(«WffWWJWMWW^WWW1*™»» if^fl^lWIIIW^I^WHKUUWIli mmmm^»m9i^^>(l!f^^,^m*n
U.S. DEPARTMENT OF COMMERCE National Technical Information Service
AD-A025 605
SPECIAL CASES OF THE QUADRATIC ASSIGNMENT PROBLEM
CARNEGIE-MELLON UNIVERSITY
PREPARED FOR
OFFICE OF NAVAL RESEARCH
APRIL 1976
V.
MWMiWfrMiiiftiiHM^^^
Best Available
Copy
'
173118
o Iß
o Carnegie-IVIel Ion University PITTSBURGH, PENNSYLVANIA 15213
GRADUATE SCHOOL OF INDUSTRIAL ADMINISTRATION WILLIAM LARIMER MELLON, FOUNDER
REPRODUCED BY
NATIONAL TECHNICAL \ INFORMATION SERVICE
U.S. DEPARTMENT OF COMMERCE SPRINGFIELD. VA. 22161
D DC
mmmm* wmmmmmm*mmm*immmmmmmmmm9m*mmmmm*<mmm
W.P.#69-75-76
Management Sciences Research Report No. 391
SPECIAL CASES OF THE QUADRATIC
ASSIGNMENT PROBLEM
Nicos Christofides* and M, Gerrardt
April 1976
r . n n c
j-
This research was pieparcd a . part of the activities of the Management Sciences Research Group, Carnesne-Mellon University. Reproduction in whole or in part is permitted for nr.v purpose of the U.S. Government.
Management Science Research Group Graduate School of Industrial Administration
Carnegie-Mellon Universif Pittshurgh, Pennsylvania 15.. i
*Carnegie-Mellon University, on leave from Imperial College, London tlmperial College, London
I.
iMinftttiiHi^tiiftiiiite^ iiiwiiimfeftrtnWifMt^fi^ mm
mmmm
REPORT DOCUMENTATION PAGF 2 äOVV ACCESSION NO.
|4
Technical Report No. 391 f i ' , r. (mnri Submit)
Special Cases of the Quadratic Assignment Problem
RE/tD INSTRJCr; ..s". "liKORE COMPLErrNt. FOKM
S «CClPlENT'S CATALOO NU^JE1*
'ft. Of REPORT ft PERIOD COVEhEC
Technical Report AgrU 1976
& ^ t*ror-viur, 0*0. REPI;M
UUHPHWUMII
I"
hr
Nicos Christoficles and M. Cerrard
Graduate School of Industrial Administration Carnegie-Mellon University Pittsburgh, Pennsylvania 15213
s ' MC". . N j Of f ICI w »Mt AUL i JD^ESS
Personnel and Training Research Programs Office of Naval Research (Code /4 34) Arlington. Virginia 22217 .. MONITORING AGENCY NAML a ADDHESSCI/ iJ//f«rw-.i Irom <. ..ntrolUnt lllllct)
NTRACV 0* 0*ANT NÜMoerir"., "'" 1
10 l»!« -.»AM tLCMCNT, CJi. i:. I ASp ARt * A WORK UNIT NUMHKH^
W RhP^RT OATB
April 1976 I XUUBCR Of PAO£5
15 15 ?rC'JRITY CLASS. Co/ -ft;- .,...r(
Unclassified
~5« beeil Assi r\c ATION '"OCS^HALTMO sent ouLir
os 'J r * rt^f I, i
Approved for public release, distribution unlimited
, 1 • I IN sT*1»M(-; I ilhtmrmnl Irrin Hmpoftt
PPL I k4f N ^ • I' • WO f '
! '6 K WOWCJ-J fC.'nr. nutf .m 'i*v*r*i. (• i/n«c»»4«/ « tt (u- '., 'j ^MJI A nuwfcer
The quadratic assignment problem, graph theory, dynamic programming
«•«•rr *rta i '
By considering the quadratic assignment problem (QA11) a^ that of minimizing the product of a distance-graph with a flow-graph sevral special cases of the QAP are investigated. A polynomial-growth algorithm Is described for the QAP when the distance and flow-graphs are Isomorphic trees. In the case when the graphs are single stars the algorithm becomes the well known rule for multiplying two sequences of numbers. The case of a complete distance- graph and a tree flow-graph becomes the travelling salesman problem when the tree is a hamlltonlan chain and the flows are all unity. A dynamic programmir
! ,r"RM 1473 tDITIOK OF I NOV 85 IS OBSOLETE » S N u ;ir;- o M • Mio 1
Unclassified (over)
^SECURITY CLASilFICATlON OF THIS PAOE (Wh»n Due Knlmr»d
^''''■■'Tay-T\\KnrfmmiM^
'''■■""-"rT^T^ '■' ■
ABSTRACT
Bv considering Lho quadratic assignment problem (QAP) as that
of minimizlnR the product of a distance-graph with a flow-graph several
special cases of the QAP are investigated. A polynomial-growth algorithm
is described for the QAP when the distance and flow-graphs are isomorphic
trees. In the case when the graphs are single stars the algorithm becomes
the well known rule for multiplying two sequences of numbers. The case
nl 11 complete distance-graph and r. tree flow-graph becomes the travelUn;-
salesman prohloin when the tree is n hami1tonian chain and the flows are
all enilv. A dynamic programming algorithm is presented for the case ol
the I low-graph being a general tree with arbitrary flows. The very special
case of "narrow" bipartite graphs is also considered.
