Quadrati Assignment Problem
FranisoChiano,GabrielLuque,EnriqueAlba
E.T.S.IngenieraInformatia
UniversityofMalaga,Spain
Abstrat
InthisartileweprovideanexatexpressionforomputingtheautoorrelationoeÆient
andtheautoorrelationlength`ofanyarbitraryinstaneoftheQuadratiAssignment
Problem (QAP)in polynomialtimeusing itselementary landsapedeomposition. We
also provide empirial evidene of the autoorrelation length onjeture in QAP and
ompute the parameters and ` for the 137instanes of the QAPLIB. Ourgoal is to
better haraterizethe diÆulty of this importantlass of problems to easethe future
denition ofnewoptimization methods. Also,theadvanethat thisrepresentshelpsto
onsolidateQAPasaninterestingandnowbetterunderstoodproblem.
Keywords: Fitnesslandsapes,elementarylandsapes,quadratiassignmentproblem,
autoorrelationoeÆient,autoorrelationlength
1. Introdution
A landsape for a ombinatorial optimization problem is a triple (X;N;f), where
f :X !Ristheobjetivefuntiontobeminimized(ormaximized)andtheneighborhood
funtion N mapsasolutionx2X to theset ofneighboringsolutions. Ify2N(x)then
y is aneighborofx. Thereisaespeialkindoflandsape,alledelementary landsape,
whih is of partiular interest in present researh due to their properties. They are
haraterizedbytheGrover's wave equation[1℄:
avgff(y)g
y2N(x)
=f(x)+ k
d
f f(x)
(1)
where dis thesize ofthe neighborhood, jN(x)j, whih weassume thesamefor all the
solutionsinthesearhspae(regularneighborhood),f istheaveragesolutionevaluation
overtheentiresearh spae,andk isaharateristi(problem-dependent)onstant. A
general landsape (X;N;f)an notalwaysbe said to beelementary, but even in this
ase itispossibleto haraterizethefuntion f asasumof elementarylandsapes[2℄,
alled theelementaryomponents ofthelandsape.
Emailaddresses: hianol.uma.es (FranisoChiano),gabriell.uma.es(GabrielLuque),
optimization problem that isat theore ofmanyreal-worldoptimization problems [3℄.
A lot of researh has been devoted to analyze and solve the QAP itself, and in fat
someother problemsan beformulatedasspeial asesof theQAP,e.g.,theTraveling
Salesman Problem(TSP).Let P beaset ofn failitiesandLaset ofn loations. For
eahpair ofloationsi andj,an arbitrarydistaneisspeiedr
ij
andforeahpairof
failitiespandq,aowisspeiedw
pq
. TheQAPonsistsinassigningtoeahloation
in L onefailityin P in suh awaythat thetotalostoftheassignmentisminimized.
Eahloationan onlyontainonefailityandallthefailitiesmustbeassignedtoone
loation. Foreah pairofloationstheostisomputedastheprodutofthedistane
between the loations and the ow assoiated to the failities in the loations. The
total ostis thesum of alltheosts assoiated to eah pairof loations. Onesolution
to this problem is a bijetion between L and P, that is, x : L ! P suh that x is
bijetive. Withoutlossofgenerality,weanjustassumethatL=P =f1;2;:::;ngand
eah solutionx is apermutation in S
n
, theset permutationsof f1;2;:::;ng. The ost
funtion tobeminimizedanbeformallydenedas:
f(x)= n
X
i;j=1 r
ij w
x(i)x(j)
(2)
In [4, 5℄ the authors analyzed the QAP from the point of view of landsapes
the-ory [6℄ and they found the elementary landsape deomposition of the problem using
themethodologypresentedin[7℄,providingexpressionsforeahelementaryomponent.
In this paper we use the elementary deomposition of the previous work to ompute
the autoorrelationlength ` andthe autoorrelation oeÆient ofanyQAP instane
in polynomialtime (Setion 2). Wealso present in Setion 3empirialevidene of the
autoorrelation length onjeture [8℄, whih links these values to the number of loal
optima of a problem, and we numerially ompute ` and for the well-known publi
instanes oftheQAPLIB[9℄.
