Graph Partitioning Using Genetic Algorithms

Graph Partitioning Using Genetic Algorithms Christian Hohn Institut fur Grundlagen Elektrotechnik/Elektronik Technische Universitat Dresden Colin R Reeves School of Mathematical and Information Sciences Coventry University Email: [email protected] Abstract The ecient implementation of parallel processing architectures generally requires the solution of hard combinatorial problems involving allocation and scheduling of tasks. Such problems can often be modelled with considerable accuracy as a graph partitioning problem|a problem well-known from complexity theory to be NP-hard, which implies that for practical purposes the global optimum will not be found by any exact algorithm, except for trivially small problems. Recently genetic algorithms (GAs) have been proposed as a means of nding good solutions to the graph partitioning problem. In this paper two methods of implementing GAs are explored. In the rst case, they are combined with a heuristic `greedy algorithm'. Unlike previous approaches the heuristic is directly linked to the recombination operator and this leads to a new `heuristic neighbourhood search' operator. Comparisons with existing approaches for combining heuristics with genetic algorithms show that the direct approach leads to improved results. In addition the heuristic neighbourhood search operator gives rise to a more computationally efcient scheme. Secondly, an approach which uses a structural crossover operator is investigated. Again, improved versions of existing approaches are developed, and nally the two forms of operator are compared using a number of di erent graphs. 1. Introduction Modern parallel processing architectures give rise to a number of interesting problems of implementation. Despite the considerable potential increase in computational eciency from using parallel processors, what is achieved in practice may be signi cantly below this potential if the overall load is not distributed in a balanced way. One of the most fundamental questions is how to al- locate computational tasks to many processors, when tasks are known to be inter-related, in such a way that the overall computational load on each processor is equalized as much as possible. Given a set of algorithms to be implemented on a processor, we can derive information on the number and type of simple arithmetic or logical operations, and consequently on how much time they need. It is also necessary to know what dependencies exist between the operations, all of which can be represented on a directed graph. We can also represent the processor information on a graph which de nes the speed of the processors, and the nature and speed of the communication links between processors. A recent survey of this problem is given in [1]. Assuming such information has been gathered, the allocation problem can be modelled quite closely by the graph-partitioning problem, which we formulate below. 1.1. The k-way Graph Partitioning Problem Given an undirected graph G = (V; E; !v ; !e ), where V = fv1; v2 ; :::; vmg represents the set of vertices, E  V  V denotes the set of edges, and the weight functions !v : V 7! Z + and !e : E 7! Z + de ne non-negative integral weights for all vertices and for all edges respectively, then for a positive integer k the k-way graph partitioning problem (GPP) consists of nding k non-empty mutual disjoint subsets V1 ; V2 ; :::; Vk such that [ki=1 Vk = V . A partition is said to be feasible if X ! (v)  q ; 8i 2 [1; k] v2Vi v i where qi represents the lower bound for the size of the ith partition. By formulating the problem in this way, we allow the possibility of partitioning intoPsubsets of di erent sizes. However, by choosing qi = v V !v =k, we can force all subsets to be of exactly equal size. In some problems, this restriction may not be necessary (or even possible), and the constraint can be relaxed by decreasing qi . 2 The cost of a partition is the sum of !e (i; j) where i and j are in di erent subsets (or the sum of all external edge weights). Then the optimization task consists of nding the partition which minimizes the external edge weights (or equivalently that which maximizes the internal edge weights). The k-way graph partitioning task has proved to be a good model for many real world problems, such as memory partitioning and circuit partitioning, as well as the problem of load balancing for multiprocessor systems. In the latter case one can think of the vertices as the tasks (or jobs) in a multiprocessor system, where !v represents the computational time required by each job, and !e expresses the amount of communication between the jobs. The problem is to assign each job to one of the k processors, such that the load is balanced according to q1; q2; :::; qk; and the communication between the processors is minimal. In this paper initially two examples of graph partitioning tasks are considered. For a simple example a two dimensional grid of 10  10 nodes, all of the same weight, is to be partitioned into k subsets, for k = f2; 4; 7; 15g. As a second example some