A note on the quadratic assignment problem

Nov. 2009, Volume 6, No.11 (Serial No.60)

Journal of Communication and Computer, ISSN 1548-7709, USA

A note on the quadratic assignment problem* Wajeb Gharib Gharibi, Omar Saeed Al-Mushayt (Computer Science & Information Systems College, Jazan University, Jazan P.O.Box 2096, KSA) Abstract: The quadratic assignment problem (QAP) is one of the most interesting and challenging combinatorial optimization problems in existence and one of the most difficult problems in the NP-hard class. Many real-life problems in several areas such as facilities location parallel and distributed computing, combinatorial data analysis and combinatorial optimization problems which have many application in computer engineering and industry can be formulated as a QAP. In this paper, the author give a short note on the QAP, giving our rounding approach with the survey of other algorithms that used to solve this important problem. Key words: combinatorial optimization; integer programming; assignment

1. Introduction The Quadratic Assignment Problem (QAP) is a discrete optimization problem which can be found in many fields of study and it constitutes one of the most important and fundamental objects in computer science and operations research. Besides these areas of their ‘‘natural habitat” assignment problems have found numerous applications in other disciplines of science and engineering, including chemistry, biology, physics, archeology, electrical engineering, sports, and others. In addition to facility location, QAPs appear in applications such as layout problems, backboard wiring, computer manufacturing, scheduling, process communications and turbine balancing. In the field of ergonomics, Burkard and Offermann[1] showed that QAPs can be applied to typewriter keyboard design. The problem is to arrange the keys on a keyboard such *

Acknowledgment: The authors would like to thank to Dr. YONG Xia for his valuable comments. Corresponding author: Wajeb Gharib Gharibi, Ph.D., associate professor; research fields: computer science, operations research. Omar Saeed Al-Mushayt, Ph.D., associate professor; research fields: computer science, management systems.

as to minimize the time needed to write some text. Let the set of integers N = {1, 2, . . . , n} denote the set of symbols to be arranged. Then aij denotes the frequency of the appearance of the pair of symbols i and j. The entries of the distance matrix bkl are the times needed to press the key in position l after pressing the key in position k. An optimal solution for this QAP minimizes the average time for writing a text. A similar application related to an ergonomic design is the development of control boards in order to minimize eye fatigue[2]. Further applications concern the ranking of archeological data[3], the ranking of a team in a relay race[4], scheduling parallel production lines[5]. (for a comprehensive review of the subject of assignment problems, their formulations and applications, see for instance[6, 7] and references there in). The quadratic assignment problem (QAP) was introduced in 1957 by Koopmans and Beckmann[8] as a mathematical model for the location of indivisible economical activities. QAP is often used to describe a location problem. The QAP belongs to the class of NP-complete problems and is considered as one of the most difficult combinatorial optimization problems. Exact solution strategies for the QAP have been largely unsuccessful for any but small problems (approximately N =20). As a result, a significant amount of efforts has been put forth by researchers in developing “inexact” or “heuristic” methods, which obtain good suboptimal solutions in reasonable CPU time (see Bazaraa and Kirca[9], Burkard and Bonniger[10] and Lih et al[11]). The traditional algorithm used for solving QAPs exactly is branch-and bound. Until now the best bound functions for the QAP have been the 30 year old 1

A note on the quadratic assignment problem

Gilmore-Lawler bound. Clausen and Perregaard[12] have chosen to use an improved version of those bounds in their groundbreaking solution of the Nugent 20 problem. The assignment problems can be stated in a variety of forms, including mathematical programming, combinatorial, or graph-theoretic formulations, and the studies of random instances of assignment problems and can be described as follows: Given two n × n matrices A = (aij ) and

