Integrating Local Search into Genetic Algorithms - CiteSeerX

Integrating Local Search into Genetic Algorithms Colin Reeves School of Mathematical and Information Sciences Coventry University UK Email: [email protected] Christian Hohn Institut fur Grundlagen Elektrotechnik/Elektronik Technische Universitat Dresden, Germany

Abstract

Genetic algorithms (GAs) have proved to be a versatile and eective approach for solving combinatorial optimization problems. Nevertheless, there are many situations in which the simple GA does not perform particularly well, and various methods of hybridization have been proposed. These often involve incorporating other methods such as simulated annealing or local optimization as a post-processor to the basic GA strategy of selection and reproduction. In this paper, rather than keeping the local optimization separate from recombination, we intend to explore the possibilities of integrating it more directly into the standard genetic algorithm. The graph partitioning problem and the no-wait owshop sequencing problem will be used to illustrate the approach.

1 Introduction Genetic algorithms (GAs) as developed by Holland [1] have frequently been applied to combinatorial optimization problems, such as bin-packing [2, 3], machine-sequencing [4], vehicle routing [5], and of course the travelling salesman problem [6, 7, 8]. All of these problems have serious practical applications, but they typically require some fairly major adjustments to the `classical' GA paradigm in order to nd high-quality solutions. In what follows, we assume the reader has a basic understanding of the way in which GAs operate|if not, we refer them to Goldberg [9] or Reeves [10] for an introduction. A discussion of GAs in the particular context of combinatorial optimization can be found in [11]. The successful application of GAs requires a good adaptation to the problem to be solved. A promising approach is when existing methods such as constructive heuristics or local search methods can be incorporated into the algorithm. This idea has been used by Jog et al. [6] (amongst others) in the context of the TSP. Other examples of this type of approach are in [12, 13, 14]. An alternative strategy is to run the GA for a xed number of cycles and then apply neighbourhood search to all the nal population. However, in all these cases, the application of local optimization is kept distinct from the recombination operation. However, recent work [15, 16] which has started to explore GAs from a more traditional neighbourhood search perspective has suggested that local search methods can be integrated in a more fundamental way. It is assumed that readers will have at least a basic understanding of both GAs and neighbourhood search (NS). Further details can be found in [15], but in the next section we brie y discuss the relation between the two approaches. This work was undertaken when the second author was with the Control Theory and Appplications Centre, Coventry University, UK

1

Traditional genetic algorithms have three main characteristics by which they are commonly distinguished from classical and modern local search techniques.

Population-based selection: In point-based NS methods selection is restricted to a

choice between the parent and one or more ospring, but with GAs the presence of a population gives a spatial dimension to selection. Recombination: By recombining two parent solutions, ospring usually inherit those variables for which both parents share the same value. Thus recombination essentially results in a reduction of the size of the search space. Crossover: Traditionally crossover involves two (often binary) strings. Having de ned a set of crossover points, the intermediate bits are swapped. While selection and recombination represent general search techniques, crossover essentially stands for a special class of operators and thus could be replaced by an abstract type of operator. In the context of local search an operator ! de nes a neighbourhood N (x; ! ) S for all points x in the search space S . This general de nition holds for a 1-bit Hamming hill climber as well as for many more sophisticated heuristics. However, crossover as introduced by genetic algorithms seems to be an exception in so far as two points, say x and y , are involved in the operation such that N = N (x; y; ! ); x; y 2 S . Some implications of this are explored in [15], but it is obvious that de ning a neighbourhood by more than one point complicates the analysis. Furthermore, in this way the crossover operation is inevitably linked to recombination leading to a superposition of both eects. As a consequence the terms are often used synonymously. However, having 2 points eectively reduces the neighbourhood size by the total of all elements the two points have in common. This is an eect of recombination which occurs before any operators are applied. Following the ideas of Culberson [16] the crossover eect can then be implemented by a unary operator which always pairs a point x 2 S with its complementary point x in the reduced neighbourhood. This operator, which we term complementary crossover, thus ts into the general de nition above. As a consequence, the traditional crossover may be decomposed into neighbourhood reduction and complementary crossover. In traditional GAs the search space de ned by both parents is usually explored by evaluating only one or two randomly selected ospring. However, this is not a necessary part of a GA| some have suggested, for example, evaluating several randomly selected ospring and picking the best. In fact, one could take this to its logical conclusion and carry out a local optimization within the reduced neighbourhood. The types of problem where this approach is likely to be most eective are those where the objective function of the neighbouring points can be evaluated quickly and cheaply: if every point has to be evaluated from scratch, this is not likely to be a computationally ecient scheme.