geometric graphs reported in [2] with di erent weights at the edges and vertices are considered. To obtain subsets of nearly equal size, in all experiments the qi were evaluated according to X (1) qi = b( !kv )c; 8i 2 [1; k]: v V 2 2. Genetic algorithms Following Holland's introduction of the concept [3], genetic algorithms (GAs) have received increasing attention as a means of solving complex nonlinear optimization problems. They have proved to be robust and well suited to many optimization tasks including problems involving multi-variable solutions for nonlinear objective functions. Essentially, GAs solve optimization problems in an analogous manner to the genetic processes of nature. The physical solution of a given problem (corresponding to a `phenotype') is coded into a string, termed a chromosome, which represents the `genotype'. Each physical variable is associated with a gene (or set of genes) and a value of a gene is called an allele. The theory was originally developed for binary representations, but often higher cardinality alphabets may be more natural. To generate a new solution two `parents' are drawn from the present population and parts of their chromosomes are exchanged. For instance, suppose 2 strings (a1 ; a2; a3; a4; a5) and (b1 ; b2; b3; b4; b5) represent two solutions to a problem. A crossover point is chosen at random from the numbers f1; . . .; 4g, and a new solution produced by combining the pieces of the the original `parents'. For instance, if the crossover point were 2, then the `child' solutions would be (a1 ; a2; b3; b4; b5) and (b1; b2; a3; a4; a5). The child is thus constructed from components which have already been tested in one context, and which can now be tested in another. If it is important to ensure that the child chromosome is a `pure' recombination the recombination operator used should possess the property of `strict transmission' [4]: all alleles of the child chromosome are chosen from one of the corresponding parent values. While this de nition appears straightforward, there are many practical problems where the solution space is restricted by a set of constraints, so that operators will di er in the degree to which they strictly transmit the parent alleles. For the reproduction process parents are drawn from the population according to their tness. The better a solution, the more likely its chromosome is to be selected for creating new o spring. Thus, it is hoped that the population will naturally be directed towards `better' regions of the solution space. However, in the case of combinatorial problems, this assumption is often not ful lled, and the basic GA needs some modi cation. Recently, Reeves [5] has shown how the GA can be regarded as a neighbourhood search (NS) procedure, which di ers from traditional NS, and its modern variants, in two important respects. Firstly, the neighbourhood is de ned by two solutions (the parents) instead of only one. Furthermore, the recombination operator used in a GA is more likely to produce a new solution that is further away from the parent solutions than the operators that are used in common NS techniques. Indeed, these are the fundamental di erences that lead to two signi cant properties of the GA as a neighbourhood search methodology: on one hand they are less likely to become trapped at local optima; on the other hand they produce a relatively poor local performance. The latter property has led to several attempts to combine a GA with local hill-climbing procedures which e ectively re ne the solution obtained by the GA as a postprocessing tandem operation, as in [6] for example. In [5] it is suggested that the neighbourhood search should be embedded within the recombination step rather than as an `add-on' extra. Another approach is to hybridize the GA with an on-line heuristic which continually assists the genetic search process. However, it is not entirely clear how best to arrange this combination such that the resulting hybrid scheme works most eciently. In this paper a novel methodology of combining the GA with an on-line heuristic is proposed and its performance demonstrated when applied to the graph partitioning problem. 2.1. Genetic algorithms and the GPP For general partitioning tasks where a set of objects is required to be decomposed into a set of subsets, `adding' heuristics are often useful. The objects are ordered in some fashion and are then assigned to one of the subsets following the rules of the adding heuristic. In the case of the GPP an attempt is made to nd the best subset for each vertex with respect to the vertices that have already been placed. This may be achieved by the `greedy algorithm' shown in Fig. 1. Obviously, the resulting solution may depend strongly on the initial sequence chosen. Thus, it would appear to be helpful to use a GA to optimize the initial sequence and this is the basis of the traditional approach [7]. For m objects and k subsets a sequence-based chromosome of length m is required. This way of encoding has been applied to many combinatorial problems where sequences are of interest, e.g. the travelling salesman problem or machine sequencing. Furthermore, many operators have been developed to combine permutations. Supposing the vertex vi is to be assigned to a subset: 1. Determine for each candidate subset the internal cost which will result if vi is assigned to it. 2. Assign the vertex to the subset which maximizes the internal cost. 3. If there is more than one possible subset assign the vertex to the one of the smallest size at present. 