B = (bij ), find a permutation π * minimizing n

n

min f (π ) = ∑ ∑ aij .bπ i π j

π ∈Π ( n )

where

Π(n)

i =1 j =1

is the set of permutations of n

elements. Shani and Gonzalez[13] have shown that the problem is NP-hard and that there is no e -approximation algorithm for the QAP unless P = NP. While some NP-hard combinatorial optimization problems can be solved exactly for relatively large instances, as exemplified by the traveling salesman problem (TSP), QAP instances of size larger than 20 are considered intractable. In practice, a large number of real world problems lead to QAP instances of considerable size that cannot be solved exactly. For example, an application in image processing requires solving more than 100 problems of size n = 256[14]. Our paper is organized as follows: In section 2, we give various formulations for QAP and define some other problems related to the QAP. In section 3, we study the different algorithms for solving QAP. Conclusions are made in the last section.

2. QAP formulation The QAP is one of the most difficult NP-hard combinatorial optimization problems. It is described as, an integer linear programming problem, as follows: Consider fij the flow between facilities i and j, and

d kp the distance between locations k and p. It is our goal to calculate: 2

n

n

min ∑

∑

i , j =1 k , p =1

n

s .t .

∑x i =1

ij

n

∑x j =1

ij

= 1,

f ij .d kp xik x jp

(1)

1 ≤ j ≤ n, (2)

= 1,

x ij ∈ {0,1}

1 ≤ i ≤ n,

(3)

1 ≤ i , j ≤ n.

(4)

If we consider the cost of assignment of activities to locations, a general form for a QAP instance of order n is given by three matrices F = [fij], D = [dkp] and B = [bik], the first two matrices defining the flows between facilities and the distances between locations, bik being the allocation costs of facilities to locations. This problem can be defined as:

min

n

n

∑∑

i , j =1 k , p =1

f ij .d kp x ik x jp +

n

∑b

i , k =1

ik

x ik

(5)

s.t. (2), (3) and (4). Since the linear term of (5) is easy to solve, most authors discarded it. By using permutations, QAP can be formulated as follows: Let Sn be the set of all permutations with n elements and π ∈ Sn . Consider fij the flows between the facilities I and j and dπ ( i )π ( j ) the distances between locations

π i and π j . If each permutation π

represents an allocation of facilities to locations, the problem expression becomes: n

min ∑ f ij dπ (i )π ( j ) . π∈S n

(6) i , j =1 This formulation is equivalent to the first one presented in (1)-(4), since the constraints (2) and (3) define permutation matrices X = [ xij ] related to Sn elements, as in (6), where, all 1 ≤ i, j ≤ n .

⎧1, if π (i ) = j ; ⎫ x ij = ⎨ ⎬ (7) ⎩0, if π (i ) ≠ j .⎭ Or by using Trace formulation, which is supported by linear algebra and exploits the trace function (the sum of the matrix main diagonal elements) in order to

A note on the quadratic assignment problem

determine QAP lower bounds for the cost This approach allows for the application of spectral theory, which makes possible the use of semi-definite programming to the QAP. The trace formulation, by Edwards[15], can be stated as:

min tr ( F . X . D . X t ).

been later studied by Burkard in 1974, Kellerer and Wirsching (1998)[17], see also[7]. The QBAP can be written down by replacing the sum in the objective function of the following QAP[8]:

QA P : min f ( X ) = tr ( A X B + C ) X

t

n

n

∑∑c i =1 j =1 n

∑x i =1

ij

n

∑x

ij

ij

xij ,

= 1, j = 1, ..., n , = 1, i = 1, ..., n ,

to an integer programming (IP) or linear programming (LP) formulations of the LAP, respectively. In the graph-theoretical setting, the LAP corresponds to finding a minimum cost perfect matching in an edge-weighed bipartite graph; another useful interpretation of the LAP presents it as finding such a permutation of rows and columns of the cost matrix c = (cij ) that minimizes the sum of the elements on the diagonal. (2) The quadratic bottleneck assignment problem (QBAP) The quadratic bottleneck assignment problem (QBAP) was first considered in 1961 by Steinberg[16 ] in application to backboard wiring problem, and had