il
■ irtVirtminii iMIiiiiimiiiiiiMliMi^^^ I' - I ■■
mmmmmmmm*** !^|^f^' pww^w^ww*^^ —■MM
1 . 1 NTRODUCTION
Consider p machines 1,...,a,•.•»P with a known flow of material
I ,, between every pair of machines (o,B) . Let there be q '% p locations OP
1, . . . , i , . ..,q ith known distances d.. between every pair of locations
(i,j). An assignment of machines to locations is a one to one mapping
p of Lhe set of machines into the set of locations, so that P(Q) is the
locntion thnt machine a is assigned to.
Tin' (ost nl' a mapping p is defined as
zip) ■- Z C .d , ^ ._ (1) a ■ ■ P
C.iven the two matrices [f | and [d. .|, tVie quadratic assignment Q-fci 1 j
probl.'in (QAP) is that of finding a mapping p* which minimizes z(p) as given
by (1).
The QAP appears in a number of spacial location problems such as
ihe allocation of machines to locations - used above to introduce the QAP -
the location of electronic components on ciniMt boards [12], the ordering
of Interrelated data on magnetic tape, etc. Other examples not involving
special location, but which can he (ormulatcd as QAP's include the trian-
gulai ization of economic Input-output matrices \'.'] , the minimization of
average job completion timt in machine scheduling [8] and extensions of
the travelling salesman problem [7].
A survey of exact algorithms for the general QAP is given by Pierce
and (rowston [11], and an improved algorithn i^ described ir [4]. Exact
algorithms, however, are unable to solve general QAP's of even moderate
size [4]. Approximate algorithms for the QAP are surveyed by Nugent et al
[lO] and Moore [9], while Sciabin and Vergin [131 demonstrate that these
are, in general, unsatisfactory.
/.
»^^,,..^.,^-^^.1.Mi.^,/^;itB^tl^ji|jtiiji.| iaiwi»wiiawi!rtria<
g-'iWBigtwumifj.wiu»w'wi^fcWTO,ffi^"'rwii'^ii^^^
In tills pa|i(M we cunsMer special cnses of the QA1' which are easier
tn solve. Bv i ecasl I ng the (JAP in graph thtnretlc terms ai the dot-product
of n di st ance-v.raph with a flow-nraph, we consider cases when these graphs
have special forms. Tn particular we describe a polynomial growth algorithm
lot the QAP when h.ith of these graphs are trees. When only one of the two
graphs is a tree and the other is a complete graph, the QAP can be solved
bv a special dynamic programming algorithm which is a generalization of a
similar algorithm for the travelling salesman problem. This last case occurs
very often in practical location problems,e.g., in the layout of an assembly
I i no .