3 No-wait Flowshop Problem We have tried this out on some combinatorial problems. Initially, the problem studied was the n=m=P; no ? wait=Cmax problem (i.e. there are n jobs, m machines and the objective is to minimize the makespan), which can be modelled as an asymmetric TSP. This has the advantage that the cost of a small change (here an exchange of two job positions) is easily calculated, so it meets the criterion described above. Nine sets of 10 problem instances were investigated, with either 20, 50 or 100 jobs, and 5, 10 or 20 machines. The neighbourhood search was carried out by a simple exchange operator as indicated above, and this was compared with the C1 and PMX crossovers|`traditional' GA crossover operators for sequence-based problems. The total amount of computation was xed at that used in [4]. The results were as shown in Table 3 (the 2

Problem group mean di stdev di

n=m

20/5 20/10 20/20 50/5 50/10 50/20 100/5 100/10 100/20

8.50 5.60 -79.90 9.0 -63.7 -94.7 -45.9 -107.5 -284.6

19.40 32.07 85.00 44.0 59.3 132.8 69.1 145.5 154.5

Table 1: Mean and standard deviation of dierences in makespan (C1-NS), averaged over 10 instances in each group results of C1 and PMX were so similar that only C1 is shown). The gures shown are the mean and standard deviation of the dierence in makespan values; a positive dierence means C1 gave better results, a negative one that the embedded NS operator was superior. It can be seen that for the smallest problem instances (20/5, 20/10 and 50/5) there was little dierence between the methods. However, for the larger cases, the average makespan obtained from the embedded NS was (statistically) signi cantly less than those obtained by C1 or PMX. What is not shown here is the relative amounts of computer time consumed; each technique was allotted what was nominally the same xed amount of time (a function of n and m|see [4] for further details). However, because the test for exceeding the stopping time was only carried out after each new string was generated, which is more frequent for the conventional GA with C1 or PMX than for the embedded NS approach, the results recorded for embedded NS were actually those on the iteration before the stopping criterion was reached. Thus, on average the time taken to achieve the results of Table 3 were 5-10% less for the embedded NS approach, so that by integrating NS into the GA we have not only found higher-quality solutions, but also achieved them in less computer time.

4 Graph partitioning What the above results do not show is how much of the performance is due to the operators used, and how much to the eect of recombination. This is not easy to disentangle in the context of the owshop problem, and for a more fundamental study, we looked at the rather simpler problem of bipartitioning a graph which, as it has a natural binary encoding, is more amenable to analysis. A fuller discussion of this work is contained in [23]. The uniform bipartitioning of a graph is a special case (i.e. k = 2) of the k-way graph partitioning problem which is as follows: given a graph G = (V; E ) with a cost function c : E V V 7! IR and a size S (u) attached to each node, partition the nodes of G into k subsets of equal size such that the total cost of the edges cut is minimized. The uniform kway partitioning problem is a constrained optimization problem, i.e. the search space usually contains an nonempty set of illegal solutions. This can pose severe problems for GAs (see [11] for a dicussion of these), but happily in this case, the problem can be quite simply transformed into a free optimization problem [17, 18]. For the purposes of this investigation we de ne the following abstract GA: 1. Initialize a population of size N . 3

stopping criterion is met. 3. Choose 2N individuals from the population according to tness and pair them (randomly) so that each pair de nes a reduced neighbourhood. This is passed to the next generation together with the complete bitstring and objective value from one parent. 4. All the ospring de ne the population for the next generation. 5. If not TERMINATED go to 2, otherwise stop. The scheme is not restricted to a particular hill-climbing technique and thus provides a high degree of exibility. Furthermore, for a given application several local hill climbing techniques may be evaluated individually before being extended by recombination and selection. In the rest of the paper such an approach is illustrated for the graph bipartitioning problem. First we describe the operators