n

n

j =1 k =1 l =1

∑x i =1

j =1

bkl x ik x jl ,

= 1, j = 1, ..., n ,

ij

= 1, i = 1, ..., n ,

n

∑x

ij

ij

(11)

where, the set of all feasible solutions of QAP can be represented as The set of permutations of {1,2,…, n} which allows for (11) to be written in the following permutation form: n

n

∑∑a

Q n (π • ) = min π ∈Π n

i =1 j = 1

ij

b π ( i )π ( j ) ,

(12)

where Qn (π * ) = Qn is the optimal value of QAP, and

π * ∈ Π n is the permutation that delivers minimum to (12). With maximization:

Z n = min max a ij bπ ( i ),π ( j ) .

(13) The quadratic bottleneck assignment problem is NP-hard, and, furthermore, it exhibits a very similar asymptotic behavior in large-scale instances. (3) The quadratic semi-assignment problem (QSAP) This is a special case used to model clustering and partitioning problems by Hansen and Lih[18]. It can be written as: π ∈Π n

(10) where the decision variables xij can be taken as either binary: x ij ∈{0,1} , or non-negative: x ij ≥ 0 , leading j =1

s .t .

(9)

s .t . X ∈ Π where A, B and C are n × n matrices, tr denotes the trace of a matrix, and Π is the set of n × n permutation matrices. Moreover, there are some important problems related to the QAP such as: (1) Linear assignment problem (LAP) This problem is polynomial and easily solved by the hungarian method. The mathematical programming formulation of the LAP has the form:

s .t .

i =1

n

Or:

Ln = min

n

∑∑∑∑a

x ij ∈{0 ,1}

(8)

X ∈S n

n

Q n = m in

i, j

m

min

n

∑∑c k =1 i , j =1

s .t .

m

∑X k =1

ik

ij

xik x jk

(14)

= 1, 1 ≤ i ≤ n ,

(15)

xij ∈{0,1} 1 ≤ i, j ≤ n.

(16) . The Other applications can be found in optimal adaptation scheme reduces to solving the general form of the QAP, wherein the problem costs cannot be expressed as products of flows and distances (see Eq. (6)). [19-21]

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A note on the quadratic assignment problem

(4) The multi-objective QAP (MQAP) In 2002, Knowles and Corne[22] presented another QAP variation considering several flow and distance matrices. This problem is a benchmark case for multi-objective meta-heuristics or multi-objective evolutionary algorithms. According to the authors, this model is more suitable for some layout problems, such as the allocation of facilities in hospitals, where it is desired to minimize the products of the flows by the distances between doctors and patients, and between nurses and medical equipment simultaneously. The mathematical expression is then, ur min C (π ) = C 1 (π ), C 1 (π ), ..., C m (π ) ; π ∈S n

{

C k (π ) =

}

n

∑

ij =1

f ijk d π ( i ) π ( j ) , 1 ≤ k ≤ m .

(17)

3. QAP algorithms There are three main exact methods used to find the global optimal solution for a given QAP: dynamic programming, cutting plane techniques, and branch and bound procedures. Research has shown that the latter is the most successful among exact algorithms for solving QAP. Even still, due to the overwhelming complexity of QAP, most problems with their sizes greater than n = 30 remain nearly intractable by exact algorithms. Generally branch and bound procedures are the most helpful for solving QAPs. In typical branch and bound (B&B) algorithms for QAP, a heuristic procedure is used to generate a suboptimal, but suitable, initial feasible solution. Let us call this solution the incumbent. Then at any node of the tree, some bounding methods are used to find a “bound” on the best possible solution that can be expected from any descendent of that node, and the “bound” is compared with the objective value of the incumbent. If the incumbent is better than what we can ever expect from any solution resulting from that node, then it is safe to stop branching from that node. In other words, we can discard that part of the tree from further consideration. 4

