Exact Solution of Emerging Quadratic Assignment Problems · 2009-10-12 · Quadratic Assignment...

Quadratic Assignment Problems typeset July 31, 2009 Hahn, Zhu, Guignard& Smith

Exact Solution of EmergingQuadratic Assignment Problems

Peter M. Hahn∗ Yi-Rong Zhu† Monique Guignard‡ J. MacGregor Smith§[email protected] [email protected] guignard [email protected] [email protected]

July 31, 2009

Abstract — We report on a growing class of assignmentproblems that are increasingly of interest and very challeng-ing in terms of the difficulty they pose to attempts at exactsolution. These problems address economic issues in the lo-cation and design of factories, hospitals, depots, transporta-tion hubs and military bases. Others involve improvementsin communication network design. In this article we sur-vey the latest and best methods available for solving exactlythese difficult problems and suggest a taxonomy that pro-vides a framework for combining existing solution meth-ods and sets of computer tools that can be modified and ex-tended to make inroads in solving this growing class of op-timization problems.

Keywords — quadratic assignment, integer programming,reformulation linearization

Acknowledgements

This material is based upon work supported by the U.S.National Science Foundation under grant No. DMI-0400155. The authors are grateful to Professor MiguelAnjos of Waterloo University for reviewing this surveyarticle, thus adding to its clarity and completeness.

1 INTRODUCTION

The Quadratic Assignment Problem (QAP) is one inwhich N units have to be assigned to N sites in sucha way that the cost of the assignment, depending on thedistances between the sites and the flows between theunits, is minimal.

It can be formulated as follows: Given two N × Nmatrices, F = [fik] with fik the flow between units iand k, and D = [djn] with djn the distance betweensites j and n, find a permutation p of the set S =

∗Department of Electrical and Systems Engineering, University ofPennsylvania, Philadelphia, PA, USA†Department of Electrical and Systems Engineering, University of

Pennsylvania, Philadelphia, PA, USA‡Wharton School, University of Pennsylvania, Philadelphia, PA,

USA§Department of Mechanical and Industrial Engineering, Univer-

sity of Massachusetts Amherst, Massachusetts 01003, USA

{1, 2, ..., N} which minimizes the global cost function,Cost (p) =

∑i=1,...,N

∑k=1,...,N fikdp(i)p(k). The QAP

is one of the most difficult NP-hard combinatorial op-timization problems. Solving general problems of sizegreater than 30 (i.e., with more than 900(0−1) variables)is still computationally challenging. If among exactalgorithms, branch-and-bound are the most successfulones, the lack of a sharp lower bound technique in thesealgorithms is one of the major difficulties. The fact thatthe QAP is NP-hard is not sufficient to explain its diffi-culty, as we can now solve exactly very large instancesof a great number of NP-hard problems. The homo-geneity of the values of the solutions for most of the ap-plications, due to the structure of the problem (scalarproduct of the two matrices) is a more convincing ex-planation. Indeed, first, we have many solutions whosevalue is close to the optimum. So, even when the bestsolution is obtained, it is very hard to prove its optimal-ity. Then, fixing one assignment has a low influence onthe average value of the solutions. Even when travers-ing the branch-and-bound tree, the problem remainsvery hard. Moreover, it is difficult to prune branchesthat contain significantly large numbers of non-optimalfeasible solutions. A recent paper [8] by Barvinok andStephen gives some insights into to the difficulty of solv-ing the QAP. They obtain a number of interesting resultsregarding the distribution of objective function valueson typical and specific QAP instances.

2 CURRENT STATUS OF QAP SOLVERS

While great progress has been made on generating goodsolutions to large and difficult QAP instances, this hasnot been the case for finding exact solutions. In thelate 1960s, it was an achievement to find the optimumsolution to a difficult instance of size n = 8. In the1970’s and 80’s, one could only expect to solve diffi-cult instances for n < 16. It was not until the mid-1990s that Clausen and Perregaard [16] were able toenumerate a difficult size 20 instance. Much progresshas been made since then. In the mid 1960’s, Nugent,Vollmann and Ruml [46] posed a set of problem in-stances of size {5, 6, 7, 8, 12, 15, 20, and 30}, noted for

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their difficulty. In these instances, the distance matrixstems from an n1 · n2 grid with Manhattan distance be-tween grid points. Most of the resulting QAP instanceshave multiple global optima (at least four if n1 6= n2 andat least eight if n1 = n2). Even worse, these globally op-timal solutions are at the maximally possible distancefrom other globally optimal solutions. The Nugent in-stances have been the benchmark, against which exactand heuristic solution algorithms have been measured.Figure 1 shows the rapid progress made in exact solu-tion QAP algorithms from 1995 until early in the 21stcentury.

0

5

10

15

20

25

30

35

40

1970 1980 1990 2000 2010

Nuge

nt S

ize N

Year Solved

Nugent Instances First Solved

exact results

Figure 1: Graph of Exact Results for Nugent Problems

Other forms of the QAP do not have thatflow/distance cost structure. One example is inbalancing hydraulic turbine runners ([35], [44] and [51],for instance). A jet engine consists of several turbines,and the objective of this engine maintenance problem isto remove unwanted vibrations. This can be formulatedas a QAP, where the 0/1 decision variable xij is 1 ifblade i is allocated to position j, 0 otherwise, and thequadratic objective function corresponds to minimizingthe distance between the center of gravity of the turbineshaft and its center of rotation.

The QAP, while still of great interest to researchers, isonly one of a growing class of assignment problems thatare increasingly of interest and even more challengingin terms of the difficulty they pose to attempts at exactsolution. It is this class of problems that we address inthis survey.

Figure 2 shows the relationships between exact solu-tion methods for several assignment problems that haveappeared in the operations research literature. Theseproblems include the Generalized Quadratic Assign-ment Problem (GQAP), the 3-dimensional AssignmentProblem (3AP), the Quadratic 3-dimensional Assign-ment Problem (Q3AP), the Generalized 3AP (G3AP)and the Generalized Quadratic 3-dimensional Assign-ment Problem (GQ3AP) and Stochastic Quadratic As-signment Problem (SQAP). A short discussion of these

problems is given in the ensuing paragraphs.

