) f rs d  ( r ) ( s ) , and define  k  {{(i , j ),(r , s)} / |{(i , j )  (r , s)}|  k} ,
where k = 0, 1, 2 and 1  i , j , r , s  n . Thus, we have the variance expression for the QAP
in O( N 2 ) as stated in Theorem 3.1.
Theorem 3.1 -  FD 2  [ S0  S1  S2 ] / n!   2 ,
where: S0  4(n  4)!  f ij f rs  d ij d rs , S1  (n  3)!  f ij f rs  d ij d rs and
0
S2  2(n  2)!
0
 f d
1i  j n
2
ij
1i  j n
1
2
ij
1
.
4 - Variance applications: Verifying isomorphism in QAP instances is a hard task
inducing to the analysis of the n! possibilities. Firstly, the study of variance costs leads to
isomorphism of QAP in the sense that given two isomorphic instances QAP ( F1 , D1 ) and
QAP( F2 , D2 ) in Relclass( Q * ), they will have the same set of feasible solutions and
2
2
  FD
consequently  FD
. In practice,  FD 2 can be view as an invariant for the instances
1
2
isomorphism study, pointing out non-isomorphic instances. The instances Chr12a and
Chr12b, available at the QAPLIB [2] are suitable examples of instances sharing the same
Relclass( Q * ). However, these instances have the respective standard deviations 4413.99
and 4269.89, characterizing non-isomorphic instances.
A second application concerns to a new statistical approach that shall guide this
combinatorial subject to some more generic and sensible difficulty measure than the flow
dominance (FD(F)) used by Mautor and Roucairol [5]. We define as solutions deviation
factor SDF(Z) = 100  /  . The SDF(z) is able to indicate harder and easier versions in
Re lclass(Q * ) , while FD(F) is not because it is constant in this family. Finally, we
suggest the use of variance as a decision parameter on a most suitable procedure to solve
the problem or even evaluate the current value obtained by a given heuristic in
optimization problems.
References
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Universitat, Graz (1994).
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E. Çela, The quadratic assignment problem: theory and algorithm, in D.Z. Du and P.Pardalos (Editors), Kluwer Academic
Publishers (1998).
4.
5.
G.W. Graves and A.B. Whinston, An algorithm for the quadratic assignment problem, Management Science, v.17, n.7, (1970).
6.
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branch-and-bound algorithm for the quadratic assignment problem, SIAM J. Optim. 7,1,(1997)280-294.
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