T. Mautor, C. Roucairol, "A new exact algorithm for the solution of quadratic assignment problems", Discrete Applied Mathematics 55 (1994) 281-293.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....method. A variety of lower bounding techniques are known, including the Gilmore Lawler Bound (GLB) bounds based on linear programming (LP) and dual LP relaxations, bounds based on eigenvalues, and semidefinite programming (SDP) bounds. Most successful B B implementations have used the GLB [11,40], which is easy to compute but unfortunately tends to deteriorate in quality as the dimension n increases. The LP [46] and SDP [51] bounds are typically tighter on large problems, but can be prohibitively expensive for use in B B. For implementation in a B B algorithm the relationship between the ....

....makes extensive use of the dual matrix U in the branching step. Note that if v is the objective value of the best known discrete solution to QAP (i.e. the incumbent value) then z U ij v implies that X ij = 0 in any optimal solution of QAP. The branching process in [4] uses polytomic branching [40], where child problems are created by either assigning a fixed facility to all available locations (row branching) or by assigning all available facilities to a fixed location (column branching) In both cases logic based on U can be used to eliminate child nodes when branching. Several different ....

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T. Mautor and C. Roucairol. A new exact algorithm for the solution of quadratic assignment problems. Discrete Applied Mathematics, 55:281--293, 1994.

....with the distributed implementation and numerical accuracy, as described in Section 4.6. 3. 3 Utilization of symmetries in Euclidean search grid Since the search spaces we are mostly concerned with are grids in Euclidean space, we may make use of symmetries, as noticed by Mautor and Roucairol [6]. 4 It should be pointed out that some authors have concluded that using branch and bound searches to generate optimal solutions is inferior to using heuristic methods [6] 5 (35 possibilities) Root Place first block (10 possibilities) Place second block Figure 1: Schematic representation ....

.... we are mostly concerned with are grids in Euclidean space, we may make use of symmetries, as noticed by Mautor and Roucairol [6] 4 It should be pointed out that some authors have concluded that using branch and bound searches to generate optimal solutions is inferior to using heuristic methods [6]. 5 (35 possibilities) Root Place first block (10 possibilities) Place second block Figure 1: Schematic representation of search tree. For instance, if we were to place the rst block in the lower right corner, the search space induced by this partial placement is identical (with respect ....

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T. Mautor and C. Roucairol. A new exact algorithm for the solution of quadratic assignment problems. Discrete Applied Mathematics, 55, 1994, pp. 281-293.

....of solution methods see [7] 10] and [31] The usual approach to optimally solving QAP(A;B;C) is to employ a branchand bound (B B) algorithm. See [24] and [30] for alternatives based on polyhedral theory. Early papers reporting results of B B algorithms for QAP include [5] 8] 13] and [29]. The most important element in the construction of a B B algorithm for the QAP appears to be the method used to obtain a bound for the subproblem (itself a lower dimensional QAP) at each node of the B B tree. Most B B algorithms for the QAP have utilized the well known Gilmore Lawler bound (GLB) ....

....LAPs (see [23] as encountered in our application. The spectral decompositions required for the construction of QPB are performed using routines from the Meschach library [33] 3 BRANCHING STRATEGY In our branch and bound implementation we employ polytomic branching, as introduced for QAP in [29] and used in numerous subsequent implementations. At a given node there is a set of fixed assignments X i; i) 1, i 2 I . Let N = f1; 2; ng, J = f(i) j i 2 Ig, I = N n I , J = N n J . If the node is not fathomed, we generate children according to one of the following schemes: Row ....

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T. Mautor and C. Roucairol (1994). A new exact algorithm for the solution of quadratic assignment problems. Discrete Applied Mathematics, 55, 281-293.