ExactAlgorithms

QAP

CAP

GCAP

BiQAP

GQAP

3AP

G3AP

Q3AP

GQ3APMSAP

CDAP

SQAP

Figure 2: Morphology of QAP Problems

3 THE GENERALIZED QUADRATIC ASSIGNMENTPROBLEM (GQAP)

The GQAP covers a much broader class of problemsthan the QAP. Problems in this class involve the mini-mization of a total pairwise interaction cost among Mdepartments, equipment, tasks or other entities, andwhere placement of these entities into N possible des-tinations is dependent upon existing resource capacitiesat each destination. These problems include finding theassignment of departments to fixed locations given lim-ited area capacities at each possible location. The Leeand Ma [37] version of the problem can be stated withreference to a practical situation where it is desired tolocate M departments among N fixed locations, wherefor each pair of departments i, k a certain traffic flow ofcommodities fik is known and for each pair of locationsj, n a corresponding distance djn is known. The two-way transportation costs between departments i and k,given that i is assigned to location j and k is assigned tolocation n, are fikdjn +fkidnj . The objective is to find anassignment minimizing the sum of all such transporta-tion costs given that the capacity or resource constraintsare met. In the general case of the GQAP, the cost oftransportation between departments is known but is notdecomposable into a product of a flow and a distancematrix.

Lee and Ma [37] only recently formulated the GQAP.However, problems that are special cases, including theQAP, have long been of interest to researchers in vari-ous fields, both because of their wide applicability andtheir resistance to reliable computer solution. Prob-lems which come under the class of GQAP include theProcess Allocation Problem of Sofianopoulous ([63] and[64]), the Constrained Module Allocation Problem of El-loumi et al. [19], the Quadratic Semi-Assignment Prob-

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lem covered by Billionnet and Elloumi [9], the Multi-processor Assignment Problem by Magirou and Milis[43], the Task Assignment and Multiway Cut Problemsof Magirou [42], the Memory Constrained Allocationproblem of Roupin [55] and the constrained Task As-signment Problem of Billionnet and Elloumi [10].

Exact solution strategies for GQAP type problemshave been successful for only small instances (approx-imately M = 30). As a result, researchers have putforth a significant amount of effort in developing inex-act, or heuristic methods that obtain good suboptimalassignments, using a reasonable amount of CPU time.Cordeau et al. [17], discusses a memetic heuristic for theGQAP. They do not report using their method to find ex-act solutions. Lee and Ma [37] were the first to devise anexact solution method for the GQAP. They introducedthree linearization approaches together with a branch-and-bound algorithm. Their lower bounding strategyinvolves solving M ×N plus 1 GAP sub-problems. Thisthey did using calls to CPLEX. Using this bound in abranch-and-bound algorithm, they were able to poseand solve exactly 27 problem instances, the largest ofwhich was of size 16x7.

Hahn et al.[29] improved upon the exact solutionmethod of Lee and Ma by introducing a Lagrangeandual for the GQAP based on a Level-1 ReformulationLinearization Technique (RLT-1) Dual Ascent Proceduresimilar to one they successfully used for solving theQuadratic Assignment Problem (QAP). A unique andvery important aspect of the RLT-1 Dual Ascent Pro-cedure is that at each stage, the GQAP is restructuredas fully equivalent to the original GQAP in a mannerthat brings it closer to solution. Their RLT-1 Dual As-cent Procedure was embedded in a branch-and-boundalgorithm that is unique in many respects. A number oftest instances, selected from a web site dedicated to theTask Assignment Problem (TAP) and the ConstrainedTask Assignment Problem (CTAP) set up by Sourour El-loumi at the Centre de Recherche en Informatique duCNAM [20], were solved in record time. Comparisonswere also made with instances of the GQAP devised byCordeau, et al. [17] and Lee and Ma [37]. The B-and-Bof Hahn et al. is generally faster than the method of Leeand Ma and is about 20 times faster than the Lee andMa runtime for the difficult 16x7 instance.

Pessoa et al.[47] provide the most recent and mostpromising solution methods for the GQAP. They de-veloped a hybrid branch-and-bound exact solutionmethod. Their lower bound calculation is based on aLagrangean relaxation that makes efficient use of the in-teger linearization property in its modeling phase andof the volume algorithm in its solution phase. Pes-soa et al.’s branch and bound is as fast or faster thanother exact solution methods on easy GQAP probleminstances, and is remarkable in that it solves difficult in-stances that could not be solved exactly with any pre-vious solvers. Their look-ahead branching strategy isbased on the same techniques as found in [29]. Thus,

the improved performance reported is due entirely tothe effectiveness of the their new lower bound.

4 THREE-DIMENSIONAL ASSIGNMENT PROBLEMS

4.1 The 3-dimensional Assignment Problem

The 3-dimensional Assignment Problem (3AP), alsoknown as the Three Index Assignment Problem, in-volves the optimization of the assignment of N type-1entities and simultaneously N type-2 entities to N des-tinations. The 3AP is applied to find the minimal idlingtime of a rolling mill, optimal location of productionplants in regions, optimal number of satellites in dif-ferent directions and orbits for maximization of thescanned regions [50], teaching schedules [21], in statisti-cal processing of measurement results [32], etc. The 3APis shown in [66] to be a special case of the QAP and thusis difficult and also NP-hard.

Branch and bound is the preferred method for solvingthe 3AP exactly. One of the first 3AP branch-andboundalgorithms was proposed by Pierskalla [49]. Bound-ing techniques using Lagrangian and subgradient op-timization were proposed by Burkard and Rudolf [15].Balas and Saltzman [7] improved on existing bound-ing techniques by using dual heuristics. Among ex-act algorithms, branch-and-bound schemes using theLagrangian dual and subgradient optimization are themost successful, but the computation time for the sub-gradient procedure has been one of the major difficul-ties. An average 65 percent of the total computationtime for branch-and-bound enumeration is spent in thesubgradient solution procedure [7].

Kim et al. [34] describe new bounding methods forthe axial three-index assignment problem (3AP). Forcalculating 3AP lower bounds, they use a projectionmethod followed by a Hungarian algorithm, based ona new Lagrangian relaxation. They also use a cost trans-formation scheme, which iteratively transforms 3APcosts in a series of equivalent 3APs, which provides thepossibility of improving the 3AP lower bound. Theirmethods produce efficiently computed relatively tightlower bounds on standard test instances.

