Jorge Carrizo, Fernando Tinetti and Pablo Moscato. CeTAD C.C. 75 ARGENTINA ABSTRACT

Transcription

1 A Computational Ecology for the Quadratic Assignment Problem Jorge Carrizo, Fernando Tinetti and Pablo Moscato CeTAD Universidad Nacional de La Plata C.C , La Plata ARGENTINA ABSTRACT A new heuristic method for the Quadratic Assignment Problem based on the use of a \Computational Ecology" and a modied version of Hillier-Connors heuristic HC-63 are presented in this paper. The method is intended to include another metaheuristic called Tabu Search to increase the diversity of solutions before recombining them to create new congurations which are used to restart the search. Its intrinsic parallelism was checked when implemented on a parallel computer (Transputer system).

2 Introduction The quadratic assignment problem (QAP) is a combinatorial optimization problem which is a representative of the important class of problems known as NP-hard, thus it is important to develop and test heuristics for it since membership in this class means that solution times are likely to be at least exponential with respect to N [16]. Heuristics are often considered by many scientists as \rules-of-thumb" which aid to the solution of specic problems but lack rigorousness. However, in combinatorial optimization where the concept of eectiveness [4] plays a fundamental role, heuristics should be regarded as part of an interplay with algorithms, an earlier stage in scientic discovery. The paper is organized as follows: the next subsections give a brief introduction to the maths involved in the formulation of the problem and some applications for it, section 2 describes the main ideas of the Steepest Descent Algorithm we have implemented and its relation with the HC-63, section 3 concerns with the population approach proposed, in section 4 results are shown, and nally section 5 presents our conclusions. The QAP is dened as follows: Denition and applications Given a vector of objects O = (o 1 ; o 2 ; : : : ; o N ) and a vector of data points (or places) Q = (q 1 ; q 2 ; : : : ; q N ) and a distance measure d : QQ! R +, the objective function we must minimize is?() given by?() = i=1 j=1 C ij D (i)(j) (1) where is a permutation of the set of indices (1; 2; : : : ; N) into the vector Q, (i) indicates the i th element of (the index of the point in Q assigned to object o i ) and C ij is a hypothesized similarity between objects o i and o j. The QAP is generally represented by two N N matrices C and D. The matrix C is called the \structure" or \cost" matrix and is dened by the terms C ij. The matrix D is called the \data" or \distance" matrix where each element D ij 2 D denotes the distance between points q i and q j. Thus D (i)(j) = d(q (i) ; q (j) ) (2) The QAP was rst formulated by Koopmans and Beckmann [9] followed by Gilmore [5] and Lawler [10] and since then it has found applications from data analysis to task scheduling on parallel computers. Applications also include the minimization of total wire length in electronic assemblies, location of machines, departments or oces within a plant so as to minimize transportation eorts and costs, ordering interrelated data on a disk, scheduling theory, distribution of medical services in a large hospital center, placement of logical modules in a chip, etc. In addition to this list, the Traveling Salesman Problem (TSP) (and with it all its applications) can be considered as a special case of the QAP in which the structure matrix is a cyclic permutation matrix.