....of GLB requires the solution of n 2 1 LAPs. However, for a problem in Koopmans Beckmann form the LAP associated with each c ij is trivial to solve, and as a result C can be obtained in a total of only O(n 3 ) operations. Several successful B B algorithms for the QAP have utilized the GLB [7, 9, 12, 31]. GLB based algorithms have proven effective for problems up to about size n = 24, but for larger problems the growth in nodes may become overwhelming. 4 3.2 Eigenvalue and related bounds A Koopmans Beckmann QAP, with an added linear term, can be written in the matrix form KBP min X2 Pi ....

....X = tr(CX T ) and z is the value of GLB. If X solves LAP(C) then X ffl U = 0. It follows that if v is the value of a known solution to QAP, then z U ij v ) X ij = 0 (4) in any optimal solution of QAP. The use of (4) to eliminate children in the course of branching was introduced in [31], and has been employed in many subsequent papers. Mautor and Roucairol [31] also introduced polytomic branching, where at any node candidate children are obtained by either (row branching) fixing one facility and assigning it to all available locations, or (column branching) fixing one location ....

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T. Mautor and C. Roucairol. A new exact algorithm for the solution of quadratic assignment problems. Discrete Applied Mathematics, 55:281--293, 1994.

....[4] makes extensive use of the dual matrix U in the branching step. Note that if v is the objective value of a known discrete solution to QAP (i.e. the incumbent value) then z U ij v implies that X ij = 0 in any optimal solution of QAP. The branching process in [4] uses polytomic branching [39], where child problems are created by either assigning a fixed facility to all available locations (row branching) or by assigning all available facilities to a fixed location (column branching) In both cases logic based on U can be used to eliminate child nodes when branching. Several different ....

....25,24,26,18,3,14,7,5,9,4 the various instances have marked advances in both processor capability and QAP solution methods. See [23] for an excellent history of these problems. Most results reported for Nugent problems of size n 24 have been based on the GLB; see for example [5, 10,39]. Prior to the work reported here the largest Nugent instance solved to optimality was the nug25 problem. Nug25 was first solved using the dynamic programming lower bounding approach of Marzetta and Brungger [38] and was subsequently solved more efficiently using the dual LP approach of Hahn et ....

T. Mautor and C. Roucairol. A new exact algorithm for the solution of quadratic assignment problems. Discrete Applied Mathematics, 55:281--293, 1994.

....of solution methods see [6] 9] and [30] The usual approach to optimally solving QAP(A;B;C) is to employ a branch and bound (B B) algorithm. See [23] and [29] for alternatives based on polyhedral theory. Early papers reporting results of B B algorithms for QAP include [4] 7] 12] and [28]. The most important element in the construction of a B B algorithm for the QAP appears to be the method used to obtain a bound for the subproblem (itself a lower dimensional QAP) at each node of the B B tree. Most B B algorithms for the QAP have utilized the well known Gilmore Lawler bound (GLB) ....

....LAPs (see [22] as encountered in our application. The spectral decompositions required for the construction of QPB are performed using routines from the Meschach library [32] 3 Branching Strategy In our branch and bound implementation we employ polytomic branching, as introduced for QAP in [28] and used in numerous subsequent implementations. At a given node there is a set of fixed assignments X i; i) 1, i 2 I. Let N = f1; 2; ng, J = f(i) j i 2 Ig, I = N n I, J = N n J . If the node is not fathomed, we generate children according to one of the following schemes: Row ....

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T. MAUTOR and C. ROUCAIROL. A new exact algorithm for the solution of quadratic assignment problems. Discrete Applied Mathematics 55:281-293, 1994.

....flow coefficients and distances and the optimal solution can hence be found by computing minimal scalar products as described above. 3.2. The branching rule of Mautor and Roucairol The processing of each node consists essentially of bounding and branching. Branching is performed as described in [11] supplemented with the forcing of assignments as described in [7] Branching is performed on facilities and is based on the reduced cost information generated when solving the LAP with cost matrix L in the GLB calculation. Comparing the reduced cost of entry (i, k) against the value 3 of the ....