T-^iBiiiMittlWMlMigitfiiiiiliiiiiiiii H ii I iiitif- ir iWtM-^HiiaT-mii^a»««»^ nWiiirwüMriMimrni
'.»W^IIWBKBW^BIiJ!^«^^ ypjWHJMllltl^Jtiilil^
-3-
2. GRAPHICAL REPRESENTATION
A graph G is defined by the doublet (X,A) where X is a set of
vertices and A a set of links. Unless otherwise specified we will use
"graph" to mean a "non-directed graph without loops." The terminology
used i s I nun [ 1 , *! •
(liven a «raph G (X ,A ) with a cost matrix [c . ] , an isomorphic
graph C" - (X , A') with a cost matrix [', .! and a mapping p of X onto
x', the dot-product graph is writttn - using the product operator ^(p) - ns;
' , V „ " G Ti(p")G
and is defined as the graph G = (X,A) isomorphic to G with costs given
by c . 1 !
cii • cp(i)p(.ir
The value of a graph G - (X,A) is defined as:
V(G) - Z <■•. . f x. , x .) eA ' '
i .1
An image of a ^raph G in a y;r.'iph G is anv pnrtial subgraph of G which
is isomorphic to G'. We will denote by M(G',G ) the family of all such
ima)',e y.r-'iphs. The cardinality of the set M(G',G I is called the image
number of G ' in G and is denoted by m(G .G ).
The QAP can now be restated in the following way.
Let G = (X ,A ) be a flow-graph, whose vertices X represent the
set of machines and the link costs are the flows between the corresponding
machines. Similarly, let G = (X ,A ") be a di.-- ance-graph, whose vertices
represent the set of locations and the link costs are the dis .luce:, between
the crrrespondinf locations. We will assume (without loss of generality)
that |xf| „ lxdl.
Inü-irm-r -11 -riifflgrima'iiaMiiiiiiiiriteiiriiiiiniii- mmm* rt l'i ■awiiTi^rfifcii' fifiii^Brffint ill' firil«^* tt'itrl i -äJ
!a??8sra^aBüw^^»ww ■a!B^i..^Pwap^J!i>ww*P^ im,Wmm*\¥''''m*m**»>™m
-4-
The QAP is then the problem of finding a graph G and a mapping
p ol C on C which minimizes the expression:
Min [Min V((: TT(PU;) i I .1 P
(?)
The nuinluT DI ilifleront mappings p nf C onto an isomorphic graph is the
i somorphic mimln'r *(C ) of C . Thus, the inner minimization of (?) is
ovor a set (if cardinality s(G ) and the outer minimization is over a set
f d of cardinality m(G ,G ).
It is reasonable to expect that as s(G ) and m(G ,G ) increase,
thr difficulty of the QAP will also increase. Cases when both s(G ) and
m((', ,G ) are small can be solved trivially by enumeration. Very few
special cases in which onlv ono of these two numbers is large can be
f <l sulvcd by pul ynomla 1 growth a ! ('.or i I Inns , The case where (. (I K Cthe
n
loinplcle grnpli on n vertices: is the problem usually considered in the
literature as the general QAP and has ni(G ,G ) = 1 and s(G ) -- n!
■tüü iJÄLSL^iiÄÜÜ > - ■ ■ '• nggmuii
MJi*,WI')jM|MJHWPMM^'W,J|^'^'w^PtW^|i>^ WtgtfW!WIBaMSiW«B!iPWIIP!!BPW^
I. CASr.S WJ TH j: _ C^_. { Imn^c- nunihe r -- 1)
Wlu'ii I lie lin/iv.«' nunilipr iii((i ,(• ) 1 the outer minimization ol (.'i
beconifs redundant and only the inner niinlmi zat Ion remains.
1. !. Trivial cases (small isomorphic number)
Trivial cases that can be solved by roraplete enumeration are:
(i) Chains: When C and G are chains, the isomorphic number
s((; ) = 2 and the inner minimization in expression (?) onlv
involves two evaluations.
(ii) Cyc les: The isomorphic number sfCi ) = ?n.
(iii) Wheels: The isomorphic numoer s(C ) = 2(n-l).
(iv) Regular graphs: Certain regular graphs (e.g., webs of low
order1) have small isomorphic numbers and can be enumerated.
There is, however, grent variation in the isomorphU numbers
of regular graphs even of the same degree as shown bv the
example in Fig. 1., and no general statement can be made.
I. f" . Solvable cases (large isomorphic number <
A,
vertex (with index IM and n outer vertices, the isomorphic number
is n! However, this special QAP can be solved by a well known
rule namely: Order the n flows f in ascending order, and the OQ
n distances ci . in descending order. The optimum mapping p*
then maps the k flow in the flow list to the k distance
in the distance list for all k = l....,n.