4.2 The Quadratic 3-dimensional Assignment Problem

Pierskalla [48] introduced the Quadratic 3-dimensionalAssignment Problem (Q3AP) in a technical memoran-dum. Since then, little on the subject has appeared.Hahn et al. [28] re-discovered the Q3AP while work-ing on a problem arising in data transmission systemdesign. The Q3AP is an extension of two NP-hardproblems, the QAP and the 3AP. Thus, it is easy to seethat the Q3AP is also NP-hard. The interest in theQ3AP stems from the fact that it is applied to problemswhere the objective is to minimize linear and quadraticcosts associated with a pair of independent simultane-ous one-to-one assignments. Such a problem arises inthe design of wireless communication systems, wherein

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a digital message is repeated two times. During each ofthe repeats, the assignment of data word to transmittedsymbols is modified. The Q3AP models the problem ofoptimizing the two assignments in such a way that thetransmission errors are minimized. See [56] and [57].

Hahn et al.[28] are the first to have solved Q3AP in-stances. They developed a branch-and- bound algo-rithm based upon one of the best techniques availablefor solving the QAP exactly, as well as four differentheuristic solution methods whose genesis came fromprevious work applied to solving the QAP. Implement-ing the exact algorithm required the development ofnew lower bounds for the 3AP. Although the compu-tational results are encouraging, they also illustrate thelevel of difficulty associated with the Q3AP. Recently,Galea et al. [22] developed a parallel version of the ex-act solution algorithm of Hahn et al. [28]. This parallelcode is not only an instrument for solving exactly largeinstances, but will also enable experimentation for im-proving the runtime of Q3AP exact solution algorithms.

Presently, the exact solutions have been demonstratedonly for Q3AP instances of size 14 or smaller. Parallelsolution experiments are planned for larger instances.Stochastic local search (SLS) techniques [28] are essen-tial for reaching high quality solutions to Q3AP in-stances of practical interest. Clearly, much more work isneeded on this challenging and yet important new com-binatorial optimization problem.

4.3 The Generalized Quadratic 3-dimensional AssignmentProblem

The Generalized Quadratic 3-dimensional AssignmentProblem (GQ3AP) is a generalization of the Q3AP andthe GQAP. This problem arises in two very importantsituations. One is the assignment of spaces withinmulti-story buildings or within multi-deck naval ves-sels, so that the movement of people and materialsbetween spaces is efficient and that the time to es-cape from the structure is simultaneously minimized.This problem is known as the Multi-story Space As-signment Problem [30]. MSAP test instances are cur-rently available at: http://www.seas.upenn.edu/˜msaplib. MSAP test instances are also available atthe Facility Layout Problem Library (FLPlib) http://FLPlib.uwaterloo.ca/. FLPlib was developedat Waterloo University by Professor Miguel Anjos andstudent Christie Kong. This web site serves as a re-source of data for developing facility layout problemsand solution methods. In addition to being an MSAPresource, FLPlib contains information and problem in-stances on the GQAP, the Single-row Facility Layoutproblem (SRFL) and the One-Dimensional Space Al-location Problem (ODSAP), also known as the linearsingle-row facility layout problem, which consists offinding an optimal linear placement of facilities withvarying dimensions on a straight line.

Another application of the GQ3AP is in the designof cross-dock facilities in the less-than-full load (LTL)

trucking business. In this situation, the GQ3AP is usedto assign incoming trucks to unloading docks (stripdoors) and simultaneously assigning outgoing trucks toshipping docks (stack doors) so that the cost of mov-ing goods from strip doors to stack doors is minimized.This problem is known as the Cross-dock Door Assign-ment Problem (CDAP). Zhu, et al. [67] recently re-port on an exact algorithm for solving the CDAP as aGQ3AP.

5 OTHER PROBLEMS

5.1 Cubic Assignment Problem

In the Quadratic Assignment class, the Cubic Assign-ment Problem (CAP) is described in a newly pub-lished SIAM Monograph by Burkard, Dell’Amico andMartello [14]. Their book provides a comprehensivetreatment of assignment problems from their concep-tual beginnings in the 1920s, through present-day the-oretical, algorithmic, and practical developments. TheCAP is also mentioned, but not discussed in detail inthe book by Du and Pardalos [18]. The CAP was firstposed by Lawler in his seminal paper on the QAP [36].The CAP optimizes the problem of placing N entities atexactly N destinations, where the cost of placement in-volves the interaction between triplets of entities, ratherthan the interaction between pairs of entities as is foundin the QAP. We have searched, but have not found an ex-act solution method for the CAP. But, in [1], it is statedthat the formulation RLT-2 of the QAP is exactly a CAP.Thus, the RLT-2 exact solution solver for the QAP iscapable of solving the CAP exactly. To the best of ourknowledge, no researchers have yet reported computa-tional experience on solving the CAP using the RLT-2form, though the RLT theory establishes the theoreticalequivalence between these representations. Winter andZimmermann [65] used a cubic assignment problem foroptimizing the movement of materials in a storage yard.Burkard et. al in [14] point out that ”the objective func-tion (2.1.1) in [65] contains some typos in the indices, butis actually the objective function of a cubic assignmentproblem”.

5.2 Bi-Quadratic Assignment Problem

The BiQuadratic or Quartic Assignment Problem(BiQAP) is a generalization of the QAP. It was alsoposed in 1963 by Lawler [36]. It is a nonlinear integer-programming (IP) problem where the objective functionis a fourth degree multivariable polynomial and the fea-sible domain is the assignment polytope. BiQAP prob-lems have an application in VLSI synthesis, where pro-grammable logic arrays have to be implemented. Dueto the difficulty of this problem, only heuristic solu-tion approaches have been proposed. For details ofthe VLSI application see Burkard, Cela and Klinz [13],who studied biquadratic assignment problems, derived

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lower bounds and investigated the asymptotic proba-bilistic behavior of these problems. Burkard and Cela[12] developed metaheuristics for the BiQAP and com-pared their computational performance.