....The process stops when each facility and each location has at least two possible assignments. The location i with fewest remaining possible assignments is then chosen for branching, and a new subproblem is created for each k among these by assigning i to k. In addition the symmetry testing of [2, 11] is implemented. Note that although branching on locations exactly as described above for facilities is possible, the symmetry testing requires that branching is performed on facilities. 3.3. Finding a first feasible solution The first feasible solution to be used as upper bound can in the case ....

T. Mautor, C. Roucairol, A new exact algorithm for the solution of quadratic assignment problems, Discrete Applied Mathematics 55 (1994) 281--293.

....flow coefficients and distances and the optimal solution can hence be found by computing minimal scalar products as described above. 3.2 The branching rule of Mautor and Roucairol The processing of each node consists essentially of bounding and branching. Branching is performed as described in [12] supplemented with the forcing of assignments as described in [8] Branching is performed on facilities and is based on the reduced cost information generated when solving the LAP with cost matrix L in the GLB calculation. If the sum of the GLB computed and the reduced cost of entry (i; k) is ....

....The process stops when each facility and each location has at least two possible assignments. The location i with fewest remaining possible assignments is then chosen for branching, and a new subproblem is created for each k among these by assigning i to k. In addition the symmetry testing of [2, 12] is implemented. Note that even if branching on locations exactly as described above for facilities is possible, the symmetry testing requires that branching is performed on facilities. 3.3 Finding a first feasible solution The first feasible solution to be used as upper bound can for QAP be ....

T. Mautor, C. Roucairol, A new exact algorithm for the solution of quadratic assignment problems, Discrete Applied Mathematics 55 (1994), 281--293.

....flow coefficients and distances and the optimal solution can hence be found by computing minimal scalar products as described above. 3.2 The branching rule of Mautor and Roucairol The processing of each node consists essentially of bounding and branching. Branching is performed as described in [11] supplemented with the forcing of assignments as described in [7] Branching is performed on facilities and is based on the reduced cost information generated when solving the LAP with cost matrix L in the GLB calculation. Comparing the reduced cost of entry (i; k) against the value of the current ....

....The process stops when each facility and each location has at least two possible assignments. The location i with fewest remaining possible assignments is then chosen for branching, and a new subproblem is created for each k among these by assigning i to k. In addition the symmetry testing of [2, 11] is implemented. Note that even if branching on locations exactly as described above for facilities is possible, the symmetry testing requires that branching is performed on facilities. 3.3 Finding a first feasible solution The first feasible solution to be used as upper bound can in the case of ....

T. Mautor, C. Roucairol, A new exact algorithm for the solution of quadratic assignment problems, Discrete Applied Mathematics 55 (1994) 281--293.

....from a j to l assignment. We take advantage of this information in the branching scheme. We branch on locations, i.e. a location among the free is chosen and a child node corresponding to locating each of the free facilities on the chosen location is formed. The scheme is originally proposed in [16]. For each combination of free location j and free facility l we use the reduced cost c jl as a lower bound on the increase in the G L bound. If this lower bound together with the incumbent is sufficient to rule out the possibility of finding the optimal solution in the subspace corresponding to a ....

T. Mautor, C. Roucairol, A new exact algorithm for the solution of quadratic assignment problems, Discrete Applied Mathematics 55 (1994), 281-293.

....been published over the last 15 years. The best results reported by traditional B B codes are those of Crouse and Pardalos [9] from 1989 solving the classical Nugent 15 problem in 2005 seconds on an IBM 3090 and of Burkard [4] from 1980 and reimplemented in 1992 on a Cray 2 by Mautor and Roucairol [14] solving the Nugent 15 in 1290 seconds. A B B code taking advantage of symmetries in the problem during branching has been developed by Mautor and Roucairol [14] It solves the Nugent 15 problem in 121 seconds, the Nugent 16 problem (extracted from Nugent 20 as a subproblem with symmetries) in ....