B. Multiple stars
The graph in Fig. 2 shows a 28-vertex 3rd order star
with vertex 1 as the center. Consider a general k-order multiple
Simple stars: When V. and C are simple stars with one central
»■^.■„■..i-^. m&imi*mm ': : " • ■ •■ u. -. i_ jy • ' v; -. ■ • •' ■,■ M A- ..- : - ■ --• ÜU^lI ^famtfe.g^^-';i;: :-.'■:: :.r: :-;^.:^r:;:- V.%- -. ■ ^■^i;:
|IPP£WWWyPi.<WJK>«^
\
N
vv
X
\ v s
\ ■ '\ '
, i
>
:. ( ( i ) -- I \;- HG) ^ 4
FlGj I Th£ llUNA^f PH1C K^KAtE-R/- Of T*JO RtlGul-AP-.
OR.^PHS or üC-^RE-C 4 (M-^
lüMteüi - r-'-
^WlilJilii^jjj^WiliW
-6-
star f.rapli arhitrarilv rooted at the center vertex and lot the
label Mxi nl vertex x be the cardinality of the path from
tin' rciitei luv. I In- label (if the iiulennouL vertlce.'i In then
k ami the label ol the center vertex is zero.
Wo describe below a dynamic programming algorithm for the
solution of QAP's involving arbitrary k-order stars.
Let the flow and distance graphs be G = (X ,A ) and
('. (X ,A 1 respectively. For any vertex x e X and x. e X
let c(j,n be the minimum cost of mapping x and all its successors a
(i.e., vertices reachable from x via arcs of the rooted tree) Q
to x. and all its successors. We will denote by x . , the i p(a)
predecessor ol vertex >: . Li
Description nl the nl^orilhm (lor k-order stars)
S'cn 1. For each >: e X and x. e X with -t.(x ) = -^ (x ) = k set a i Q i
P(Q\, P(i)i
Set LEVEL - k - 1
Step 2. For each x e X and x. e X with -t.(x ) = -t(x.) = LEVEL calculate a i (v i
c(a , i ) as follows:
(i) Let [ai,...,tr] -- iß|p(ß) = a]
and ij1,...,,ir] = [j \?< j'l - i]
(ii) Set up the linear assignment problem with cost matrix
(iii
c(B1>i1)
0(3^1,)
• c(P j ) 1 r
c(ß ,1 ) r r
and let V . be the value of the solution of this problem, ai
Update c(a,i) = f , x . d .... + V ., P(a)a p(i)i ai
ruMiiiHiirinrHii-ir -^—^-^
WpilPlPIBTO^w^ »WWW»i^wj«W!W«w«TOW«-w«W!!
XT'
/ O
V
>' of(x) = 5
p <X (. LNTtTR
/fc):2
\ -O n:"
XD
Ö
f id P.. ■ ri\iki> 'iv.i. ^ .\AW-. wriH hi = 2.^
T ^OMOKV^WC Mu.At.i k 4SS
^ ^
^^^^^^^..^u^^.^^ua^^ii^^^^
wmmmmm^^mmm^^^mmmmmmw^ ipiPPIPPPPipipPBi .T>v/!rf/i-^''-'r :"■ ■ ■, fPPPPBiipppppiM
-7-
Step 3. Set LEVEL = LEVEL -1. If LEVEL = 0 go to (4) else go to (2).
Step A. Stop. If x and x. are the center vertices of G and G Q i o o
respectively, C(Q ,i ) is the value of the solution to the QAP.
(Note: The mapping corresponding to this solution value can be
found by backtracking in the usual dynamic programming manner.)
The above algorithm is good, exhibiting polynomial rate of growth
with the total number of vertices in the k-order star. Thus, if each
k+1 vertex of the star has exactly m successors, there are (m -l)/(m-l)
k-1 2k-2 vertices in all. The algorithm involves 2m sortings, m" " evaluations
2k-2 0
and the solution of (m -IV(m -1) assignment problems of size m X m.
It should be noted here that in the first pass through step 2 of the above
algorithm, the solution of the assignment problems defined by C is unnecessary
since these assignment problems correspond to simple stars and L.I.. he
solved by the simple ordering described earlier. Thus, in the case of the
simple star the above algorithm disintegrates to the well known ordering
rule of i. '(A) .