5.3 Generalized Cubic Assignment Problem

The Generalized Cubic Assignment Problem (GCAP) isa generalization of the Cubic Assignment Problem andthe Generalized QAP. The GCAP optimizes a situationwhere M entities have to be placed at N destinations,such that the placement of entities at each possible loca-tion is limited by the capacity of the destination to ac-cept entities, where the cost of placement involves theinteraction between triplets of entities, rather than theinteraction between pairs of entities as is found in theQAP. This problem is introduced in Zhu [66] and a so-lution method is suggested. However no papers havebeen published on this subject. For this problem, onewould have to generate potential test instances, as noneexist.

5.4 Stochastic Quadratic Assignment Problem

Li and Smith ([38], [39], [40], [62]) have developeda formulation and associated heuristic solution algo-rithms for the Stochastic Quadratic Assignment Prob-lem (SQAP) which involves the examination of randomflows in a facility layout. The random flows are mod-elled with an additional node in the layout to accom-modate the dynamic flows of customers or products inthe layout where congestion occurs. These models havemany applications in facility planning, manufacturingsystems and other QAP problems with dynamic flows.Traditionally, these models are solved with heuristics,so it would be worthwhile for someone to solve themexactly. While the objective function in the SQAP prob-lems are nonlinear, there are lower bounds availablefrom some of the queueing models that will effectuatethe exact solution of these problems. See the latest Smithand Li paper for further details [62].

5.5 Summary

Table 1 and Table 2 summarize the achievements madein solving the various assignment problems consideredin this survey. The tables give the applications for whichthe problems were originally posed and list the problemsizes that can be solved exactly, the method of solutionand the publication describing the solution method.

6 DESCRIPTIONS OF ASSIGNMENT PROBLEMS

Until now, assignment problems have been dealt withindependently. Little attention was given to developinga unified approach. This survey emphasizes the com-mon structures and suggests a common framework for

dealing with these very difficult problems. Since the ap-proach that has been most successful for solving assign-ment problems exactly is to develop tight lower boundsbased upon Lagrangean relaxations, we divide the indi-vidual problems by the following descriptive character-istics:

• Dimensionality of the objective function:(2-dimensional, 3-dimensional, etc.)

• Degree of the objective function: (linear, quadratic,cubic, bi-quadratic, etc.)

• One-to-one versus many-to-one assignment: (APversus GAP)

• Linearization and relaxation options (level of Re-formulation Linearization)

6.1 2-dimensional Assignment Problems

6.1.1 The Linear Assignment Problem

The LAP is given by:

Min

{ N∑i=1

N∑j=1

Bijxij

}(1)

subject to the following constraints on X:

N∑i=1

xij = 1 (j = 1, 2, · · · , N), (2)

N∑j=1

xij = 1 (i = 1, 2, · · · , N), (3)

xij = 0, 1(i = 1, 2, . . ., N ; j = 1, 2, . . ., N). (4)

The LAP is an easy problem, even though it has N !feasible solutions, as are found in the QAP. It is easy,not so much because the objective function is linear, butespecially because the LP relaxation optimizes preciselyover the convex hull of the feasible 0-1 integer points ofthe solution space. Figure 3 shows a typical LAP solu-tion, superimposed on the square objective function costmatrix of a size 5 LAP. The solution shown is optimal.Linear Assignment Problem!

12 8 7 15 4

7 9 17 14 10

9 6 12 6 7

7 6 14 6 10

9 6 12 10 10

Figure 3: Typical LAP Solution

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Prob. Application Size Method Who/WhenLAP Assign jobs to machines >2000 Sparse instances Goldberg-Kennedy 1995 [23]

” ” >2000 Dense instances Jonker-Volgenant 1987 [33]QAP Facility loc./Ckt. layout >30 Quadratic Prog. Brixius-Anstreich. 2001 [11]

” ” >30 QAP-RLT-2 Adams et al. 2007 [1]CAP Solve QAP-RLT-2 >30 QAP-RLT-2 Adams et al. 2007 [1]

BQAP Solve QAP-RLT-3 20 QAP-RLT-3 Hahn et al. 2008 [31]GAP Assign jobs to machines 200x5 Branch and Bound Haddadi-Ouzia 2004 [26]

” ” 100x5 Branch and Cut Nauss 2003 [45]GQAP Assign tasks to processors 16x7 GLAP-LB Lee-Ma 2004 [37]

” ” 30x20 GQAP-RLT-1 Hahn et al. 2006 [29]” ” 35x15 Volume Algorithm Pessoa et al. 2009 [47]

GCAP Solve GQAP-RLT-2 24x8 GQAP-RLT-2 Hahn et al., unpublishedSQAP Random Flows in a layout > 30 Steepest-descent Li and Smith 1998 [40]

Table 1: 2-dimensional Exact Solution Methods

Prob. Application Size Method Who/When3AP Assigning Jobs 26 Lagrangean Dual w/ Subgr. Opt. Balas and Saltzman 1991 [7]

Q3AP Symbol Mapping Diversity 14 Q3AP Branch and Bound Hahn et al., 2006 [28]C3AP Solving the Q3AP 8 RLT-2 Q3AP Branch and Bound Hahn 2007, unpublishedG3AP Subproblem of GQ3AP TBD Volume Algorithm Hahn et al. (in progress)

GQ3AP Solve MSAP and CDAP 17x17x4 GQ3AP Branch and Bound Hahn et al. [30]

Table 2: 3-dimensional Exact Solution Methods

6.1.2 The Quadratic Assignment Problem

The QAP is given by:

Min

{ N∑i=1

N∑j=1

Bijxij +N∑

i=1

N∑j=1

N∑k=1k 6=i

N∑n=1n 6=j

Cijknxijxkn

}(5)


N∑i=1

xij = 1 (j = 1, 2, · · · , N) (6)

N∑j=1

xij = 1 (i = 1, 2, · · · , N) (7)

xij = 0, 1(i = 1, 2, . . ., N ; j = 1, 2, . . ., N), (8)

X are said to be a ‘solution’. Figure 4 shows a typ-ical feasible QAP solution, superimposed on the 9x9objective function cost matrix of a size 3 QAP. Certainelements of the objective function cost matrix are des-ignated by asterisks, indicating that they cannot con-tribute to any feasible solution. There are N ! feasiblesolutions to the QAP.