2. Autoorrelation ofQAP
Letusonsideraninniterandomwalkfx
0 ;x
1
;:::gonthesolutionspaesuhthat
x
i+1 2N(x
i
). Therandom walk autoorrelationfuntion r:N !R isdened as[10℄:
r(s)= hf(x
t )f(x
t+s )i
x0;t hf(x
t )i
2
x0;t
hf(x
t )
2
i
x0;t hf(x
t )i
2
x0;t
(3)
wherethesubindiesx
0
andtindiatethattheaveragesareomputedoverallthe
start-ing solutionsx
0
and along the omplete random walk. The autoorrelation oeÆient
of a problem is a parameter proposed by Angel and Zissimopoulos [11℄ that gives a
measureofitsruggedness. Itisdenedafterr(s)by =(1 r(1)) 1
[12℄. Another
mea-sureofruggednessistheautoorrelationlength `[13℄whosedenitionis`= P
1
s=0 r(s).
The autoorrelationoeÆient fortheQAPwasexatlyomputed byAngeland
Zis-simopoulos in [14℄. However,reentresults(see [4℄)suggestthat theexpression in [14℄
present(withoutproof)theresultsof[5℄thatarerelevanttoourgoal.
Proposition 1 (Deompositionof theQAP). For the swap neighborhood, the funtion
f denedin (2) an be written asthe sum of atmost three elementary landsapes with
onstants k
1
=2n, k
2
= 2(n 1), and k
3
= n: f = f
1 +f
2 +f
3
. The elementary
omponents an bedenedas
f
1 =
n
X
i;j;p;q=1
i6=j;p6=q ijpq
1
(i;j);(p;q)
2n
(4)
f
2 =
n
X
i;j;p;q=1
i6=j;p6=q ijpq
2
(i;j);(p;q)
2(n 2)
(5)
f
3 =
n
X
i;j;p;q=1
i6=j;p6=q ijpq
3
(i;j);(p;q)
n(n 2) +
n
X
i;p=1 iipp
'
(i;i);(p;p)
(6)
where
ijpq = r
ij w
pq , '
(i;i);(p;p)
is the funtion dened using the Kroneker's delta by
'
(i;i);(p;p)
(x) = Æ p
x(i)
, and the funtions are partiular ases of the parameterized
funtionsdenedas:
;;;";
(i;j);(p;q) (x)=
8
>
>
>
>
<
>
>
>
>
:
ifx(i)=p^x(j)=q
ifx(i)=q^x(j)=p
ifx(i)=px(j)=q
" ifx(i)=qx(j)=p
ifx(i)6=p;q^x(j)6=p;q
(7)
Thedenitionofthefuntionsisasfollows: 1
(i;j);(p;q) =
n 3;1 n; 2;0; 1
(i;j);(p;q)
, 2
(i;j);(p;q) =
n 3;n 3;0;0;1
(i;j);(p;q)
,and 3
(i;j);(p;q) =
2n 3;1;n 2;0; 1
(i;j);(p;q) .
Proof. See[5℄fortheproof.
Proposition 2 (Autoorrelation measures). The autoorrelation oeÆient , the
au-toorrelation length `, and the autoorrelation funtion r(s) an be omputed from the
atual problem data(instane) usingthe expressions:
=
W
1 4
n 1
+W
2 4
n +W
3 2
n 1
1
=
n(n 1)
2n(1+W
1 )+2W
2 (n 2)
(8)
`=d
W
1
2n +
W
2
2(n 1) +
W
3
n
= W
1
(1 n)+W
2
(2 n)+2(n 1)
4
(9)
r(s)=W
1
1 4
n 1
s
+W
2
1 4
n
s
+W
3
1 2
n 1
s
(10)
where the oeÆientsW
i
for i=1;2;3are denedby
W
i =
f 2
i f
i 2
f 2
f 2
(10)isprovenin[2℄. WealsousedthefatthatW
1 +W
2 +W
3
=1toremoveW
3 inthe
expressionsfor and`.
Asaonsequene,weonlyneedtoomputeW
1 andW
2
toobtain and`. Thus,we
provideinthispapersomepropositionsthatallowustoeÆientlyomputeW
1 andW
2 .
Aordingto (11)weneedtoomputef 2
,f 2
,f 2
1 ,f
1 2
,f 2
2 ,andf
2 2
. Letus startwith
f
1 andf
2 .