3 Task Scheduling and Computational Load Balancing on Parallel Computers One of the advantages of studying heuristics for combinatorial optimization problems using large QAP instances is that many results can be evaluated using them as \test-beds" and the insights gained can be applied into other settings. This is the case for the load balancing issues arising in parallel computation. It can be viewed as a combinatorial optimization problem in which a number M of objects (Processes) must be assigned to a number N of places (Processors). With more generality we can consider it as a graph partitioning problem: given a graph G composed of 1. P vertices, from 0 to P? 1, which represent subproblems (tasks, processes, instructions, etc.) which will be processed concurrently, each one with an individual processing time (weight) w(p); 2. P (P? 1) communication links C(p; p 0 ) such that C(p; p 0 ) is the data amount (measured in bytes, words, etc.) to be sent from p to p 0 before the execution of the p 0 -th process. We wish to seek a partition of G in N subgraphs G 0 ; G 1 ; : : : G N?1 such that G = G 0 [ G 1 [ : : : [ G N?1 (3) where N is the quantity of nodes numbered from 0 up to N? 1 of a concurrent computer where the processes will be executed. 3. The computational load is well balanced, that is W n = X p w(p) Const (p 2 G n ); (4) 4. Communication time is minimal Comm = X p;p 0 C(p; p 0 ) t[n(p); n(p 0 )] = min (p 6= p 0 ) (5) where n(p) is the node that contains subprocess p and t[n; n 0 ] is the minimal time required to send data from the node n to node n 0. As we can see, the QAP is related with the communication part of the objective function. In this case, we will assume that communication time between processors only depends upon the distance between them so we can replace time by distance. We will mainly restrict ourselves to the study of QAP instances in which the P vertices lay on a grid (a rectangular array) due to the fact that we have large instances with known global minima. At rst glance, if no restrictions are given, the number of assignments of processes to processors is N! thus even for small sized problems (more than N = 15 or N = 20) the time needed for nding an optimal solution is prohibitive. Branch and bound was also applied but its implementation is also limited due to the computer time it requires [5] [10]. Iterative Improvement Strategies Heuristics may be classied in two main groups: 1.- Constructive. 2.- Iterative Improvement.

4 Constructive techniques start from either a partial or null assignation and they go on assigning new elements as long as restrictions or the objetive function constraints are satised. Iterative Improvement techniques start from an initial solution that includes all the elements to be placed and continue to make changes until a certain condition is reached. Among them we can remark those that belong to the SDPI class (Steepest Descent Pairwise Interchange), that is to take pairs of objects and try to interchange them regarding the cost function. For the QAP we must cite the H63 due to Hillier, the HC-66 due to Hillier and Connors [15], the CRAFT due to Armour and Bua and the `Biased Sampling' proposed by Nugent et al. With the exemption of the latter, they are all deterministic procedures. H63 and HC63-66 The HC63-66 is an iterative improvement heuristic (a modied version of the H63), which uses a so called \Move Desirability Table" (MDT) to select which pair of objects will be moved. The elements of the MDT indicate the cost changes that would result from \unilaterally" moving an object to a dierent location. This means that the net cost change of interchanging two elements is not evaluated at the time the MDT is made. The HC63-66 selects which objects will be moved just by inspection of the MDT which somewhat measures the individual \desire" of pair of objects to move. Once the objective function is evaluated, the move is accepted if it eectively produces a cost reduction (when the interchange considered is between objects at maximum distance, a minimum allowable cost reduction is imposed; a kind of \threshold"). The main dierence between the H63 and the HC63-66, is that the latter starts trying to move an object to locations which are as far as possible from the current location on the grid, while the former only considers up and down, and right or left moves. Eventually it also considers diagonal moves [15]. We can see other dierence in that H63 accepts \every" positive cost reduction, while H63-66 does not, some changes are discarded due to the \threshold". These heuristics are not only eective regarding the low-cost solutions it yields, but it also gains from the use of the MDT. In the case of H63, O(8 N) interchanges are tried instead of a naive O(N 2 ). The HC63-66 searches the O(N 2 ) neighborhood in an \ordered way" so most of the times this will conduce to a cost reduction interchange in less than O(N 2 ) attempts. We must recognize that this is an appealing characteristic but there is an obvious tradeo, since when we are near to local minima most of the moves that try to interchange the positions of objects far apart signicantly increase the cost. HC-91 When selecting what kind of method will be used for the rst stage, where local optimality is the goal, an iterative improvement procedure was chosen due to its solution quality and computational complexity. Among the collection of techniques described above, HC63-66 from Hillier & Connors was selected, showing a good performance in the average nal cost in several dierent instances of the QAP with sizes from ve to thirty department [15] along with a reduced computing time. This heuristic was modied for a new approach in which objects are selected according to its relative contribution to the objective function. Now we introduce a new heuristic which has many similarities with the previously cited methods. Up to a certain extent we can consider them as formal similarities. To avoid misinterpretations we will explicitly describe the main dierences. We will call the method \HC?91" since we think of it as a direct descendant of the strategies employed by Hillier & Connors. We have chosen a criterion to create the MDT which is a list based on the individual contribution of each object to the objective function. The MDT will give the interchange order between objects. For each object J.