C. Generil trees
Any tree can be arbitrarilv rooted and considered as a k-order
star with oo-cost (flow/distance) arcs. The algorithm described above
can therefore he used to solve QAP's with general trees. This is equivalent
to slightly motlifying the above algorithm so that c(Q,i) is finite only
for those pairs of vertices [x ,x.] for which the subtree defined by x a i Q
and its successors in G is isomorphic to the . ibtree defined by x. and i
its successors in G .
söiaaJsfiiÄaSiäaSÄa msämäMä^imääm ^fiiiiaMBiitiiii^^
gjg^wi^'w.ww^^iyw^tw^wv*-'"^^
1). Narrow Bipartite Cr.iphs
Consider the QAP when (I and (I are complete bipartite graphs,
say of the form K . Using the normal notation we will express G in terms i , s
f f f of its two independent vertex sets, i.e., G = (X ,X ) and similarly
G = (X ,X ). Wo call K narrow if min (r,s) •'• max (r,s). Let us r s r,s
assume that r = min (r,s) and specifically consider the case when r is small.
The isomorphic number of K _ is r!s!, however, if r is small enough, the r! r, s
possible mappings of X on X could be enumerated. For each such mapping p
we would then compute ¥ x G X and x. e X : a s is
c(a,i) = E f . d xf aß ip(ß)
ß r
and solve the s by s linear assignment problem with cost matrix [c(a,i)l.
The least cost assignment solution over all r! mappings p i^ then Llr solution
to the QAP.
Obviously, such a procedure is only practical when r is very small
7 5 (.say _ 5) but with a given r the complexity as ;i function of s is 0(s' ' )
since it only involves the linear assignment problem.
■Mmtfirtiritiiffi^
«flfSWWWWWTOSSWW ippwpwjwiippg^^^
-9-
4. CASI'S WIIKN C IS COMPTJ'TK
Wi* wl I 1 IIDW I .'ike C ID In' .i complete' ^rMpli on n vc'rticoH )m<l
conn I dor cases when ('. Is ol illlferont forms.
^ • Ü _ LK_ ü si"iplc sfnr with n vortices
The imaKP number m((; ,('• ) is n. Kach image corresponds to n
star partial graph of G with a specific center vertex. Once an image graph
G is chosen the optimum mapping p of G onto G can be found as in section
2 3.2(A) earlier. The total complexity of the procedure is therefore 0(n log n).
B, General trees with n vertices
Although the procedure for simple stars given in section 3 for the
case in which C - G is generalized above to the case in which G is a
complete praph, the corresponding algorithm of section 3.2(B) for k-order
stars (or arbitrary trees) does not generalize. The fact that such .i generaliza-
tion is not possible can be demonstrated by considering G to be a simple
chain of n vertices numbered consecutively Irom an end vertex and take f , = 1 o.crH
for all a = l,...,n-l. Wc now have s(G ) -- 2 and m(G ,G ) - l/2n! In fact,
the image graphs of G in G are all the hamiltonian paths of G , and since
we have taken all flows to be unity, the value of the product graph of G
with an image graph Is simply the length of the hamiltonian path forming the
image graph. (Note that the 2 possible mappings of G onto the image graph
give the same value.) Thus, the QAP with G a simple chain and G the complete
graph becomes equivalent to the open-ended travelling salesman problem.
Although the algorithm of section 3.2(R) do s not gener.iH ze to the
present case, this case (of G being an arbitrary tree and G the complete
graph) is possibly the most important of all cases of the QAP - as far as
['(inTinirtrt^ni-ininnK-d-n.iiiindi Ü^M^^^^ rnuäiM aüftüü
jirv^Mmmv??** mmmMMmmmmmammmmmmmmmimmimmmmimmmmmmm^^
practical applications are concerned - since in many situations (e.g., in
assembly line layout, pipeline design, etc.) the flow graph is of this
form. In view of its importance we present here a specialized algorithm
which can solve QAP's of considerebly larger size than any algorithm for the
general QAP.
The algorithm is a generalization of the dynamic programming algorithm
in [6] for the travelling salesman problem. The generalization is in two
directions: (i) it considers different flows between machines, and (ii) it
can accommodate nrMtrary trees instead of simply chains.