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6.1.3 The Cubic Assignment Problem

The CAP is given by:

Min

{ N∑i=1

N∑j=1

Bijxij +N∑

i=1

N∑j=1

N∑k=1k 6=i

N∑n=1n 6=j

Cijknxijxkn

+N∑

i=1

N∑j=1

N∑k=1k 6=i

N∑l=1l 6=j

N∑m=1

m6=i,k

N∑n=1n 6=j,l

Dijklmnxijxklxmn

}(9)


N∑i=1

xij = 1 (j = 1, 2, · · · , N) (10)

N∑j=1

xij = 1 (i = 1, 2, · · · , N) (11)

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xij = 0, 1(i = 1, 2, . . ., N ; j = 1, 2, . . ., N), (12)

Figure 5 shows a typical feasible CAP solution, su-perimposed on the 64x64 objective function cost matrixof a size 4 CAP. The objective function cost matrix has16x16 submatrices, each with 4x4 cost elements. Only16 of these submatrices are involved in a feasible solu-tion. Within a solution submatrix only four of the six-teen elements are in the feasible solution. Again, certainelements in the objective function cost matrix are disal-lowed from any feasible solution. As before, these ele-ments are designated by asterisks. As in the LAP andQAP, there are N ! feasible solutions to the CAP.

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################################################

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&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

where, for example

D2134

=

' D213412 D213413 '' ' ' '' ' ' '' D213442 D213443 '

!

"

####

$

%

&&&&

Figure 5: Typical CAP Solution

Where are we going with this? We notice the follow-ing patterns:

• Expansion involves multiplication by the binaryvariables X .

• Expansion provides a multi-level hierarchy of 0-1assignment problems.

• There is a direct relation with the Reformulation-Linearization Technique (RLT), which we discuss inthe next subsection.

6.2 Assignment Problems and the Reformulation-Linearization Technique

The Reformulation-Linearization Technique (RLT) wasdevised by Hanif Sherali and Warren Adams for thesolving of zero-one quadratic assignment problems andlater extended to other problems. It recasts an assign-ment problem as a mixed 0-1 LP, via two steps: a refor-mulation step in which additional nonlinear valid in-

equalities are automatically generated through a mul-tiplication process involving the binary variables anda linearization step in which each product term is re-placed by a single new continuous variable. Depend-ing on the product factors used, different formulationsemerge. See [3], [4], [59], [60] and [61]. The first two ofthese articles deal specifically with quadratic programs.The latter three articles are more general, dealing withprograms that involve cubic and higher order objectivefunctions. The result is a multi-level hierarchy of mixed0-1 linear representations, RLT-1, RLT-2, etc., of the orig-inal problem. Each level of the hierarchy provides a pro-gram whose continuous relaxation is at least as tight asthe previous level. The highest level gives a convex hullrepresentation.

RLT has been used to achieve significant advancesin the solvability of classical and newly posed assign-ment problems (APs) and generalized assignment prob-lems (GAPs). Applying RLT to Quadratic AssignmentProblems was first done by Adams and Johnson [2].QAP lower bounds for a level-1 RLT (RLT-1) formula-tion were first calculated by Adams and Johnson [2] andResende et al. [54] and for a level-2 RLT (RLT-2) formu-lation were first calculated by Ramakrishnan et al. [53].

Appearing in the literature more recently are theRLT-1 QAP exact algorithm by Hahn et al. [27], theRLT-2 QAP exact algorithm by Adams et al. [1], theRLT-3 QAP lower bound calculations by Hahn et al.[31], the RLT-1 exact algorithm for the Q3AP by Hahnet al. [28] and the RLT-1 parallel exact Q3AP solver byGalea et al. [22] and the RLT-1 exact algorithms for theGQAP by Hahn et al. [29] and by Pessoa et al. [47].

Reformulation- Linearization Techniques achieve sig-nificant advances in the solvability of the QAP. ProblemRLT-2 for the QAP, in particular, provides sharp lowerbounds and consequently leads to very competitive ex-act solution approaches [1]. A striking outcome, docu-mented in Table 2 of Loiola et al. [41], is the relativelyfew nodes considered in the binary search tree to verifyoptimality. This leads to marked success in solving dif-ficult QAP instances of size 30 in record computationaltime. Hahn et al. [31] used the level-3 RLT in order toget even tighter bounds. The challenge was to take ad-vantage of the additional strength, without being hurtby the substantial increment in problem dimensions.

In preparing this survey, it became clear to us that theRLT formulations connect all these problems into a tax-onomy, wherein connections are made between prob-lem types, objective function degree, problem dimen-sion and whether mappings are one-to-one or many-to-one. We illustrate one aspect of this taxonomy by con-sidering an artificial example, namely the RLT-2 formu-lation for the Linear Assignment Problem. There is, ofcourse, no reason for applying RLT to the LAP, since itis a special case of a linear program and relatively easyto solve [14].

DocNum 7 . 6. 2. 0


The RLT-2 for the LAP is given by:

Min

{ N∑i=1

N∑j=1

Bijxij +N∑

i=1

N∑j=1

N∑k=1k 6=i

N∑n=1n 6=j

Cijknyijkn

}(13)

subject to the following constraints on X and Y:

N∑i=1k 6=i

yijkn = xkn (j, k, n = 1, 2, · · · , N), j 6= n (14)

N∑j=1

n 6=j

yijkn = xkn (i, k, n = 1, 2, · · · , N), i 6= k (15)

yijkn = yknij(i, j, k, n = 1, 2, . . ., N), i < k, j 6= n (16)

yijkn ≥ 0(i, j, k, n = 1, 2, . . ., N), i 6= k, j 6= n (17)

N∑i=1

xij = 1 (j = 1, 2, · · · , N) (18)

N∑j=1

xij = 1 (i = 1, 2, · · · , N) (19)

xij = 0, 1(i = 1, 2, . . ., N ; j = 1, 2, . . ., N), (20)

We recognize this formulation to be a linearization ofthe QAP. Specifically, it is identical to formulation LPfrom Hahn and Grant [27]. This is not an accident. Itis an important fact about the relationship between RLTrepresentations of assignment problems and the degreeof the assignment problem objective function. Namely,each level of RLT representation of an assignment prob-lem, results in a representation of a higher degree as-signment problem. This fact is further illustrated laterin Figure 13. A more detailed discussion of the advan-tages of RLT in solving QAPs is given in §7.