4. If there is still more than one possible subset assign the vertex randomly. Figure 1. The greedy algorithm This type of hybrid method was used by Jones and Beltramo [7], whose experiments suggested the best approach for partitioning problems was to use the PMX operator combined with a greedy decoding of the sequenceencoded chromosome. Although the combination of the GA and an on-line heuristic has also been successfully applied to other hard combinatorial problems [8, 9], there still remain some strong objections. In particular, it can be argued that by interposing an on-line heuristic the phenotype might become too remote from the present coding to nd good `building blocks'. If such fears are well-founded, the recombination of `good' genotypes will not necessarily lead to a chromosome which represents a `good' phenotype. Thus, much of the information provided by the parent solutions may be lost. To investigate this further, an experiment to analyze the constitution of a newly generated solution was set up. The parent solutions were compared with the child to ascertain the degree of `strict transmission' by recording the number of `randomly' placed vertices within the child (i.e. vertices that are placed in subsets to which neither of the parents assigned them). Two di erent GAs were run, one using the PMX operator combined with the greedy algorithm shown in Fig. 1 and the other a `strict transmission' (ST) operator inspired by the random assortment operator proposed in [10]. The ST operator uses group-number encoding, where the ith gene encodes the subset to which vertex vi 2 V is assigned, and it constructs a child by recombining the values of the parent solutions as long as the o spring remains within the solution space (see Fig. 2). In the experiment the 1. Choose two chromosomes x1 and x2 . 2. For (i = 0; i < m; i ++) if (x1 [i] == x2 [i]) then put 2 copies of x1 [i] into the bag. Otherwise put one copy of x1 [i] and one copy of x2 [i] into the bag. 3. Draw randomly an allele x[i] from the bag. 4. If equation (1) is satis ed then y[i] = x[i] and all remaining values x[i] are removed from the bag. 5. Go back to 3 until the child is completed or the bag has become empty. 6. If the bag is empty ll the child chromosome randomly, but obeying equation (1). Figure 2. The ST operator ST approach serves as an ideal comparison because although the transmission may not be completely strict, in practice only a very small number of `randomly' chosen values can be expected. For the problem under consideration a 10  10 grid was to be partitioned into k subsets, k 2 f2; 4; 7; 15g, and a population of 100 chromosomes was used. The number of `randomly' placed vertices was averaged over all solutions generated in 5000 trials and the results are given in Table 1. k 2 4 7 15 PMX+Gr 24.9% 56.2% 73.3% 86.3% ST 1.3% 2.5% 4.7% 11.9% Table 1. Percentage of `randomly' placed vertices averaged over 30 runs It was found that the number of `randomly' placed nodes increased with the number of subsets. In fact, for the hybridized GA the values for 25-85% of the variables were found to be set exclusively by the greedy algorithm, with the information o ered by the parent solutions being ignored. It appears that the heuristic substitutes for rather than assists the GA. In recognition of the potential shortfalls of the traditional approach the hypothesis put forward in this paper is that the performance of the `hybridized' GA may be improved if a better gene transmission can be achieved. 3. Heuristic neigbourhood search To circumvent the problems discussed so far the heuristic can be combined directly with the recombination operator as suggested in [5]. Again group-number encoding is used. Assuming that the loci of the chromosomes are numbered from 1; . . .; m, two mutually exclusive and exhaustive index sets Ifix and Ifree can be constructed by assigning the index of each gene either to Ifix or Ifree . The child is then created by xing the alleles for all genes de ned by Ifix while the indices of the remaining positions build up an initial sequence for the greedy algorithm to complete the child solution. In this way the operation of the heuristic is restricted to a neighbourhood search in the hyperplane de ned by Ifix . Therefore, the group of operators working on this basis are termed heuristic neighbourhood search (HNS) operators. In essence, this approach o ers the opportunity to control the in uence of the heuristic by manipulating the sets Ifix and Ifree . However, since the greedy algorithm acts on a particular subset only once, the initial sequence cannot be optimized (even if that were possible in many trials). It would thus seem reasonable to present the sequence in a random order, which also reduces the possibility of the greedy algorithm becoming trapped in local optima. However, under such conditions the HNS operators are likely to perform worse than the traditional approach if the range of the neighbourhood becomes too large. Attention must therefore be focused toward a proper adjustment of the neighbourhood size in order to optimize the performance of the heuristic. In what follows methods for constructing Ifix and manipulating the neighbourhood size in a sensible manner are discussed. 