Consider a general tree graph G and suppose it is arbitrarily
rooted at some vertex x . Let T(x ) be the directed subtree reachable from o a
x , including x itself as the root of the subtree. We will also use T(x ) a a a
to mean the set" of vertices of this subtree. Let x . v be the immediaLe Pia;
predecessor of vertex x in the rooted tree G . o
For a given vertex x e X let S c X and x. e S where Is I = |T(X )1.
Let S -be rhe induced subgraph on the subset S of vertices of the distance a a
graph G . Define the function g(S ,x.) as:
g(Sa,x.) Min GeM(T(x ), S ■)
Min V
-p(x )=x Q' 1
(T(X )TT(p)G) (3)
i.e., g(S ,x.) is the solution to the QAP defined on the subgraphs T(x ) and a i a
• S ^ with the restriction that the optimum mapping should have p(x ) = x. , a a i
The function g(S ,x.) can be computed recursively as follows: o i
(i) If x is the predecessor of only one vertex x«:
g(S ,x ) = Min [g(S x ) + f Rd..1 a i v -c P 1 W ij
wl lere S = SD U [x.}
(4)
jjiiigatiiittiiaBigii^^ ÄS^toä
^fPg»!P!P!»i!^B*^pi^^^ W/W^ftgmft'.SWWjtpi^
-11-
(11) IT x Is I he proilrci'SHor of r ( ■ 1) vertices x , a P,
fY 1 Mln E Min Ig(S ,x ) + f d
k=l x. eS„ ßk Jk ^ ij' \ K
k 'k
where the o uter minimization is over all possible sets SQ ,....SD with ßl ßr
lSß 1 - |T(xß )1, and
(5)
S = SR U ... US U [x }, Q P1 Pr 1
Sft OS, = 0 ¥ k.-tell.-.-.r}.
llie initial values of g(S ,x.) are taken to be 0 for S = fx,}, ¥ x. e X a i a 1 i
and for all terminal vertices x of the directed tree G . a.
It may be worthwhile to indicate the order in which the computation of
the functions g(S ,x.) would take place for the example of Fig. 3.
x is a terminal vertex of G , We can start from g(S ,x.) = 0 with
S = [x } ¥ x. e Xd, and then calculate g(S ,x.) for each S ex with JS j = |T(3)| =5
and for each x. e S by using expression (4) iteratively. Similarly we can
compute: g(S8,x.) ¥ Sg with jSgj = 3; gCS^x.) ¥ S11 with JS^ = 2;
g(S13,x.) ¥ S13 with |S13| = 2; g(S16,x.) ¥ S^ with JS^J = 2; and gCS^.x.) ¥ S^
with 1S18| = 3.
The next computation would be g(S ...x.') from equation (5) ¥3^ with
|S I = JS | + JS |+1=6. The next computation would be g(S ,x ) also
from equation (S) ¥ S with |S?1 =- JS | + |s | + JS | + 1 - 11. Finally
K(S ,x ) Is computed from equation (5) for S = X (i.e., JS | ^ |S,| + |s | I
|s j.! + 1) and ¥ x. e S . The value g(S ,x.) then is the solution to the
t^^.^^^^^«^^ Mjaaa.sa.^a mm* i^ü^S^M^..^^.
P^PWfPPP^W^^ igppBWigWWBSIP'PSP^
O -^-Q,
-* o -*- o
IZ
-o n Zi)
rui f LOW jW./^l. FCl.' -Tt-\t: e.XAS\^L.t:
I
lltLs
WJiSiSäiiäsilibiiiSrfäii .^^...^-iY-^.^^ i%AiäKkM^^t^«^A^^Mä^mmili
p^pippiigipiywi^^ . —
12-
QAP with vertex 1 of the flow graph G mapped onto vertex x of the distance
graph G . The value of the solution to the QAP would then be:
z = Min [g(S x )] x.eXd 1 '
i
It is interestii-t, to note that the case of G being a single star-
graph involves only one application of iteration (5) to calculate g(S ,x. )
for each x and even for that single application, the linear assignment problem
can be used to solve the outside minimization as mentioned earlier. This is the
simplest case mentioned in section 4.A. above. The computationally most difficult-
case is the casf of G being a hamiltonian chain which, as mentioned earlier,
leads to the travelling salesman problem. Cases of trees G with values of
graph diameter 1etween these two extremes are of intermediate complexity.