Thus, the connection between assignment problemdegree and RLT level has been introduced. In the nextsection we introduce the generalized (many-to-one) ver-sions of the assignment problems discussed above andillustrate the similarities and differences with the one-to-one versions by displaying their solution spaces.

6.3 Generalized Assignment Problems

6.3.1 Generalized (linear) Assignment Problem (GAP)

A GAP may be given by:

Min

{ M∑i=1

N∑j=1

Bijxij

}(21)


M∑i=1

aixij ≤ Aj (j = 1, 2, · · · , N), (22)

N∑j=1

xij = 1 (i = 1, 2, · · · , M), (23)

xij = 0, 1(i = 1, 2, . . ., M ; j = 1, 2, . . ., N). (24)

where ai is a need associated with entity i and Aj isa resource associated with location j, which limits theamount of i entities that can be assigned at location j.

Note that we have used a simplified definition of theGAP. In many representations the needs are more gener-ally represented by aij , which accommodates problemsets that involve needs that are dependent on locationassignment.

Figure 6 shows a typical solution for the GAP, super-imposed on the 7x4 objective function cost matrix of a7x4 size GAP. In this problem, the feasible solution con-tains one element for every row of the objective functioncost matrix. The amount of feasible solution elementsin a column are dictated by the capacity constraints forthat column. These are determined by the needs of eachrow and the resources allocated to each column. Thenumber of feasible solutions is instance dependent.

Generalized (linear) Assignment Problem!

15 4 7 7

13 5 9 11

8 13 5 17

9 12 10 19

11 22 6 14

20 14 6 15

8 9 11 24

4

7

6

10

3

8

9

20 12 15 10

needs!

<- resources!

Figure 6: Typical GAP Solution

6.3.2 Higher Degree Generalized Assignment Problems

The formulation of the GQAP is given in Lee and Ma[37], Hahn, et al. [29] and Pessoa et al. [47], so we do notre-introduce it here. But, we illustrate in Figure 7 the ob-jective function cost matrix of a size 4x3 (4 facilities into3 locations) GQAP superimposed with a typical feasiblesolution to this relatively easy to solve problem. We fol-low this in Figure 8 with an illustration of the objectivefunction cost matrix of a 4x3 Generalized Cubic Assign-ment Problem (GCAP). We could detail the characteris-tics of the objective function cost matrices and superim-posed solutions in the same detail as for the QAP andCAP. But, instead, we merely mention the fact that the

DocNum 8 . 6. 3. 2


objective function cost matrices for the generalized as-signment problems (GAPs) are similar to those for theirrelated assignment problems (APs).

Generalized Quadratic Assignment Problem!

C =

C11 C12 C13

C21 C22 C23

C31 C32 C33

C41

C42

C43

!

"

####

$

%

&&&&

=

C1111 ' ' ' C1212 ' ' ' C1313

C1121

C1122 C1123 C1221 C1222

C1223 C1321 C1322 C1323

C1131

C1132 C1133 C1231

C1232

C1233 C1331 C1332 C1333

C1141

C1142

C1143

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C1342

C1343

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C2312

C2313

C2121

' ' ' C2222

' ' ' C2323

C2131

C2132

C2133

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C2142

C2143

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C2242

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C3112

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C3312

C3313

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C3122

C3123

C3221

C3222

C3223

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C3322

C3323

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' ' ' C3232

' ' ' C3333

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C4132

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C4333

C4141

' ' ' C4242

' ' ' C4343

!

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#########################

$

%

&&&&&&&&&&&&&&&&&&&&&&&&&

Figure 7: Typical GQAP Solution

6.4 Three Dimensional Assignment Problem representa-tions

6.4.1 3-dimensional Assignment Problems (3AP andQ3AP)

As mentioned earlier in §4.1 and §4.2, three-dimensionalassignment problems involve the simultaneous and in-dependent assignments of two different entities to acommon location. It is important to be able to visualizethe objective function space and the selection of thoseobjective function coefficients that are summed for theobjective function value of a specific problem instance.Thus, we have devised a means of drawing the threedimensional objective function cost space as an inter-section of two two-dimensional objective function ma-trices. If the two two-dimensional objective functioncost matrices are square, then the three-dimensional ob-jective function cost matrix is a cube. This, in fact, isthe case for the 3AP and the Q3AP. Superimposed onone side of the cube is the assignment pattern that con-stitutes a feasible solution for one of the independentassignments. Superimposed on another side is the as-signment pattern that constitutes a feasible solution forthe other independent assignment. The actual solutioncost for the pair of feasible solution patterns is the sumof costs that reside inside the cube at the intersectionof the two independent assignment patterns. The re-sulting representation of the three-dimensional objec-tive function cost matrices and a typical feasible solu-tion are shown in Figure 9 for the 3AP and in Figure10 for the Q3AP. Figure 9 shows the inside of the 3APcube, so that the actual feasible solution elements areidentified. Whereas, in Figure 10 the inside of the cubeis not shown. Instead, the two independent assignmentpatterns are projected on two faces of the cube. In the

D1142

=

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! ! !

"

#

$$$$$$$$$

%

&

'''''''''

C,D =

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D31

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!

"

###########

$

%

&&&&&&&&&&&

=

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' '

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' ' C1313

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' '

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' '

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' C4242

'

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Figure 8: Typical GCAP Solution

remaining description of objective function costs andproblem solutions, we shall continue to use the latterillustrative technique.

6.4.2 Generalized 3-dimensional Assignment Problems

Here we introduce the Generalized 3-dimensional As-signment Problem. This problem arose as a subproblemof the GQ3AP. By studying exact solution methods forthis linear problem, we hope to gain insights that willsignificantly improve our methods for solving the muchmore difficult quadratic version.

The Generalized 3-dimensional assignment problemis given by:

Min

{ M∑i=1

N∑j=1

P∑p=1

Bijpxijyip

}(25)


M∑i=1

sixij ≤ Sj (j = 1, 2, · · · , N), (26)

DocNum 9 . 6. 4. 2


11!