3.1. The respectful HNS operator Radcli e [4] suggests that an important property of a recombination operator is that it is `respectful', by which he means that if both parents share a common characteristic the child should inherit it. In the context of this paper, if the HNS operator is to pay full `respect' to both parents then it is essential that Ifix includes the indices of all positions where the parent chromosomes share the same allele. The resulting operator is termed the respectful HNS (rHNS) operator. Obviously, the size of the neighbourhood area jIfree j will depend on the cardinality of the alphabet used. The higher the cardinality becomes, the less likely it is that two parents will share a common value. Consequently, the neighbourhood becomes larger and the task for the greedy algorithm increases. Redundancy arises in the GPP since there are k! possible ways to encode the same solution (the labelling of groups is arbitrary). This will also a ect the size of the neighbourhood, as the greater the redundancy, the fewer positions the parents will share. Since the redundancy increases factorially, large values of k may reduce the cardinality of Ifix signi cantly, thus preventing the GA from nding good `building blocks'. Redundancy can be removed by renumbering the chromosomes before applying recombination as suggested in [7]. Here renumbering is indicated by `+R', e.g. rHNS+R. 3.2. The escape mechanism As with traditional GAs the hybridized version runs the risk of becoming trapped at local optima. As soon as the whole population has converged to a region which excludes global optima, the GA gets trapped because the rHNS operator o ers no way to escape. To prevent the GA from getting trapped a mutation operator is often used which randomly alters the value of a variable so as to escape. A similar operator could be introduced which acts on Ifix , such that each index belonging to Ifix can be removed and inserted into Ifree with a given probability. This approach has the possibly useful advantage that the in uence of the mutation operator increases as the population converges. However, in this paper the avoidance of local optima is attempted in a deterministic manner which makes use of problem-speci c knowledge. Fig. 3 shows four plots representing solutions for the partitioning of a 4  4 grid into four subsets. The rst plot shows one of the optimal solutions, the second and the third show two potential parent solutions and the fourth plot represents the resulting neighbourhood. From the last plot it may be seen that this neighbourhood excludes the global optimum because (for example) the node in the third row and second column is misplaced. To establish access to the global optimum this node must be removed, which involves moving the associated index from Ifix to Ifree . To achieve this an algorithm is implemented which scans Ifix for nodes blocking the global optimum and subsequently removes them. Several characteristics can be used to recognize such nodes. In the algorithm proposed here the index of a vertex is moved from Ifix to Ifree if the vertex possesses an external weight but does not possess an internal weight. Thus the neighbourhood is extended in some sensible manner and the GA will be less likely to become trapped. However, this is achieved at the cost of slower convergence. In the following experiments the use of the escape mechanism is labeled by `+E', e.g. rHNS+E. It should also be recognized that any redundancy within the code acts on the neighbourhood. As stated above, the greater the redundancy, the larger the neighbourhood, thus reducing the possibility of the GA becoming trapped. Whilst for large values of k the redundancy has a detrimental e ect on the GA there might be a range of small values of k where it is advantageous, helping the GA to avoid local optima. However, to make use of the redundancy one has to nd a parameter to tune its in uence properly. 3.3. The neighbourhood reduction operator It has been argued that poor performance will result if the neighbourhood becomes too large. To reduce the neighbourhood it has been suggested that all redundancy should be removed. However, this e ect is limited and it may be desirable to make a further reduction of the neighbourhood size. This will involve a corresponding increase in the cardinality of the set Ifix . To achieve 1 1 3 3 1 1 3 3 2 2 4 4 2 2 4 4 1 3 3 3 2 3 1 4 2 2 1 4 2 1 4 4 1 3 3 3 1 1 1 3 2 2 4 4 2 2 4 4 1 2 2 3 2 3 1 4 3 4 4 Figure 3. 