Proposition3. Two expressionsfor f
1 andf
2 are:
f
1 =
r
t w
t
2n
(12)
f
2 =
r
t w
t (n 3)
2(n 1)(n 2)
; (13)
where r
t andw
t
aredenedas:
r
t =
n
X
i;j=1
i6=j r
ij
; w
t =
n
X
p;q=1
p6=q w
pq
(14)
Proof. Theaveragevalueof 1
and 2
is 1
= 1,and 2
=(n 3)=(n 1)[4℄. Using
these averagevaluesweanomputef
1 andf
2
withthehelp of(4)and(5)as:
f
1 =
1
2n n
X
i;j;p;q=1
i6=j;p6=q ijpq
; f
2 =
n 3
2(n 1)(n 2) n
X
i;j;p;q=1
i6=j;p6=q ijpq
(15)
Takingintoaountthat
ijpq =r
ij w
pq
andusing thenotationr
t , w
t
dened above
weantransform(15)in (12)and(13).
Bothexpressions(12)and(13)anbeomputedinO(n 2
). Beforegivinganexpression
forf letusrstintrodueanewfuntion t
n
denedas:
t
n
:P(f1;:::;ng 2
) ! N
Q 7! t
n (Q)=
X
x2Sn Y
(i;p)2Q Æ
p
x(i)
(16)
Thisfuntionwillbeusefullaterintheomputationoff,f 2
,f 2
1 ,andf
2
2
. Aording
to its denition,the evaluation of t
n
is noteÆient sineit requiresasummation over
all thepermutations in S
n
. However,weansimplify theexpression oft
n
to makethe
omputation moreeÆientasthefollowingpropositionstates.
Proposition4. The funtion t
n
satisesthefollowing equality:
t
n (Q)=
(n jQj)! if jQ
1 j=jQ
2 j=jQj
0 otherwise
; (17)
where Q
1 (Q
2
n
elementsin S
n
that fulll theondition V
(i;p)2Q
x(i) =p. Now,wemust observethat
if we nd twopairs(i;p) and (j;q)in Qsuh that i =j and p6=q, then the valueof
t
n
(Q) must be zerobeauseit isnot possibleto satisfy at thesametime x(i)=pand
x(j)=q. Wean haraterizethissituation usingtheondition jQ
1
j6=jQj. That is, if
thenumberofpairsinQisnotequaltothenumberofrstelementsofthesepairs,then
there existinQatleasttwopairsoftheform(i;p)and(i;q)withp6=qandt
n
(Q)=0.
Forthesamereason,t(Q)=0ifjQ
2
j6=jQj. IfjQj=jQ
1 j=jQ
2
jthen thepairsinQx
the valuefor jQj omponentsof thesolution vetorand the number of solutionsin S
n
withthexedomponentsist
n
(Q)=(n jQj)!.
One wehavedenedthet
n
funtion and weknowaneÆientwayofomputingit
weanprovideanexpressionforf.
Proposition5. An expressionfor f is:
f = r t w t n(n 1) + r d w d n (18) where r d = P n i=1 r ii andw d = P n p=1 w pp .
Proof. Usingthedenition off andt
n
weanwrite:
f = 1 jS n j n X i;j;p;q=1 ijpq X x2Sn Æ p x(i) Æ q x(j) ! = 1 n! n X i;j;p;q=1 ijpq t n
(f(i;p);(j;q)g) (19)
Ifwetakeintoaountthatt
n
anonlytaketwodierentvalues,weanrewritethe
previousexpressionas:
f =
(n 2)!
n!
n
X
i;j;p;q=1
i6=j;p6=q ijpq + (n 1)! n! n X i;p=1 iipp = r t w t n(n 1) + r d w d n (20)
With thehelp ofthefuntion t
n
weanalsoprovideanexpressionforf 2
.
Proposition6. An expressionfor f 2 is: f 2 = 1 n! n X i;j;p;q=1 n X i 0 ;j 0 ;p 0 ;q 0 =1 ijpq i 0 j 0 p 0 q 0t n
(f(i;p);(j;q);(i 0 ;p 0 );(j 0 ;q 0 )g) (21)
whih anbeomputedinO(n 8
).