....15 problem in 2005 seconds on an IBM 3090 and of Burkard [4] from 1980 and reimplemented in 1992 on a Cray 2 by Mautor and Roucairol [14] solving the Nugent 15 in 1290 seconds. A B B code taking advantage of symmetries in the problem during branching has been developed by Mautor and Roucairol [14]. It solves the Nugent 15 problem in 121 seconds, the Nugent 16 problem (extracted from Nugent 20 as a subproblem with symmetries) in 969 seconds, and the Amour Buffa problem of size 20 (which was hitherto unsolved) in 1189 seconds, all on a Cray 2. Of parallel codes solving large QAPs, only a ....

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T. Mautor, C. Roucairol, A new exact algorithm for the solution of quadratic assignment problems, Discrete Applied Mathematics 55 (1994) 281-293.

....In practice, the lack of efficiently computable, tight lower bounds for the QAP has been the key factor in the problem s difficulty. There are a variety of approaches to obtaining lower bounds. One of the oldest methods, the Gilmore Lawler bound (GLB) is still widely used; see for example [8] and [19]. The GLB is closely related to a certain linear programming (LP) relaxation of QAP, proposed by Adams and Johnson [1] Several other bounding techniques can also be shown to be methods for approximately solving the dual of this LP relaxation; see [16] The exact solution of the LP relaxation ....

.... programming generalization of the well known technique of variable fixing based on reduced costs in linear programming relaxations of discrete optimization problems; see for example [9] Fixing logic based on reduced costs from the master LAP associated with the GLB for QAP is described in [19]. 6 Computational Results In this section we describe the performance of the bound QPB on a suite of test problems from QAPLIB [6] In order to facilitate a comparison with other bounding methods we examine the same test problems used in [27] In Table 1 we give, for each problem, the optimum ....

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T. MAUTOR and C. ROUCAIROL. A new exact algorithm for the solution of quadratic assignment problems. Discrete Applied Mathematics 55:281-293, 1994.

....problem) The only known methods to solve exactly the QAP are branch and bound ones. However, due to time constraints, they remain limited to problems of size around fifteen, and only recently, exploiting symmetries in locations distribution, problems of size up to twenty have been solved exactly [MR94]. In all other cases, one must use approximate methods, i.e. heuristics. Recently a comparison of eight heuristics [MDC95] has led to a particularly interesting result: the remarkable effectiveness of greedy local search, which sometime outperforms more elaborate heuristics like simulated ....

T. Mautor and C. Roucairol. A new exact algorithm for the solution of quadratic assignment problems. Discrete Applied Mathematics, 55:281-- 293, 1994.

....lorsque le cout d epend des distances entre les sites et des flots entre les unit es. Il est l un des probl emes classiques de l Optimisation Combinatoire, ses applications re elles etant nombreuses et vari ees en electronique, informatique, ergonomie, architecture, Mautor et Roucairol [25, 26] : ffl conception et implantation des circuits VLSI, ffl architecture et conception d ateliers, ffl l analyse de r eactions chimiques, et ffl allocation de processus a des processeurs, etc. L identification et la formulation du probl eme sont dues a Koopmans et Beckman en 1957 [35] De ....

....un etat inactif avant le traitement, et passent a un etat actif apr es avoir re cu un sous probl eme. A la fin du traitement, les processeurs peuvent repasser a un etat inactif. Chaque processeur ex ecute le meme algorithme B B (se r ef erer a Mautor pour plus de d etails sur l algorithme, [26]) qui fonctionne selon les principes ci dessous. Proc edure pour un processeur Pr(i) S election : c est une strat egie meilleur d abord. Le sous probl eme choisi de la liste locale, est celui qui poss ede la plus petite evaluation. S eparation : c est une s eparation polytomique. Une fois le ....

T. Mautor and C. Roucairol. A new exact algorithm for the solution of quadratic assignment problems. Discrete Applied Mathematics, 55:281--293, 1994.