92!

3!

93!

85!

51!

40!

10!

31!

36!

86!

83!

18!

81!

43!

19!

63!

58!

80!

44!

43!

42!

71!

90!

44!

58!

8!

3!

35!

90!

31!

31!

12!

14!

60!

80!

75!

7!

25!

73!

17!

51!

98!

3!

97!

39!

24!

92!

29!

85!

52!

44!

56!

25!

68!

6!

36!

28!

22!

20!

20!

98!

65!

79!

i

j!

p

Figure 9: Typical 3-dimensional Assignment ProblemSolutionQuadratic 3-Dimensional Assignment Problem!

i,k!

j,n! p,q!

i j; k n!

i p; k q!

Figure 10: Typical Q3AP Solution

N∑j=1

xij = 1 (i = 1, 2, · · · , M), (27)

xij = 0, 1(i = 1, 2, . . ., M ; j = 1, 2, . . ., N). (28)

and subject to the following constraints on Y:

M∑m=1

rpymp ≤ Rp (p = 1, 2, · · · , P ), (29)

P∑p=1

ymp = 1 (m = 1, 2, · · · , M), (30)

ymp = 0, 1(m = 1, 2, . . ., M ; p = 1, 2, . . ., P ). (31)

Figure 11 shows a typical solution for the G3AP, su-perimposed on the 4x3x2

The Generalized Quadratic 3-dimensional Assign-ment Problem is adequately described in [30]. So, wewon’t give the formulation here. But, to illustrate thecomplexity, we show in Figure 12 a typical solution toa size 4x3x2 problem instance superimposed on its 3-dimensional objective function cost matrix. Note thatthe projections of the cost elements contributing to the

i!

j! p!

Figure 11: Typical G3AP solution

problem cost consist of a 4x3 GQAP solution matrix onone face of the cost matrix solid and a 4x2 GQAP solu-tion matrix on an adjoining side of the cost matrix solid.

i,k!

j,n! p,q!

Figure 12: Typical GQ3AP solution

Now that we have introduced the family of quadraticassignment problems that are emerging, we point outthe relationships between all these problem types. Thisdefines the taxonomy of the models we have comeacross. Extrapolating from these relationships to in-clude other models is intuitive. Consider Figure 13which contains a table of all problem types reported onso far. The top half of the table contains 2-dimensionalproblem types and the bottom half 3-dimensional prob-lem types. Problem types are connected with the subsetsymbol. The table tells us that the LAP may be solvedby any algorithm for the following problems: LAP, QAP,CAP, BQAP, GAP, GQAP, GCAP or GBQAP. The tablealso tells us that the GQAP, can be solved only by algo-rithms for the GQAP, GCAP and GBQAP, assuming thatall of these algorithms exist. Of course, using an algo-

DocNum 10 . 6. 4. 2


rithm that is designed to solve a more complex problem,may or may not save computation time. Additionally,the more complex algorithms will certainly take morecomputational resources such as disk and RAM.

Relationships among various assignment problems!

Linear! Quadratic! Cubic! Bi-quadratic!

2-Dim! LAP! QAP! CAP! BQAP!

GAP! GQAP! GCAP! GBQAP!

Linear! Quadratic! Cubic! Bi-quadratic!

3-Dim! 3AP! Q3AP! C3AP! BQ3AP!

G3AP! GQ3AP! GC3AP! GBQ3AP!

Shading indicates an exact solution method exists. !

!

! ! !

! ! !

! ! !

! ! !

!

!

! !

! !

!

Figure 13: Relationships among various assignmentproblems

7 PROMISING RESEARCH

We describe here the promising areas of research thatjudiciously applied will not only allow significant ad-vances in the art of solving the more complex versionsof the Quadratic Assignment Problem, but will also im-prove the speed and efficiency with which the more fa-miliar forms of the QAP can be solved.

7.1 Subgradient-Modification

The lower bound improvement provided by em-bedding a dual ascent method in a subgradient-modification algorithm, namely the volume algorithm,is a significant algorithmic progress for the GQAP(Pessoa, et al. [47]). Similar techniques should ap-ply equally well to other difficult problems for whichdual ascent steps permit the immediate improvementof the Lagrangean objective function value. Prob-lems that could be addressed immediately are the3-dimensional Assignment Problem (Zhu [66]), theQuadratic 3-dimensional Assignment Problem (Hahn etal. [28]) and the Generalized Quadratic 3-dimensionalAssignment Problem (Zhu [66]).

7.2 Grid Computing

The University of Versailles PRISME Laboratory devel-oped a parallel implementation of branch and boundsolvers for the RLT-2 version of our QAP and the RLT-1Q3AP algorithms. See [22]. These parallel implemen-tations were built using the Bob++ library. See https://software.prism.uvsq.fr/bobpp/. Bob++ is anopen-source framework for the implementation of par-allel solvers based on tree-search algorithms such as

branch-and-bound. Its supported parallel environ-ments include POSIX threads for multi-core and multi-processor SMP machines, and Kaapi for large scale com-putation grids.

For Bob++, PRISM implemented asynchronous fullydistributed branch and bound algorithms. Their algo-rithms are asynchronous because they offer more effi-cient parallelization; allocation of work is not delayedwhile waiting for the next synchronization moment,and essential information, such as upper bound values,is distributed in a timely manner. The distributed ap-proach for grid computing is essential in order to 1) re-duce contention for the master processor and 2) reducethe impact of the computational grids poor and unpre-dictable communication proprieties. However, the dis-tributed approach is complex, because it requires theimplementation of a load balancing algorithm and acheckpoint mechanism to save not only the state of thecomputation of the master process, but of all the pro-cesses. There are also grid computing facilities in theU.S. These include the Condor Project at the Universityof Wisconsin (http://www.cs.wisc.edu/condor/)and the The NASA Advanced Supercomputing (NAS)Division located at the NASA Ames Research Centerin Moffett Field, California. Seehttp://www.nas.nasa.gov/.