Proof. Usingthedenition off weanwrite:
f 2 = 1 jS n j X x2Sn 0 n X i;j;p;q=1 ijpq Æ p x(i) Æ q x(j) 1 A 2 = 1 n! X x2Sn n X i;j;p;q=1 n X i 0 ;j 0 ;p 0 ;q 0 =1 ijpq i 0 j 0 p 0 q 0 Æ p x(i) Æ q x(j) Æ p 0 x(i 0 ) Æ q 0 x(j 0 ) (22)
whih anbetransformedinto(21)byommutingthesumsandusing thedenitionof
t
The omputation of f
1 , f
2
requires amore omplextreatment. Wepresent their
expressionsin thefollowing
Proposition7. Two expressionsfor f 2 1 andf 2 2 are: f 2 1 = 1 4n 2 n! n X
i;j;p;q=1
i6=j;p6=q
n X i 0 ;j 0 ;p 0 ;q 0 =1 i 0 6=j 0 ;p 0 6=q 0 ijpq i 0 j 0 p 0 q 0 7 X m=1 7 X m 0 =1 1 m 1 m 0t n v i;j;p;q m [v i 0 ;j 0 ;p 0 ;q 0 m 0 ! (23) f 2 2 = 1 4(n 2) 2 n! n X
i;j;p;q=1
i6=j;p6=q
n X i 0 ;j 0 ;p 0 ;q 0 =1 i 0 6=j 0 ;p 0 6=q 0 ijpq i 0 j 0 p 0 q 0 7 X m=1 7 X m 0 =1 2 m 2 m 0 t n v i;j;p;q m [v i 0 ;j 0 ;p 0 ;q 0 m 0 ! (24)
where the 7-dimensional parameterized vetors v 2 P(N 2
)
7
and 2 R 7
are given in
Table 1and
1
and
2
denote the vetors whose parameters;;;"; arethose of
1
and 2
,respetively,that is,
1
=
n 3;1 n; 2;0; 1
and
2
=
n 3;n 3;0;0;1
.
Component(m) v i;j;p;q
;;;";
1 ;
2 f(i;p)g ( )
3 f(i;q)g (" )
4 f(j;q)g ( )
5 f(j;p)g (" )
6 f(i;p);(j;q)g ( 2+)
7 f(i;q);(j;p)g ( 2"+)
Table1:Contentofthevetorsv i;j;p;q
and ;;;";
.
Proof. Afterthedenition off
1 andf
2
weanwrite:
f 2 1 = 1 4n 2 n! n X
i;j;p;q=1
i6=j;p6=q
n X i 0 ;j 0 ;p 0 ;q 0 =1 i 0 6=j 0 ;p 0 6=q 0 ijpq i 0 j 0 p 0 q 0 X x2Sn 1 (i;j);(p;q) (x) 1 (i 0 ;j 0 );(p 0 ;q 0 ) (x) ! (25) f 2 2 = 1 4(n 2) 2 n! n X
i;j;p;q=1
i6=j;p6=q
n X i 0 ;j 0 ;p 0 ;q 0 =1 i 0 6=j 0 ;p 0 6=q 0 ijpq i 0 j 0 p 0 q 0 X x2Sn 2 (i;j);(p;q) (x) 2 (i 0 ;j 0 );(p 0 ;q 0 ) (x) ! (26)
Inthisaseitisnotsosimpleto writetheinnersummationasafuntion oft
n . We
willwritethefuntionsaslinearombinationsofKroneker'sdeltasusingthedenition
of the funtions and the following haraterization of the funtions, whih anbe
(i;j);(p;q)
(x)=Æ
x(i) Æ x(j) +Æ x(i) Æ x(j) +(Æ x(i) Æ x(j) ) 2 + +"(Æ q x(i) Æ p x(j) ) 2
+(1 Æ p x(i) )(1 Æ q x(i) )(1 Æ p x(j) )(1 Æ q x(j) )= =( )(Æ p x(i) +Æ q x(j)
)+(" )(Æ q x(i) +Æ p x(j) )+ +Æ p x(i) Æ q x(j)
( 2+)+Æ q
x(i) Æ
p
x(j)
( 2"+)+ (27)
Thus, ;;;";
(i;j);(p;q)
isasumofsixtermswithÆ andoneonstant,and thesummation
X x2S n ;;;"; (i;j);(p;q) (x) ;;;"; (i 0 ;j 0 );(p 0 ;q 0 ) (x) (28)
an be written as a weighted sum of 49 t
n
terms. In order to write this summation
in a ompat way we dene one vetordenoted with v i;j;p;q
ontaining the sets to be
onsidered in the t
n
termsand a vetor ;;;";
ontaining theoeÆients for the t
n
terms. Theontentofthe previousvetorsis shown in Table 1. Usingv and wean
write thesummationoftheprodutoffuntionsinthefollowingway:
X x2S n ;;;"; (i;j);(p;q) (x) ;;;"; (i 0 ;j 0 );(p 0 ;q 0 ) (x)= 7 X m=1 7 X m 0 =1 ;;;"; m ;;;"; m 0 t n v i;j;p;q m [v i 0 ;j 0 ;p 0 ;q 0 m 0 (29)
and usingthepreviousequalityin(25)and(26)weobtain(23)and(24).