4.1 The LPK heuristic

One of the traditional operators for solving the k-way graph partitioning problem is the heuristic by Lee, Park and Kim [17] which is based on the Kernigham/Lin heuristic [19]. Given an initial partition, = (A; B ), consisting of the blocks A and B , a pass starts with moving the vertex having the highest gain to its complementary block. Once moved, the vertex becomes locked for the remainder of the pass and the gains of the remaining vertices are updated. The pass proceeds with the next best move and terminates after all vertices have been moved, i.e. when the initial partition is reached again. Then the best partition encountered during the pass is taken as initial partition for the next pass. The algorithm stops when no further improvement can be achieved, i.e at a local optimum. The LPK-heuristic uses a steepest ascent strategy where the neighbourhood searched forms a closed path. For problems involving l vertices the path is of length l starting and ending at the initial point and always follows the maximum gain. Initializing and maintaining the gains require O(l2) time per pass.

4.2 Complementary Crossover

Using group-number encoding a solution for the bipartitioning problem of l vertices requires a binary string of length l. Usually when crossover is applied to partitioning problems two common pitfalls arise The operator is not closed with respect to the search space. Thus illegal solutions may result. The coding is redundant, multiplying the number of local and global optima. Because of the graph transformation the solution space is extended to the whole search space, so that no illegal solutions occur. Furthermore, since in this study the investigations are restricted to the bipartitioning problem any in uence of redundancy can be eliminated fairly easily.

4.2.1 1-Point Complementary Crossover (1CX)

The implementation of the 1CX is based on the following de nition:

y = 1CX (x; i); x; y 2 S = f0; 1gl; 1 i l yk =

xk if k i xk otherwise 4

inversion of the complete bit string. 1CX produces a neighbourhood structure similar to the LPK-heuristic. It forms a closed path of length l starting and ending with the initial partition. Unlike the LPK-heuristic the path is determined by the order of the vertices. A steepest ascent strategy will follow all the path where each step requires the update of the associated gains. Thus the whole pass requires O(l2) time. Obviously, although the crossover does not process additional information the time required is the same magnitude as of that of LPK heuristic, clearly an advantage for the more sophisticated heuristic.

4.2.2 2-Point Complementary Crossover (2CX)

The 2CX is de ned as follows:

y = 2CX (x; i; j); x; y 2 S = f0; 1gl; 1 i < j l + 1 yk =

xk if i k < j xk otherwise

De ned in this fashion 2CX resembles the 1-bit Hamming hill climber (if j = i + 1) as well as 1CX (if? j = l + 1). Since there are l + 1 feasible sides for a cross point the neighbourhood 2 contains l+1 2 points. Thus the size of the neighbourhood is l(l + 1)=2, i.e bounded by O(l ). Consequently, for larger problems an exhaustive search is out of question. Instead a next ascent search is performed with an upper limit of tmax = l trials per pass. Since the complete evaluation of the objective value involves l(l ? 1)=2 operation the time bound is given by O(l2) for a single operation and O(l3) for a complete pass.

4.3 1-bit Hamming Hill Climber (HC)

The 1-bit Hamming hill climber can be de ned as follows:

y = HC (x; i); x; y 2 S = f0; 1gl; 1 i l if j = i yj = xxj otherwise j

Obviously, applying a 1-bit Hamming hill climber combined with a steepest ascent search strategy to the graph bipartitioning problem corresponds to a strategy that always moves the vertex of the highest gain. Since only a single vertex is moved it is convenient to maintain the gains and update the solution after each move which requires O(l) time. Since spotting the best vertex can be done in linear time too the computational complexity of a steepest ascent Hamming hill climber is bound by O(l). The search terminates after a local optimum has been reached.

4.4 Mutation (MUT)

Mutation traditionally ips a single bit with a given probability p. Thus

P (xi ! xi) = p; 1 i l The eect of mutation corresponds to moving several vertices at a time. Thus the computational complexity for single operation will range between O(l) for a low mutation rate and O(l2) for a high mutation rate. If only a single vertex is moved the move can be suspended if the associated gain is nonpositive. In the following experiments the mutation rate has been set to p = 1=l. The search terminates after the maximum number of evaluations per pass, tmax = l, has been exceeded. Thus the time required for one pass is bounded by O(l3). 5

Op N'hood Time Strategy Termination LPK l O(l2) SA local optimum 2 1CX l O(l ) SA local optimum 2CX l2 O(l3) NA < l trials/pass HC l O(l) SA local optimum MUT 2l O(l3) NA < l trials/pass Table 2: Certain characteristic of the operators considered. SA and NA stand for steepest and next ascent.