C. Narrow bipartite graphs
When G = (X ,X ) is the complete bipartite graph K with r -'• s the
QAP can he solved when r is very small by the method of section 3.D. , i.e.,
simply enumerating all r!(r) mappings of the set X into a set of r vertices
of G and solving an s by s assignment problem for each such mapping. Obviously
this would only be practical for r = 2 or at most 3 even for graphs G with
only 20 or 30 vertices,
D. G is a collection of links
For Instructional purposes it may be worthwhile to note that if G
is a disconnected graph composed of q components each of which is a single
link, then if all flows are unity the QAP becomes a. matching pii.'ilem and
can be solved as such. It is not at all clear if the problem with non-unitv
fatodj^^^ftma*^
WH^PSpBIBWWW^sst^^ T!!»SWW«?S?"?"fiP"^S!SBSW?^^
-13-
flows could be solved as a matching problem. Although the matching algorithm
can also bt used to solve general degree-constrained partial ^raph problems
defined on C , the lorm of the solution cannot be guaranteed to correspond
to any a priori defined flow graph G .
: : ^^.^ii^^^i.^^^i.^^^i.^i^'j^^^^ gMaajgaaiiiiiiigaiiaaiig
wmmimmmmmmimmmmmm>mmmmm^^ sppsppppspip WWPWSP^B^a?
•14-
5. CONCLUSION
We have expressed the GAP in terms of graph multiplication and
classified and investigated rases depending on the form that the distance
graph C and flow graph C. Lake. The case when C and C. are hofh simple
star grapiis was known as .1 solvable case of the QAP. A polynomial-growth
n 1 gorithin lias now been given for the solution of QAP's when both C and G
are arbitrary trees. Al tliough the algorithm for simple star graphs generalizes
to the case of one graph G being a star and the other G being a complete
graph, the new algorithm for arbitrary trees does not, since any such generaliza-
tion implies the travelling salesman problem. However, a specialized algorithm
for solving QAP's where G is any tree and G the complete graph, is described
which can solve considerably larger problems than any general QAP algorithm.
^^a^^ad^a^^
HWfSHBpHWSS^»
-15-
REF2RENCES
|1| C. Berge, The Theory of C r.-iphs, Methuen, London (1962).
[2] V. J. i'owman, I). A. Pierce and F. Ramsey, "A Linear Programming Formula- tion of <i Special Quadratic Assignment Problem," Carnegie-Mellon University, Management Sciences Research Report No. 277 (1971).
[3] N. Christof ides. Graph Theory-An Algorithmic Approach, Academic Press, London (1975) .
[41 N. Christofides and M, Gerrard, "An Improved Algorithm for the QAP," Imperial College, Report MS 76-1 (1976).
[5] P. C. Gilmore, "Optimal and Suboptimal Algorithms for the Quadratic Assignment Problem," J, SIAM, 2£> 305-313 (1962).
\6] M. Held and R. Karp, "A Dynamic Programming Treatment of the Travelling Salesman Problem," .1. SIAM, 10, 196-210 (1962) .
[71 E. L. Lawler, "The Quadratic Assignment Problem," Man. Sei,, 9, 586-59A (1963).
[8| W.L. Maxwell, "The Scheduling of Single Machine Systems: A Koviow," 1. I. Prod. Kes., 3, I7 7-I'>(» (1 Of,/,) .
I'M .1. M. Moore, "Compulei Ai<lcil I'.ic I 1 i t i es Design: An International Survey," 1 . .1. Prod. Res. , UL :'I-/.A ( 1974) .
[10| C. E. Nugent, T. K. Vollman and J. Runil, "An Experimental Comparison of Techniques for the Assignment of Facili;i^ to Locations," O.R., 16, 150-17 3 (1968).
[Ill J. F. Pierce and W. B. Crowston, "Tree-search Algorithms for the Quadratic Assignment Problem," N.R.L.Q., _18, 1-36 (1971).
[12] T. Pomentate, "On Minimization of Backboard Wiring Functions," SIAM Review, 9, 56A-568 ^1967).
| 131 M. Scriabin and R. C. V'ergin, "Rila(i"e Effectiveness of Computer and Manual Methods for Plant Layout," Man. Sei., 2J_, 564-568 (1975^.
mmmmmm ...'v..^^rf1-,-i:JA-..ii^,iit:wiiiiAiti,