NowwehaveeÆientexpressionsforomputingf,f 2 ,f 1 ,f 2 1 ,f 2 ,andf
2
2
. Withthis
expressionsweareinonditionsof eÆientlyomputingtheautoorrelationmeasures
and `. Thisresultissummarized inthefollowing
Theorem1(EÆientomputationofand`). IntheQAP,thevaluesofand`related
tothe swap neighborhood anddenedby
=
n(n 1)
2n(1+W
1 )+2W
2 (n 2) [eq:(8)℄ `= W 1
(1 n)+W
2
(2 n)+2(n 1)
4
[eq:(9)℄
anbeomputedinpolynomialtimeoverthesizeof the problemn usingequations(12),
(13), (18),(21), (23),and(24).
Proof. Afteromputingf,f
1 ,f 2 ,f 2 , f 2 1 ,andf
2
2
usingtheequations(18),(12),(13),
(21), (23), and (24) we should ompute W
1
and W
2
using equation (11). Then, the
autoorrelationoeÆient an beobtainedwith(8) and` anbeomputed with(9).
Noneofthepreviousequationsrequiresmorethaneightnestedsummationsovernand,
thus, theomputationanbedoneinO(n 8
).
We have gone one step further and we haveexpanded the expressions for f 2 , f 2 1 , and f 2 2
in ordertomakeamoreeÆientomputation. TheresultisaO(n 2
)algorithm
(whih we omit due to spae onstraints) to ompute ` and . It is not diÆult to
provethatsuhalgorithmisoptimalin omplexity,sinethedataof aQAPinstaneis
omposedof2n 2
numberswhihhaveto betakenintoaountinorder toomputethe
Theautoorrelationlengthisspeiallyimportantin optimizationbeauseof the
au-toorrelationlengthonjeture,whihlaimsthatinmanylandsapesthenumberofloal
optima M an beestimated by theexpression M jXj
jX(x
0 ;`)j
[8℄, where X(x
0
;`) is the
set of solutionsreahablefrom x
0
in ` (the autoorrelation length) orlessloal
move-ments (jumps betweenneighbors). Thepreviousexpression is notanequation, but an
approximation. Itanbeusefultoomparetheestimatednumberofloaloptimaintwo
instanes ofthesameproblem. Ineet,foragivenprobleminwhihtheonjetureis
appliable, the higherthevalue of` (or ) thelowerthenumberof loal optima. Ina
landsapewith alownumberof loal optima, aloal searh strategyan apriori nd
the global optimum using less steps. This phenomenon has been empirially observed
fortheQuadratiAssignmentProblem(QAP)byAngelandZissimopoulosin[14℄.
InordertohektheautoorrelationlengthonjetureintheQAPwehavegenerated
4000randominstanesofQAPwithsizesvaryingbetweenn=4andn=11(500foreah
value ofn)usingarandomgeneratorwhere theelementsof thematriesareuniformly
seletedfromtherange[0,99℄. Foreahinstaneweomputedtheautoorrelationlength
` using (9) and the numberof loal optima(minima) by omplete enumeration of the
searhspae. WeomputedtheSpearmanorrelationoeÆientofthenumberofloal
optimaand`fortheinstanesofthesamesize. TheresultsareshowninTable2. Wean
observeaninverseorrelation(around 0:3)betweenthenumberofloaloptimaandthe
autoorrelationlength. Althoughthisfatisinagreementwiththeautoorrelationlength
onjeture,theorrelationoeÆientislow. However,AngelandZissimopoulos[14℄used
asimulatedannealing algorithmbasedontheswapneighborhoodandreportedabetter
performane of the algorithm asthe autoorrelation length inreased. Assuming that
thenumberofloaloptimaisaparameterwithanimportantinueneonthesearh,we
onludethat eveninproblems inwhihthenumberofloaloptimaislowlyorrelated
with `(likeQAP)theautoorrelationmeasures( and`)anbeusefulasestimatorsof
theperformaneofloalsearhalgorithms.