The extreme difficulty of QAP has made it an ideal problem for the development of heuristic search methods. Local searches, simulated annealing, tabu search, genetic algorithms, GRASP (Greedy Randomized Adaptive Search Procedure), and other specialized methods have all been applied to QAP. The performance of different heuristics tends to vary with certain problem characteristics. 3.1 Lower bounds Since QAPs are NP-hard, good lower bounds are of eminent importance for solving these problems by implicit enumeration procedures like branch and bound. The QAP lower bound presented in 1962 by Gilmore[23] and by Lawler in 1963[24] is one of the best known. Its importance is due to its simplicity and its low computational cost. However, it shows an important drawback as its gap grows very quickly with the size of the problem, making it a weak bound for bigger instances. The Gilmore and Lawler lower bound (GLB) is given by the solution of the following linear assignment problem (LAP): n

∑ (b

min

ij =1

n

∑x

s.t.

ij

i =1

n

∑x

+ lik ). x jk

= 1, 1 ≤ j ≤ n;

=1, 1 ≤ i ≤ n;

ij

j =1

ij

(18) (19) (20)

xij ∈{0,1} 1 ≤ i, j ≤ n. (21) In order to solve (18)–(21), it is necessary to find the coefficients lik , as below:

lij = min

n

∑

k , p =1

cijkp . y ijkp k ≠ i , p ≠ j

n

∑y

s .t .

k =1

n

∑y p =1

ijkp

ijkp

=1 1 ≤ i , j , p ≤ n ,

=1 1 ≤ i , j , k ≤ n ,

y ijkp ∈ {0,1} 1 ≤ i , j , k , p ≤ n .

(22) (23)

(24) (25)

A note on the quadratic assignment problem

One can see[25-30] some presented improvement methods for the GLB and its application to algorithms used to solve QAP. 3.2 Local search A local search starts from some initial assignment and repeatedly tries to improve the current assignment by local changes. If in the neighborhood of the current assignment a better assignment is found, it replaces the current assignment and the local search continues. In the QAP case, the neighborhood of a permutation J is typically defined by the set of permutations which can be obtained by exchanging two facilities. The simplest local search algorithm based on the above described neighborhood is an iterative improvement, which is referred to as 2-opt. 3.3 Simulated annealing methods Simulated Annealing (SA) is one of the first available meta-heuristics. Therefore it is not astonishing that it was also the first one to be applied to QAP[31]. Following this implementation, some few others were proposed and currently the one due to Conolly[32] appears to be the best in performing. Thonemann and Bölte[33] have proposed an improved SA algorithm for the QAP. A meta-heuristic closely related to SA, was also applied to QAP by Nissen and Paul[34]. 3.4 Tabu search There are several types of tabu searches which are classified according to construction of the tabu list. (In the case of assigning an element i to some location j, there are n × n ways for selecting i and j, where n is the size of the problem. Then, n × n kinds of vectors could appear in the tabu list). One of tabu search applications to QAP was due to Skorin-Kapov[35], but probably the best known tabu search algorithm for QAP is the robust tabu search (RoTS) algorithm of Taillard[36]. This algorithm is based on the 2-opt best improvement local search algorithm. As tabu attributes, the algorithm uses assignment of facilities to specific objects, that is, a

tabu attribute t (i, j) refers to the fact that it is forbidden to assign facility i to location j. 3.5 Genetic algorithms Genetic algorithms also receive their name from an intuitive explanation of the manner in which they behave. This explanation is based on Darwin’s theory of natural selection. Genetic algorithms store a set of solutions and then work to replace these solutions with better ones based on some fitness criterion, usually the objective function value. Conventional genetic algorithms did not find the best known solution for the Nugent’s problems of sizes 20 and 30. For larger problems of size up to 100, they seldom really compete with tabu search procedures. In 2000, Ahuja, Orlin and Tiwari[37] obtained very promising results on large scale QAPs in QAPLIB by applying a version of GA called a greedy genetic algorithm. Recently, Drezner[38] designed a new GA with a problem-specific crossover rule and a tabu search, and obtained even better results than those obtained by Ahuja et al. This new genetic algorithm is currently one of the best heuristics to solve QAPs. 3.6 Linearization Linearization is the first attempt to solve QAP achieved by introducing new variables and new linear (and binary) constraints. Then, existing methods for (mixed) linear integer programming (MILP) can be applied, see references[39, 40]. The current smallest linearization in terms of the number of variables and constraints for QAP is XIA and YUAN linearization[40], which requires ( n − 1 )