4  4 grid to be partitioned into 4 subsets this, additional variables are required to be xed in their values. One could use random values within the feasible range, but this would impair the aim of a stricter transmission. A simple approach is to make use of the parent solutions by taking the values in a random fashion from both parents or strictly from only one of the parent solutions. The latter approach will bias the neighbourhood towards one of the parents and is more likely to improve the local performance. Therefore, an algorithm is implemented which moves half of the elements belonging to the set Ifree to the set Ifix , adjusting the corresponding position in the child chromosome to the values of one parent solution. However, by reducing the neighbourhood in this manner, the GA may be forced directly into a local optimum. To avoid this case the neighourhood reduction operator was only used in connection with the escaping algorithm described earlier. In the following experiments the use of the neighbourhood reduction operator in conjunction with the escape mechanism is marked by `E+N', e.g. rHNS+E+N. 4. Experimental design To compare the traditional approach with the proposed HNS operators the rst experiment used the 10  10 grid. For this task a steady state GA was used where an o spring replaces an individual randomly drawn from the lower half of the present population. The population size was set to 100 and the number of trials to 5000. The parents for the recombination operators were chosen randomly and no mutation operator was used. The initial population was obtained in two steps. Initial permutations for the traditional approach were generated by `shuing' the numbers 0; . . .; 99; then, to get an initial population, these were translated into partitions by randomly assigning each vertex to a subset obeying eq. (1). All experiments were run 30 times using di erent initial populations. k PMX rHNS rHNS rHNS rHNS rHNS +Gr +E +R +E+R +E+N 2 16.0 11.1 10.6 11.6 10.6 10.6 4 38.1 22.2 22.4 23.9 22.2 21.4 7 51.8 37.9 37.5 38.6 37.3 37.6 15 73.6 77.8 75.3 74.0 72.5 68.9 Table 2. Average best value Table 2 shows that for k < 15 all HNS operators produced roughly the same perfomance and converged toward better solutions than the traditional approach. For k = 15, apart from the rHNS+E+N operator, the HNS operators did not achieve better results. However, they di er in the computational time required. The results in Table 3 show that in the cases where the neighbourhood is reduced either by using a non-redundant code or by the rHNS+E+N operator, the computational time decreases signi cantly. Furthermore, one can observe that for k < 15 the rHNS operator is slightly quicker on average than rHNS+R, whilst for k = 15 rHNS+R is better. This illustrates the di erent in uences of the redundancy on a GA as argued in the previous section. k PMX rHNS rHNS rHNS rHNS rHNS +Gr +E +R +E+R +E+N 2 47 10 17 10 14 12 4 58 13 17 15 17 18 7 76 25 27 25 26 27 15 140 101 100 77 76 78 Table 3. Computational time in seconds 4.1. Geometric graphs To demonstrate the performance of the proposed operators when applied to graphs with a less regular structure, geometric graphs with di erent weights at the edges and vertices were considered. The rst to be investigated was a geometric graph like the G 600 graph reported in [2]. Using the same set of parameters, i.e. 600 nodes and an expected degree of 10, a graph was generated with 2844 edges, the sum of vertex weights was 1779, and the sum of edge weights 20096. The graph was to be partitioned into 20 subsets. Besides the operators used in the previous experiment, a standard GA using the ST operator was also used. To set up a fair competition, noting the fact that the algorithms di er in their computational complexities, the computational time was limited to 100 seconds per run using a SPARC station IPX. The population size was set to 50. Table 4 summarizes the results obtained after 30 runs. It shows the best value and the average best values in the rst two rows, and the number of trials completed in 100 seconds in the third row. To highlight the di erent in uences of the heuristic, the number of `randomly' PMX+Gr G 600 10 graph Best 6721 Aver 7207 Trials 350 Rstart 90.2% Rend 90.2% G 600 4 graph Best 1948 Aver 2070 Trials 300 Rstart 90.5% Rend 90.3% rHNS rHNS+E rHNS+R rHNS+E+R rHNS+E+N ST 6378 6976 550 76.6% 74.5% 6270 6806 500 78.4% 73.1% 5729 6408 650 72.7% 56.6% 5856 6495 600 76.5% 57.4% 3263 3692 1000 57.1% 15.0% 18653 18718 650 3.5% 3.5% 1848 2016 450 78.1% 77.6% 1570 1876 400 80.3% 78.5% 1627 1868 450 76.4% 69.8% 1532 1775 450 78.3% 51.8% 484 749 850 54.2% 20.5 % 7487 7570 650 3.5% 3.4% Table 4. Results for the geometric graphs placed vertices at the start and end of a typical run are given in the fourth and fth rows. In this experiment the rHNS+E+N operator clearly outperforms the other operators. The number of `randomly' placed nodes seemed to fall in an `optimal' range of roughly 60% at the beginning, decreasing rapidly to 15% at the end of the run. In comparison the ST