....2000 iterations, which produced a bound of 2177. For larger instances, instead of branching to generate all possible assignments, we use the branching rule utilizing information from the reduced cost obtained by computing GLB in the root node (for a description of the branching rule we refer to [9, 17]) and branch on only one facility. We also make use of symmetries in the coefficient matrices to reduce the number of subproblems considered. Taking the minimum bound of the subproblems created, we obtain a lower bound as argued above. We have computed this for several instances from QAPLIB. In ....

T. Mautor and C. Roucairol, A new exact algorithm for the solution of quadratic assignment problems, Discrete Applied Mathematics 55, 1994, 281--293. 34 S.E. Karisch, E. C¸ ela, J. Clausen, and T. Espersen

....against the current best solution. If the node cannot be fathomed based on its own bound, a branching step is performed generating the children of the node. These are then stored in the pool of live subproblems with the bound calculated for the current node. Branching is performed as described in [21] and is based on the reduced cost information generated when solving the LAP with cost matrix L in the GLB calculation. Comparing the reduced cost of entry (i; k) against the value of the current best solution it can be determined if GLB for the subspace generated by assigning facility i to ....

....by assigning facility i to location k exceeds this value thus implying that the optimal solution will not belong to the subspace. Hence a number of assignments can be ruled out, and the location with fewest remaining assignments is chosen for branching. In addition the symmetry testing of [3, 21] is implemented. The hardware for which the code is written is an MIMD MEIKO system with 16 Intel i860 processors each with 16 MB of internal memory. Each processor has a peak performance of 30 MIPS when doing integer calculation giving an overall peak performance of approximately 500 MIPS for ....

T. MAUTOR and C. ROUCAIROL. A new exact algorithm for the solution of quadratic assignment problems. Discrete Applied Mathematics, 55:281--293, 1994.

....The branch selected is the facility location assignment whose alternative cost is maximal. This strategy, while effective on small Nugent instances, was not effective for the larger ones. What seemed to work for our purposes, was the polytomic depth first search strategy of Mautor and Roucairol [32]. This strategy extends a node by all assignments of an unassigned facility to unassigned locations based upon the counting of forbidden locations. A forbidden location is one where the addition of the corresponding leader 11 element would increase the lower bound beyond the upper bound. We ....

.... Conclusions Our work on tree elaboration (branching) strategies for the Quadratic Assignment Problem has extended the concept of forbidden locations as originally discussed by Lawler, et al., 28] in connection with the travelling salesman problem and later implemented by Mautor and Roucairol [32] for the QAP. In forbidden locations, the leader value of the next possible branching decision is used as a bound estimate for deciding whether or not to take that branch. We have developed for purposes of tree elaboration, a series of bound estimates ranging from rather poor estimates that are ....

Mautor, T. and Roucairol, C., "A New Exact Algorithm for the solution of Quadratic Assignment Problems", Discrete Applied Mathematics, Vol. 55, (1994): 281-293

....the QAP and thus were directed toward the avoidance of high cost assignments. These strategies are not designed for solving large QAPs in reasonable time and thus were summarily rejected. What seemed to work for our purposes, was the polytomic depth first search strategy of Mautor and Roucairol [31]. This strategy extends a node by all assignments of an unassigned facility to unassigned locations based upon the counting of forbidden locations. A forbidden location is one where the addition of the corresponding leader element would increase the lower bound beyond the upper bound. We found, ....

.... Conclusions Our work on tree elaboration (branching) strategies for the Quadratic Assignment Problem has extended the concept of forbidden locations as originally discussed by Lawler, et al., 29] in connection with the travelling salesman problem and later implemented by Mautor and Roucairol [31] for the QAP. In forbidden locations, the leader value of the next possible branching decision is used as a bound estimate for deciding whether or not to take that branch. We have developed for purposes of tree elaboration, a series of bound estimates ranging from rather poor estimates that are ....