5 MDT (J) = max J 0 (C JJ 0 D (J)(J 0 )) (J 6= J 0 ) (6) Another dierence can be found in the way the elements are selected to be interchanged. An auxiliary array is used to keep the sorting of the objects regarding the MDT. Using this auxiliary array, we take the rst object of the MDT, that is the object that has the biggest entry, and we try to interchage it with all the other elements. We follow the arbitrary order previously assigned, so if object o s has been selected, we try to interchange it with o 1 ; o 2 ; : : : until we nd an interchange that decreases the cost. We note o s as s and o v as v. Let's suppose we select object o v, then the decrement is given by (o s ; o v ) = +?? C ks D (k)(v) + C sk D (v)(k) + C ks D (k)(s)? C sk D (s)(k)? C kv D (k)(s) C vk D (s)(k) C kv D (k)(v) C vk D (v)(k) (7) From this formula we can see that the cost update requires O(N) operations. Finally, we do not forbid changes if they are smaller than a given threshold as implemented by Hillier & Connors. Each change that decreases the cost is performed no matter its magnitude. This lead us back from the 8 N of H63 to O(N 2 ) neighborhood, but in return it oers better solutions than the original method. We must remark that this approach also diers from CRAFT since not all pairwise exchanges are considered. This is true as long as an improvement can be made. In addition, we should note that the calculation of the cost reduction is performed in O(N) steps. Some possible changes in the denition of the MDT have been considered, with and without a non-zero minimal cost reduction step, but although they seemed promising the nal congurations found were not better than those found with the HC? 91. Tabu Search for the QAP Our basic heuristic strategy (HC-91) does not include backtracking, that is it halts after nding a local minimum. We have decided to include Tabu Search (TS) to explore neighboring solutions [6]. From a given conguration, TS seeks among the neighboring congurations (this depends on the move set), trying to nd that one that has the best evaluation (regarding the objective function, for example). If there are no improving moves possible, we are in a kind of local optimum and the one that least degrades the objective function is chosen. Two other elements are part of this \metaheuristic", the \tabu list" and the \aspiration criteria". The former is a data structure that forbids certain moves and attempts to avoid \cycling" while the second are rules that override the \tabu status" of some moves if they are relevant or interesting. An example can be those that lead to a better solution than the best found so far. We have not yet dened what are the elements of the Tabu list. Skorin-Kapov has selected them to be the exchange of two units that were exchanged during the preceding s iterations. Taillard has a dierent approach: \for every unit and location, the latest iteration at which the unit occupied that location is recorded. A move is tabu if it assigns both interchanged units to