which placed only 3.5% of the vertices `randomly', performed very poorly. Furthermore, the performance of a standard GA was no better than that produced by any other random technique, so that the `pure' recombination of two solutions cannot substitute for the knowledge incorporated by a heuristic (at least not in the time allowed). On the other hand the PMX/greedy algorithm placed nearly 90% `randomly' (i.e. according to the heuristic) which led to poor results although the initial sequences for the greedy algorithm were optimized during the run. Therefore, it appears that the best results will be obtained if the in uence of the heuristic can be tuned properly as in the case of the rHNS+E+N operator. In order to increase the con dence of the results a second geometric graph was generated with 600 nodes and an expected degree of 4. The resulting graph had 1171 edges, the sum of node weights was 1757 and the sum of edge weights 8229. The results for this graph generally con rm the previous observations, although there is an interesting di erence between the best value and the average best value for the rHNS+E+N operator, indicating a more complex search space in which the operator becomes trapped more often. This observation is supported by the fact that the operators which made use of the escaping mechanism achieved signi cantly better results than those which did not. 5. Structural crossover The above results are encouraging, but it is clear that the HNS operators do not really perform any recombination as traditionally understood in the realm of genetic algorithms. The second parent merely determines the nodes belonging to Ifix but contributes minimal addi- tional information concerning the partition. Although the results appears to be fairly good, one has to recognize that the GA does not use a proper recombination operator, which is usually regarded as one of the hallmarks of the GA approach. 5.1. Redundancy The main problem in the construction of appropriate recombination operators is caused by the highly redundant nature of the group-number encoding. For example, if solution A assigns the vertices vi and vj to subset a, while solution B assigns the vertices vi and vj to subset b then both solutions put vi and vj into a common subset. However, the di erence in labelling hides this similarity, so that the redundancy prevents the GA from nding similarities among the solutions and increases the diculty of nding good `building blocks'. In the following, a procedure is given which allows the recombination of two partitions, while preserving this kind of structural information. Given G = (V; E; !v ; !e ), two partitions can be written as V1 = V11 [ V12 [ ::: [ V1k = V and V2 = V21 [ V22 [ ::: [ V2k = V . Then V = V 1 \ V2 = [k [k V i=1 j =1 1i \ V2j : The resulting subsets V1i \ V2j contain only those vertices which are assigned by both parents to a common subset. Let V be the set of all resulting subsets. Then a feasible partition of V will also be a feasible partition of V , and a respectful combination of V1 and V2 , in the sense that all vertices assigned to a common subset by both parents will remain in one. However, to implement this procedure directly would be computationally expensive, because the cutset of k2 subsets has to be calculated. In the following, several approaches will be discussed for realizing such a procedure more e ectively. 0 0 5.2. The SR operator The structural recombination operator proposed by von Laszewski [11] can be considered as an approximation to the procedure given above. One subset, say V2j , is chosen randomly from the second parent and copied into the partition of the rst parent. This temporary and probably illegal partition is then repaired as follows. If V1j is the subset replaced by V2j then a repairing mechanism detects all vertices belonging to V1j \ V2j and assigns them randomly to the remaining subsets to obtain a legal solution. However, in this approach, the operator still identi es the subset V1j as the corresponding subset to V2j according to the group-number. Although von Laszewski did propose a modi ed procedure to remove the in uence of the redundancy, it has a high computational cost. In this paper a modi ed version of this structural crossover is used. The `SR' operator di ers from the original in the following respects:  For generating the initial population a clustering algorithm is used to partition the graph into k clusters using some randomly chosen vertices as initial points.  The subset which is copied is chosen according to some partial tness value, to prevent a bad partial solution from being inherited by the o spring. The only candidate subsets are those which have an above average internal cost within the partition.  The subset V2j does not simply replace the subset bearing the same group-number, but the one which has the most nodes in common, say V1i . In this way, as much as possible of the information encoded in the parent solutions is preserved.  