Mautor, T. and Roucairol, C., "A New Exact Algorithm for the solution of Quadratic Assignment Problems", Discrete Applied Mathematics, Vol. 55, (1994): 281-293

....VLSI design [36] architecture design [14] The objective of QAP is to assign n units to n sites in order to minimize the quadratic cost of this assignment, which depends both on the distances between the sites and on the flows between the units. We use the algorithm of Mautor and Roucairol [32]. Their algorithm uses the depth first strategy, we modified it to use the best first strategy. 0 50 100 150 200 250 300 350 400 450 500 0 2 4 6 8 10 12 14 16 Time Number of processors Skew Heap 3 3 3 3 3 3 3 3 3 Single Semi Splay Semi Splay 2 2 2 2 2 2 2 2 2 Simple Semi Splay ....

T. Mautor and C. Roucairol. A new exact algorithm for the solution of quadratic assignment problems. Discrete Applied Mathematics, to appear, 1993. MASI-RR-9209 - Universit'e de Paris 6, 4 place Jussieu, 75252 Paris C'edex 05.

....The Quadratic Assignment Problem (QAP) consists of assigning n units to n sites in order to minimize the quadratic cost of this assignment. This cost depends both on the distances between the sites and on the flows between the units. We modified the algorithm developed by Mautor and Roucairol [43] which uses the depth first strategy, to use best first. Figure 4 shows the results obtained on a KSR1 machine with virtual shared memory. The programs solve the nugent15 problem which was obtained from the QAP Library. We can see that the Splay trees and the funnels PQs are more efficient than ....

T. Mautor and C. Roucairol. A new exact algorithm for the solution of quadratic assignment problems. Discrete Applied Mathematics, to appear, 1993. MASI-RR92 -09 - Universit'e de Paris 6, 4 place Jussieu, 75252 Paris C'edex 05.

.... Gamma i such that Gamma i is not used ffl (i) Gamma i ffl Set Gamma i as used ffl Insert d(i; Gamma i) in TabuList ffl i = Gamma i (1) i Figure 4: Computation of . the QAP the distance matrix plays a central role, inducing the existence of symmetric solutions (see [6] for a discussion on this topic) Therefore, in order to obtain a good cover of the feasible set, the permutation is computed in such a way that it prohibits whenever it is possible for a couple of firms to be located at the same distance in two different solutions. 6 Numerical results We first ....

T. Mautor and C. Roucairol. A new exact algorithm for the solution of quadratic assignment problems. Discrete Applied Mathematics, 55:281--293, 1994.

....approach, developed by Wang, in 1990, 25] on a microVax computer, which simulates n 2 processing units performing nonlinear transformations. 3 Sequential Branch and Bound Let us briefly recall the main principles of the sequential algorithm introduced by Mautor and Roucairol, in 1992, [16], 15] which currently provides the best sequential results for the quadratic assignment problem. Meanwhile, we discuss important features, such as lower bounds and branching strategies, concerning any Branch and Bound algorithm. 3.1 Bounding Undoubtedly, the computation of the lower bound is ....

....which several symmetric or isometric equivalences can be detected. We want to avoid creating, visiting and bounding these different but equivalent nodes in different branches of the search tree. For this purpose, a simple test which is quickly computed, has been introduced by Mautor and Roucairol [16]. For any partial solution, this test identifies the different isometric classes. Thus, a unit can be assigned to only one site of each isometric class. Reduction test using the search gap This classical test forbids some assignments and, therefore, reduces the size of the B B tree. Moreover, it ....

[Article contains additional citation context not shown here]

T. Mautor and C. Roucairol. A new exact algorithm for the solution of quadratic assignment problems. Discrete Applied Mathematics, to appear, 1993. MASI-RR-92-09 - Universit'e de Paris 6, 4 place Jussieu, 75252 Paris C'edex 05.

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T. Mautor, C. Roucairol, "A new exact algorithm for the solution of quadratic assignment problems", Discrete Applied Mathematics 55 (1994) 281-293.

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