n 4 5 6 7 8 9 10 11
0:3256 0:2317 0:2126 0:3195 0:3032 0:2943 0:2131 0:1640
Table2: Spearman orrelation oeÆient for the number of loal optimaand the autoorrelation
length.
InFigure1weplotthenumberofloal optimaagainsttheautoorrelationlength`
foralltheinstanesofsizen=10. Weanobserveaslighttrend: astheautoorrelation
lengthinreasesthenumberofloal optimadereases. Thetrendisthesamein allthe
instanes withdierentsizes(weomittheirplots).
Inaseond experimentwehekthat theautoorrelationmeasuresprovided bythe
elementarylandsapedeompositionarethesameastheonesomputedusingstatistial
methods. ForthisexperimentwehavehosensixinstanesoftheQAPLIB[9℄: twosmall,
twomedium andtwolargeinstanes. Foreahinstanewehavegeneratedonerandom
walk of length 1 000 000 and we haveomputed the r(s) values for s 2 [0;49℄. This
proess hasbeen repeated 100 times and we haveomputed the average value for the
100independentruns. Theresultsempiriallyobtainedandthosetheoretiallypredited
200
300
400
500
600
700
800
900
1000
1100
1200
1300
4.34
4.36
4.38
4.4
4.42
4.44
4.46
4.48
4.5
# Local optima
replaemen
ts
`
Figure 1: Numberof loal optimaagainst theautoorrelation length `for randominstanes ofQAP
withn=10.
between the empirial and the theoretial value, as expeted. The advantage of the
theoretial approahis thatitis muh faster. Theexperimental resultsofTable3were
obtainedafter157783seondsofomputation(morethan43hours). However,theexat
valueswereobtainedevaluatingEquation(10) in0:4 seonds, nearhalf amilliontimes
faster.
Finally,wehaveomputedthevaluesof and`forthe137QAPinstanesfoundin
theQAPLIBdatabase[9℄. Theresults,showninTable4inalphabetialorder,ouldbe
helpfulforfutureinvestigationsontheQAP.Inthetableweanobservesomeinteresting
behaviours,likethatoftheesinstanes,whihhavealwaysavalueofn=4for and`.
This happens beausein those instanes W
1 =W
3
= 0and W
2
= 1,that is, theyare
elementary landsapeswith k =2(n 1). All theelementary landsapeshave avalue
fortheautoorrelationmeasuresthatdoesnotdependontheinstanedata,butonlyon
the problemsize. Intheaseof es16f,theobjetivefuntion isaonstant,that is,it
takesthesamevalueforeverysolutionandtheautoorrelationmeasuresmakenosense.
Weshouldalsonotiethat thevalueof`and dependonn,thesizeoftheproblem
instane. Ineet,thevaluesarebounded(see[4℄)by
n 1
4
;`
n 1
2
(30)
Thus, the values of and ` usually inreasewith the problem size n. As a
onse-quene,theautoorrelationlengthonjetureanbeappliedonlywhen theomparison
is performed over instanes with the same size n and, in general, it is not true that
the higherthe valueof `theeasierto solvetheinstane,sinethelargestinstanesare
tai10a
E 0.624255 0.393489 0.250810 0.161890 0.106102 0.070590
T 0.624380 0.393590 0.250903 0.162013 0.106129 0.070617
es16a
E 0.749984 0.562424 0.421759 0.316365 0.237300 0.177939
T 0.750000 0.562500 0.421875 0.316406 0.237305 0.177979
es64a
E 0.937402 0.878700 0.823668 0.772063 0.723672 0.678292
T 0.937500 0.878906 0.823975 0.772476 0.724196 0.678934
lipa70a
E 0.943369 0.890041 0.839723 0.792267 0.747507 0.705296
T 0.943479 0.890170 0.839890 0.792466 0.747735 0.705545
tho150
E 0.975680 0.951974 0.928863 0.906338 0.884384 0.862981
T 0.975722 0.952060 0.928997 0.906518 0.884607 0.863251
tai256
E 0.984364 0.968983 0.953843 0.938935 0.924256 0.909805
T 0.984375 0.968994 0.953854 0.938950 0.924279 0.909837
Table3: Experimental(E)andexat(T)valuesfortheautoorrelationfuntionr(s)insixinstanesof
theQAPLIB(sfrom1to6).