All vertices belonging to V1i \ V2j are re-assigned by means of a greedy algorithm instead of randomly. As a nal variation, the `SR+Gr' operator additionally re-assigns all those vertices whose internal costs are less than or equal to the external costs by means of a greedy algorithm. 6. Results In this section, some computational experiments are reported for the operators `SR' and `SR+Gr' as described above. The results of the `PMX+Gr' and `rHNS+E+N' approaches (as de ned earlier) are repeated for comparison. 6.1. The grid instances The initial experiments again used the 10  10 grid with equal weights at the edges and vertices, to be partitioned into k = f2; 4; 7; 15g subsets. Tables 5 and 6 display the performances achieved. The undoubted winner of this competition is the structural recombination operator `SR+Gr'. The speed of this approach is particularly impressive. It appears that the very regular structure of the graph favours this approach. k PMX+Gr rHNS+E+N SR+Gr 2 4 7 15 16.0 38.1 51.8 73.6 10.6 21.4 37.6 68.9 10.0 20.0 37.7 66.3 SR 10.0 20.0 39.8 75.3 Table 5. Average best value k PMX+Gr rHNS+E+N SR+Gr SR 2 4 7 15 47 58 76 140 12 18 27 78 3 4 4 8 3 4 4 9 Table 6. Computational time in seconds 6.2. Geometric graphs Table 7 shows the best and the average best solutions found within 30 runs for the geometric graphs. Furthermore, the number of trials performed within 100 seconds highlights the computational e ort required. Finally, the fourth and fth rows of the tables display the percentage of vertices placed by the heuristic at the beginning and end of the search respectively, in order to illustrate the in uence of the heuristic during the search. PMX+Gr rHNS+E+N SR+Gr SR G 600 10 graph Best 6721 3263 2964 3781 Aver 7207 3692 3087 4063 Trials 350 1000 1800 2200 Hstart 100% 79.5% 12.6% 3.5% Hend 100% 39.1% 9.3% 3.3% G 600 4 graph Best 1948 484 368 854 Aver 2070 749 459 1004 Trials 300 850 3600 4000 Hstart 100% 73.5% 10.0% 5.9% Hend 100% 41.0% 3.3% 2.9% Table 7. Results for the geometric graphs The basic SR operator is inferior here to the HNS operator, but SR+Gr dominates again, presumably because of the reduced in uence of the heuristic, and the corresponding increased in uence of recombination. 6.3. Random graph In the nal set of experiments, a random graph was generated with 400 vertices and an expected degree d = 8. The nal graph had 1762 edges and was to be partitioned into 8 subsets. For all algorithms the initial conditions were as above. PMX+Gr rHNS+E+N SR+Gr SR 3000 2593 2638 2806 3040 2655 2697 2831 1100 3300 2100 4300 100% 50.4% 32.2 % 4.5% 100% 5.0 % 54.0 % 11.3% Best Aver Trials Hstart Hend Table 8. Results for the R 400 8 graph The main di erence between this experiment and the previous one was that the number of vertices placed by the greedy algorithm increased for SR+Gr, considerably slowing down the operator. It appeared that the random graph o ers less structural information, so that recombination disturbs the parent solutions rather than leading to an improvement. Thus the HNS operator, which actually performs no recombination, seems to cope better with the random graph than the SR operators. However, a more thorough analysis has revealed a problem with the selection of the vertices which are to be re-assigned by the greedy algorithm. This aspect is currently under investigation, but preliminary results seem to indicate that the performance of SR+Gr can be enhanced by using a more exible approach. 7. Conclusion A new group of heuristic neighbourhood search operators has been proposed and investigated for the GPP. They are constructed by combining a heuristic directly with the recombination operator. In contrast to the traditional hybridized GAs the operation of the heuristic is restricted, so that the HNS operators are less computationally intensive and preserve the parent solutions better. Several operators have been investigated for controlling the in uence of the heuristic by manipulating the size of the neighbourhood. This approach demonstrates better performance than the simple `PMX+Gr' approach of [7], in terms of computational time and quality of results when applied to two examples of the graph partitioning problem. On the whole, the results are fairly encouraging and it is believed that the approach may be applicable to a wider range of combinatorial problems. However, there are reasons for believing that in many cases a GA can only outperform other search techniques if full use of recombination is made. Therefore, an alternative recombination operator has been investigated to nd one which best preserves the information gathered View publication stats by the parent solutions while being computationally efcient. The results achieved suggest that some version of the structural recombination operator is likely to produce the best results overall, although for random graphs it does not do quite as well as the best HNS approach. Overall, the results obtained indicate that the performance of a GA approach to graph partitioning can be considerably enhanced if account is taken of problemspeci c factors in designing and implementing operators. 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