2

additional nonnegative continuous variables and 2 ( n − 1 ) additional linear constraints. 3.7 Rounding approaches Formally the QAP can be written as QAP : min f (x ) = tr (AX B + C )X T s .t . X ∈Π

(26) (27) Where A, B and C are n × n matrices, t r denotes the trace of a matrix, and Π is the set of n × n permutation matrices. Throughout we assume

5

A note on the quadratic assignment problem

A and B are symmetric. For comprehensive survey of QAPs, please refer to refs[41, 42]. The stand way to reach the goal is to find out X which is the closest to Xc , i.e., we have to solve min tr (X − X c ) (X − X c )T

(28) (29) Since XX T = 1 , it reduces to the following linear assignment problem (LAP) s .t . X ∈Π

max tr X c X T

s .t . X ∈Π

(30)

(31) We notice that the linear assignment problem (30) – (31) can be exactly solved in O (n 3 ) . This optimal solution is denoted by X o . Other speed-up approaches [43-44]

are based on inexactly solving (30)–(31). They are also applicable in our cases.

s. t. θ ≥ 0

(39)

It is easy to verify: Proposition: f(X(θ)) is a piecewise constant function with respect to θ. From this proposition, f(X(θ)) is discontinuous except the trivial cases. In this article we inexactly solve it by the canonical 0.618 search method and obtain the final solution (θ1 , X 1 ) . Generally we cannot guarantee θ1 to be optimal. A number of important problem instances have been solved to optimality in recent years. The details can be found on the QAPLIB homepage.

4. Conclusions

β ≥ 0 . For any α > 0 (32)-(33)

For a long time, the extreme difficulty of QAP has made it an ideal problem for the development of heuristic search methods. However, some large problems are still challenging. On the other hand, new exact algorithms and novel computing structures developed in recent years make it possible to solve a number of long-open QAPs to optimality, including those posed by Steinberg in 1961, Nugent et al. in 1968 and Krarup in 1972. For this reason, there has been a large amount of research on both fields of exact algorithms and heuristics. Finally, what the researchers have done in the recent years shows that the interest in the algorithms for QAP continuous to be very strong, which is a cyclical trend in theoretical development. The interest of the applications of QAP is still going on, especially in the field of optimization and telecommunications networks.

can be further reduced to a one-parametric model min tr ( ∇f ( X c ) − θ X c ) X T (36) s .t . X ∈Π (37) where θ ≥ 0. The optimal solution of (36)-(37) is denoted by X(θ). Now we have to solve the following one-variable problem min f ( X (θ )) (38)

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3.8 A one-parametric model Let ∇f ( x) = 2 AXB + C be the gradient of f .

Concerning the information of ∇f ( xc ) we can find out a better feasible solution from a linear bi-objective programming

⎛ tr ∇f ( X c ) X T ⎞ min ⎜ ⎟ T (32) ⎝ tr − X c X ⎠ s .t . X ∈Π (33) which can be approximately solved by a parameterized LAP min tr (α ∇f ( X c ) − β X c ) X T

(34) (35) And α ≥ 0, β ≥ 0. Denote the optimal solution of

s .t . X ∈Π

(32)-(33) by X (α ,

β ) for each fixed (α , β ) . It is

easy to verify that X (0, β ) is the optimal solution of (30)-(31) for any

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A note on the quadratic assignment problem

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(Edited by Jane, Sang

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