Instane ` Instane ` Instane ` Instane `
bur26a 11.825 12.130 es32b 8.000 8.000 nug16a 4.475 4.796 tai100b 35.472 39.613
bur26b 11.727 12.073 es32 8.000 8.000 nug16b 4.472 4.792 tai10a 2.662 2.774
bur26 12.109 12.291 es32d 8.000 8.000 nug17 4.836 5.220 tai10b 3.002 3.253
bur26d 12.050 12.258 es32e 8.000 8.000 nug18 5.111 5.516 tai12a 3.419 3.674
bur26e 12.032 12.248 es32f 8.000 8.000 nug20 5.800 6.311 tai12b 3.358 3.586
bur26f 11.962 12.208 es32g 8.000 8.000 nug21 6.218 6.807 tai150b 40.458 42.947
bur26g 12.323 12.407 es32h 8.000 8.000 nug22 6.751 7.446 tai15a 3.858 3.946
bur26h 12.296 12.392 es64a 16.000 16.000 nug24 7.067 7.737 tai15b 7.000 7.000
hr12a 3.096 3.171 had12 3.743 4.092 nug25 7.308 7.987 tai17a 4.402 4.526
hr12b 3.201 3.346 had14 4.319 4.732 nug27 8.023 8.813 tai20a 5.211 5.385
hr12 3.044 3.079 had16 4.405 4.690 nug28 8.181 8.949 tai20b 6.866 7.582
hr15a 3.917 4.049 had18 5.084 5.477 nug30 8.613 9.373 tai256 64.000 64.000
hr15b 4.126 4.388 had20 5.830 6.352 rou12 3.158 3.275 tai25a 6.373 6.482
hr15 3.843 3.920 kra30a 9.131 10.089 rou15 3.927 4.066 tai25b 6.896 7.374
hr18a 4.585 4.658 kra30b 9.086 10.031 rou20 5.354 5.628 tai30a 7.779 8.021
hr18b 4.632 4.742 kra32 9.848 10.908 sr12 3.407 3.657 tai30b 7.599 7.689
hr20a 5.105 5.195 lipa20a 5.072 5.135 sr15 4.303 4.650 tai35a 8.922 9.077
hr20b 5.035 5.067 lipa20b 5.196 5.358 sr20 5.514 5.885 tai35b 9.382 9.895
hr20 5.260 5.469 lipa30a 7.622 7.732 sko100a 27.800 29.985 tai40a 10.216 10.413
hr22a 5.763 5.980 lipa30b 7.652 7.787 sko100b 28.106 30.470 tai40b 10.583 11.074
hr22b 5.672 5.819 lipa40a 10.154 10.295 sko100 27.548 29.578 tai50a 12.675 12.839
hr25a 6.490 6.693 lipa40b 10.355 10.669 sko100d 27.535 29.557 tai50b 12.824 13.119
els19 5.178 5.494 lipa50a 12.684 12.855 sko100e 27.600 29.663 tai60a 15.292 15.563
es128 32.000 32.000 lipa50b 12.854 13.174 sko100f 27.346 29.247 tai60b 17.837 19.691
es16a 4.000 4.000 lipa60a 15.111 15.217 sko42 11.559 12.378 tai64 16.000 16.000
es16b 4.000 4.000 lipa60b 15.124 15.243 sko49 13.413 14.331 tai80a 20.214 20.419
es16 4.000 4.000 lipa70a 17.693 17.876 sko56 15.598 16.817 tai80b 24.021 26.612
es16d 4.000 4.000 lipa70b 17.785 18.052 sko64 17.504 18.706 tho150 41.190 44.174
es16e 4.000 4.000 lipa80a 20.102 20.201 sko72 19.929 21.436 tho30 8.326 8.938
es16f lipa80b 20.191 20.373 sko81 22.739 24.629 tho40 11.492 12.531
es16g 4.000 4.000 lipa90a 22.610 22.716 sko90 25.046 27.024 wil100 28.362 30.868
es16h 4.000 4.000 lipa90b 22.733 22.957 ste36a 10.954 12.122 wil50 13.832 14.860
es16i 4.000 4.000 nug12 3.135 3.237 ste36b 11.821 13.177
es16j 4.000 4.000 nug14 3.892 4.155 ste36 11.270 12.525
es32a 8.000 8.000 nug15 4.029 4.234 tai100a 25.195 25.383
Table4:AutoorrelationoeÆientandautoorrelationlength`forthe137instanesoftheQAPLIB.