7.3 Level-3 Reformulation-Linearization Technique

As mentioned in §6.2, Reformulation-LinearizationTechniques achieve significant advances in the solvabil-ity of Quadratic Assignment Problems. Problem RLT-2,in particular, provides sharp lower bounds and con-sequently leads to very competitive exact solution ap-proaches. A striking outcome, documented in Table 2of Loiola et al. [41], is the relatively few nodes con-sidered in the binary search tree to verify optimality.This leads to marked success in solving difficult QAPinstances of size 30 in record computational time. Basedon this success, Hahn et al. turned attention in [31] tothe level-3 form in order to get even tighter bounds,knowing that we would have to pay a price for the in-creased model size. The challenge was to take advan-tage of the additional strength, without being hurt bythe substantial increment in problem dimensions. Thisrequired novel computational steps, better adapted tothe much larger formulation size. They showed that, asfor level 2, the level-3 RLT for the QAP can be handledvia a Lagrangean approach to obtain subproblems withnested structure. This time, however, they had manymore dualized constraints and decomposable subprob-lem blocks.

The work on RLT-3 for the QAP, shows promise butit also runs into severe limitations. The encouragingpoint made in [31] is that in all but three test instances,the RLT-3-based algorithm reached lower bounds suffi-ciently close to the optimal solution that the best knownsolution was confirmed as optimal. But, just because alower bound is tight, does not mean that one can count

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on it to be useful in a branch-and-bound algorithm. Thebound has to be calculated quickly. Fortunately, thedual ascent bounds that Hahn et al. developed for RLT-3 are calculated iteratively. The graph in Figure 1 of[31] shows the fraction of the Nug 22 optimum solutionvalue that is reached by the RLT-3-based lower bound,as a function of runtime. Excellent lower bounds werereached in only a few minutes. To verify that RLT-3bounds are indeed effective requires that they be testedin a branch-and-bound implementation.

A severe limitation is that the number of variablesgrows dramatically with RLT level. The RLT-2 branch-and-bound solver code already runs into memory lim-its of the current generation of computers for probleminstances larger than N = 36. Memory limits of ma-chines available to researchers today make it difficult,if not impossible to calculate RLT-3 lower bounds forproblem instances larger than N = 25 using the cur-rent Fortran code. On the positive side, experimentshave demonstrated promise for reducing the numberof nodes that must be considered for proving optimal-ity using branch-and-bound [1]. A potential approachfor dealing with RLT-3 memory requirements is to dis-tribute computation load to multiple CPUs, thus mar-shaling the RAM from several computers.

7.4 Convexification Methods

Another area of investigation tries convexificationmethods for quadratic optimization problems having atleast some linear equality constraints. Convexificationof nonconvex 0-1 integer programming problems hasbeen done, for instance, using SDP [52] or techniquessuch as those proposed by Floudas and used by Zlobec[68]. One needs to identify at least one such techniquethat results in strong bounds for QAP-type problems,for the continuous or possibly some other relaxation. Sofar continuous bounds based on Plateau’s method havebeen disappointing for most nonconvex GQAPs. How-ever the tools that have been developed jointly by ourteam and GAMS Dev Corp [58] can be transposed toother quadratic 0-1 problems, such as those studied byPlateau, and provide strong bounds.

Nonlinear convex 0-1 integer programs may havean objective function that is too convex, i.e., with adeep valley that produces lower than needed continu-ous lower bounds. As long as the alternate convex ob-jective function chosen coincides with the original oneat all integer feasible solutions, it produces a valid lowerbound. Thus, this ”de-”convexification can be appliedto convex problems in order to improve the continuousrelaxation bound, and this can be done using the toolsmentioned above.

For convex models, the Convex Hull Relaxation(CHR) bound is the best among all Lagrangean-typebounds for a given objective function. One could inves-tigate the possibility of generating strong lower boundsby combining some de-convexification of the objectivefunction, and more general relaxation models than the

continuous relaxation, for instance the CHR relaxation,for convex problems.

Current experiments ([6] and [25]) show that the CHRapproach applied to nonconvex 0-1 quadratic problemsis a very fast way of generating optimal or near-optimalsolutions for several types of nonconvex 0-1 integerproblems, such as quadratic knapsack problems andsome assignment-type problems. The impact of addingmore starting points to the CHR approach to increasethe probability of reaching points close to the optimal0-1 solution for these problems has been substantial.

8 SUMMARY AND PREDICTIONS

We have presented a survey of a rapidly growing classof 0-1 integer programming problems related to theQuadratic Assignment Problem. Not all of these prob-lems have quadratic objective functions, but those thatdo not are closely related to those that do.

We reported the latest and best methods availablefor solving exactly these difficult problems and suggesta taxonomy that provides a framework for combiningexisting solution methods and sets of computer toolsthat can be modified and extended to make inroadsin solving this growing class of optimization problems.The message we wish to impart is that the set of prob-lem types we describe herein are closely related andwhat can be learned from the exact solution method forone of them carries over to most of the others. More-over, a solution technique for the more complex of theseproblems automatically becomes a solution method formany of the simpler problems.

We predict that future research on this class of prob-lems will include exact solution methods for the QAPand similar 0-1 integer problems that heretofore havebeen too large to solve exactly. For instance, solving aQAP of size 40 will be a practical everyday event. Thereis even hope that it will be possible to solve exactlyproblems of size 50 and larger with reasonable compu-tational effort.

We predict that there will be increased interest in RLTtechniques as a means of solving 0-1 mixed integer pro-grams of all sorts, as well as in optimizing more gen-eral families of nonlinear, non-convex programs. TheRLT process of recasting problems into higher-variablespaces can be generally applied when, for example,products of continuous variables arise. It has also beenrecently extended to nonlinear integer programs byAdams and Sherali [5] that can treat discrete variablesthat are not necessarily binary. Though the convex hullresults differ in such cases, the basic constructs of re-formulation and linearization continue to be applicable.We hope that there will emerge a new trend in optimiza-tion that encourages the use of this heretofore underuti-lized technique.

Finally, we hope that there will be renewed interestin problems such as the QAP by the research commu-nity and its funding agencies. This is turn should help

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to make it possible for developed as well as undevel-oped countries to become more efficient in such areas asfacilities planning, layout of communication and trans-portation networks and design of wireless communica-tion devices. Similar benefits should accrue to various0-1 mixed integer problems, such as those found in air-line scheduling, organization of manufacturing opera-tions and supply chain management.

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