6 locations that had occupied within the s most recent iterations." Nothing has been said about how long an element must remain \tabu" or the length of that list. Both issues have been studied in previous works [17] [18]. The implementation of a circular list of objects without a xed length can be found in Ref. [18]. Some of the instances studied in this paper have been also employed to study the behaviour of a \deterministic update" for simulated annealing [12]. The \Memetic" Algorithm With the name of \Genetic Algorithms" (GA), we recognize a number of methods, generally applied to operations research and articial intelligence problems, which make use of a \population approach" for the search. They borrowed the name from the analogies with biological mechanisms and they are part of what constitutes the eld of \evolutionary programming" [7]. Very recently, a number of techniques started to depart from traditional GA methods although they also shared two of the basic ingredients of GA, Reproduction and Crossover [14] [1]. These techniques have been tentatively called \memetic algorithms" due to the introduction of the word \meme" by R. Dawkins to designate the \unit of imitation". The \meme" would play the analogous role of the gene but in cultural evolution [11] [3]. These new approaches for optimization can also be viewed as \computational ecologies" where processes engage in competitive and cooperative behaviour [8] [13]. The memetic approach has proved to be a really powerful tool in the search of suboptimal solutions in other NP-hard problems like the TSP where interaction between individuals enhances the performance of individual heuristics [13]. Our basic strategy can be summarized with the following piece of pseudocode. Begin RANDOM INITIALIZATION; (* N a agents are created *) GenNum := 0; MaxGenNum := 40; Repeat HC-91 TO LOCAL MINIMA; (* all agents in the population *) SORT AGENTS; (* regarding the objective function *) TEST LOSS OF DIVERSITY; (* some agents may be discarded *) GENERATE NEW POPULATION; (* a new set of N a agents *) GenNum := GenNum + 1; Until (DiversityCrisisF lag = 1) or (GenNum > MaxGenNum) End. A certain number of agents (usually small, from 4 to 16 have been used in the tests) are initialized randomly and they undergo a period of local optimization until all of them reach local minima congurations. Then the agents are sorted regarding the value of the objective function. This ranking is used in generating the new population. Since we want to avoid \premature convergence" we test for diversity loss in each generation. We discard those agents which appear twice or more times in the population. That is, we eliminate exact duplicates but we leave two congurations if at least two objects have been assigned to dierent places. If the rst half of the sorted population has exactly the same conguration, then the population is judged to have a \diversity crisis" and we set DiversityCrisisF lag := 1. Selection criteria and Crossover The recombination operator (Crossover) is intended to help to explore the region of conguration space between good quality congurations. It is generally viewed as a mechanism that

7 interchanges information. In a \memetic" approach, the parents are well-developed congurations or even local optima, thus we expect it to exploit the correlation of local optima of the objective function landscape. Selection criteria for the parents are always needed. We also follow the guidelines of GA in selecting as parents the best agents (regarding the objective function) currently in the population. We took as parents, the best N a =2 dierent congurations. A new set composed of N a? 1 new agents is created at each generation using the crossover mechanism and the best local minima found in the previous step completes the list of N a agents. In this implementation, we have always selected the best local minima found in the previous step as one of the parents in order to have a fast convergence and to study general performance characteristics of the method but other type of selection criteria may be possible which would yield to better nal solutions. The process is continued until there is no diversity within the individuals. The best and the second best agents create three new agents, then the best with the third create two agents, then the best with the fourth creates two agents up to the eight agent in the list. For the QAP, H. Muhlenbein has recently introduced a recombination called N p -sexual voting recombination [14]. For each place, the object placements of N p parents are considered. If the same object is assigned to a particular place more times than a given threshold, then it is also assigned in the same place in the ospring. The remaining places will be completed with \mutations", that is random assignments of the other objects not yet placed. They presented results with N p = 7; 4 neighbours, the global best solution with a weight of two and the current conguration. We used the method proposed by Brown, Huntley & Spillane which is similar to the PMX operator used by Goldberg and Lingle [1]. It creates a new cyclic permutation O (for ospring) using two parent congurations (P 1 ; P 2 ) and the following procedure 1. Each of the parent structures is partitioned in three sections by randomly drawing two breaking points. 2. The second section of the O's structured is copied from the corresponding section of parent P The rst and third sections of the O structure are obtained by copying terms from corresponding positions in P 2 's structure which have not yet been assigned before. 4. The remaining empty positions in the rst and third sections of O are then obtained from a permutation of the terms not yet allocated. Results We have used two dierent population sizes in many runs of the heuristic. The population size is a parameter that takes an important role by giving better average results regarding the nal value of the cost function. The optimal size for the population (if any) was not studied in this paper. Instead, we select this value ad hoc regarding some preliminary runs. Although the SDPI heuristic of each individual is used many times during a run, we must remark that the population approach is dierent from a typical stochastic \multistart" algorithm. We have observed that the crossover operator contributes to drive the population towards better regions of conguration space. It is also remarkable that, as a consequence of that property, the time involved in using an individual SDPI decreases, that is the local minima are reached faster from recombined congurations in each generation. Seeking a comparison, we implemented a stochastic \multistart" algorithm using HC-91 a number of times which was approximately 5500; and the results were:

8 1. Average: Standard Deviation: Minimum: 6124 (The global minimum). More information about these results and the ones found adding Tabu Search can be found in [2]. With the population strategy, we made a preliminary study by doing 78 runs with a population size of 8, and the results found were: 1. Average: Standard Deviation: Minimum: 6124 The values given above are the best agent found at the end of a run, that is when a \diversity crisis" is reached. We should remark here that a run with a population size of 8 (or generally speaking, with a population size of N a ), is more expensive (in computer time) than 8 (or N a ) local minima found with the HC-91. The number of runs (5500 with HC-91 against 80 with the population approach), in some way \equalizes" the dierence regarding the computer time employed. As a deterministic SDPI method, the HC-91 has no freedom, given a starting point, it will end up in a certain, but a priori unknown, minimum. A characterization of its performance is possible given an idea of what will be the nal average cost. The bigger number of times it starts form a dierent point, the bigger the chances of nding a better solution. With this belief, more independent searches were added in order to improve the nal costs. So we try with a number of 16 HC-91 independent searches to embrace a wider range in the conguration space. As expected, better results were found reaching the known minima very easily and both the average cost and the standard deviation of the conguration cost have shown a reduction: 1. Average: Standard Deviation: Minimum: 6124 The selection of results was made in the same way that for the population size of 8. For a more accurate comparison between the results obtained with the two population sizes (8 and 16), we include frequency histograms made in the same way (the same cost values range: and the same \step" of 20) (see Fig. 1). At this time we should say that the increment in the population size from 8 to 16 will not double the computer time when the population is implemented on a sequential computer, (e.g. 2N a will not take twice the time taken by N a individual searches). We nd this type of approach to be particularly appropriate for implementation on parallel computers, like a Transputer network. To see the eect produced by discarding those agents which appear twice or more times in the population, we made some runs with a population size of 30 without such a \control". The population size of 30 seems to be better than the previously cited 8 and 16 (30 involves a wider range in the conguration space) and is appropriate to avoid \premature convergence". The results found were: 1. Average: 6152

9 2. Standard Deviation: Minimum: 6128 These results show the importance taken by such a control, because they are similar to those obtained with a population size of 8 in which the \duplicated" agents were discarded (and with less computer eort). Conclusions and further work As general conclusions on the results we can say: 1. The memetic approach searches better than a typical stochastic \multistart" approach. 2. The population size has an important eect on the results. 3. The control of diversity (and its relation with convergence) in the context of a memetic approach is necessary to obtain better results with a small population. Regarding the computer eort necessary to implement this kind of approach, we have mentioned before the relevance of a parallel implementation. There are (at least) three things to note about this: 1. SDPI heuristics are polynomial-time from the point of view of computational complexity. 2. If there are enough resources available, it will be possible to use a bigger population size, with at least two consequences: (a) A wider range in the conguration space will be explored. (b) A higher speed to reach good local minima will be obtained. 3. By improving the asynchronousness of the strategy we will improve the speedup of the global optimization process. These statements should be carefully studied to see the magnitude of each one, and their interaction. Early tests performed on problems of bigger sizes show an acceptable behavior of this heuristic. The Skorin-Kapov's instance of size 90 was tested and some results have been obtained. They were approximately 0:2 % above the best minimum found. This approach gives new breath to SDPI algorithms in the context of Memetic Algorithms. We can use dierent SDPI procedures in dierent agents and combine them (better said, the congurations they obtain) via the crossover operator to see how are the congurations in quality (cost) and how quickly they are found. These dierent SDPI procedures should be studied regarding their own behavior and how they interact in the population approach. The method described in this paper halts a run when a diversity crisis is reached. We are considering a method in which the population is restarted after the agents undergo a period of independent Tabu Search. This will increase diversity while preserving the congurations of the best local minima previously found. We expect that this blend would combine the strengths of both approaches. The inuence of population size must also be studied and it could be interesting to nd a relation between the population and the problem size. Acknowledgements We thank the rm MicroSistemas S.A. for the computer facilities and the CeTAD Director, A.A. Quijano for the support of this project. We specially thank J. Skorin-Kapov and E. Taillard. This work was conducted within the framework of the CICPBA's Program for Technology Transfer ( ).

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