Inthisartilewegiveanoptimalwayofexatlyomputingtheautoorrelation
mea-sures and` for theQAP. These twoparametersareimportant to better haraterize
QAPandtoguidepratitionersintherelativediÆultyoftheexistingprobleminstanes.
These resultsanbeautomatiallyapplied toallthesubproblemsofQAP,likedeTSP.
Themain ontributionsofthisworkare:
An exatexpression foromputingtheautoorrelationoeÆient andthe
auto-orrelationlength`of theQAPin polynomialtime.
EmpirialevideneoftheautoorrelationlengthonjetureinpratiefortheQAP,
byusingarbitrarilygeneratedinstanes.
Thenumerialvalueof and`foralltheinstanesintheQAPLIBdatabase.
Asafutureworkweplantoobtainexatexpressionsfortheautoorrelationmeasures
inotherproblems,andstudytheatualpratialappliationsoftheinformationobtained
from them.
Aknowledgements
ThisworkhasbeenpartiallyfundedbytheSpanishMinistryofSieneandInnovation
and FEDER under ontrat TIN2008-06491-C04-01(M
projet) and the Andalusian
Governmentunder ontratP07-TIC-03044(DIRICOMprojet).
[1℄ L.K.Grover,LoalsearhandtheloalstrutureofNP-ompleteproblems,OperationsResearh
Letters12(1992)235{243.
[2℄ P.F.Stadler,Landsapesandtheirorrelationfuntions,JournalofMathematialChemistry20
(1996)1{45.
[3℄ M. R. Garey, D. S. Johnson, Computers and Intratability: A Guide to the Theory of
NP-Completeness,W.H.Freeman,1979.
[4℄ F.Chiano,G.Luque,E.Alba,Elementarylandsapesdeompositionofthequadratiassignment
problem,in:ProeedingsofGECCO2010,ACM,Portland,OR,USA,2010,pp.1425{1432.
[5℄ F.Chiano, G.Luque, E. Alba, Elementaryomponents of the quadrati assignment problem,
arXiv:1109.4875v1,availablefromhttp://arxiv.org(september2011).
[6℄ J.Barnes,B.Dimova,S.Dokov,A.Solomon,Thetheoryofelementarylandsapes,Applied
Math-ematisLetters16(3)(2003)337{343.
[7℄ F.Chiano,L.D.Whitley,E.Alba,Amethodologytondtheelementarylandsapedeomposition
ofombinatorialoptimizationproblems,EvolutionaryComputationJournal19(4).
[8℄ P.F.Stadler,BiologialEvolutionandStatistialPhysis,Springer,2002,Ch.FitnessLandsapes,
pp.183{204.
[9℄ R.Burkard,S.Karish,F.Rendl,QAPLIB- aquadrati assignmentproblemlibrary,Journalof
GlobalOptimization10(1997)391{403.
[10℄ E.Weinberger,Correlatedandunorrelatedtnesslandsapesandhowtotellthedierene,
Bio-logialCybernetis63(5)(1990)325{336.
[11℄ E.Angel,V.Zissimopoulos,AutoorrelationoeÆientforthegraphbipartitioningproblem,
The-oretialComputerSiene191(1998)229{243.
[12℄ E.Angel,V.Zissimopoulos,OnthelassiÆationofNP-ompleteproblemsintermsoftheir
orre-lationoeÆient,DisreteAppliedMathematis99(2000)261{277.
[13℄ R.Gara-Pelayo,P.Stadler,Correlationlength,isotropyandmeta-stablestates,PhysiaD:
Non-linearPhenomena107(2-4)(1997)240{254.
[14℄ E.Angel, V.Zissimopoulos,Onthe landsaperuggednessof the quadrati assignment problem,