5 Evaluation Of The Operators In the rst step 1-point and 2-point complementary crossover, the 1-bit Hamming hill climber and mutation are compared against the LPK-heuristic. Therefore, they are implemented in a simple local hill climbing template that uses both next ascent and steepest ascent respectively and terminates when a local optimum has been reached or the maximum number of trials per pass has been exceeded. For evaluation a set of random graphs has been generated. All graphs have uniform weights at the edges and at the vertices. The parameters are summarized in Table 3. Acronym R.50.5 R.50.10 R.100.5 R.100.10 Vertices 50 50 100 100 Mean Degree 5 10 5 10 Edges 124 252 244 472 Max degree 10 18 11 16 Min degree 2 5 0 3 Table 3: Parameter of the graphs The results of 50 runs are displayed in Tables 4 - 7. Besides the standard measures (best result, worst result, average and standard deviation) the average number of moves per pass has been measured as well as the time required. While the rst gives a rough idea of the distribution of local optima the latter highlights the computational complexity. The results can be summarized as follows

The LPK-heuristic performs best in terms of accuracy and requires a reasonable amount

of time. 1-bit Hamming hill climbing is the fastest operator and achieves a similar performance as 1CX and 2CX. Mutation shows the second best performance after the LPK. However, it suers from high computational costs mainly caused by evaluation procedure. 2CX requires the most computational time which however is not justi ed by the performance achieved. To sum up, operators have to be judged according to the search space they induced (number of local optima, speed of convergence) and the degree up to which they support an ecient evaluation of the objective function. Concerning graph bipartitioning the LPK-heuristic exhibits an ecient tradeo between both measures. 6

Operator LPK 1CX 2CX MUT HC Operator LPK 1CX 2CX MUT HC Operator LPK 1CX 2CX MUT HC Operator LPK 1CX 2CX MUT HC

max 572 532 530 552 544

min average deviation min cut average moves time in s 562 569 3 26 2.280 2.0 488 511 8 46 1.720 1.0 498 515 8 47 6.340 8.0 498 531 13 36 13.060 7.0 482 515 13 40 2.920 0.0

max 1142 1058 1060 1116 1082

min 1124 972 1002 1040 972

Table 4: R.50.5 average deviation min cut average moves time in s 1137 5 51 2.500 7.0 1019 14 93 1.820 2.0 1029 12 92 7.540 55.0 1081 16 64 32.460 46.0 1030 26 81 4.500 1.0

min 572 478 498 490 478


min 1176 972 1012 1064 966


max 600 540 542 580 558 max 1208 1074 1080 1164 1128

Table 7: R.100.10

7

Recombining two binary strings of length l results in a search space Sd = f0; 1gd; 0 d l where d is the Hamming distance between both strings. Since the time required for performing an operation on a bit string is usually a (polynomial) function of l a reduced search space will speed up the operator and thus save computational time. This eect has been described elsewhere [15, 20]. However, for some operators recombination exhibits an even more interesting phenomenon which is now considered in more detail. Consider two bit strings, say x; y , both of length l. Applying a 1-bit Hamming hill climber the neighbourhood of both points contains l neighbors. Assuming that the Hamming distance between x and y is 0 < d < l, recombination of x and y generates a neighbourhood N (x; y; HC ) containing d points. Obviously, N (x; y; HC ) N (x; HC ) and N (x; y; HC) N (y; HC ). In contrast, as an easy example reveals (Figure 1), the last relation does not necessarily apply for 1-point complementary crossover. Here recombination is able to introduce new points into the reduced neighbourhood and thus it may help out of a local optimum. The same observation may be made for the 2-point complementary crossover or the LPK-heuristic. 1000

0100

6

1010

0101

6

-

0101

0110

?@ ? @@ ? @@R ? ? 0000 0100

?

0010

Figure 1: Full and reduced neighbourhood of some bit string when 1CX is applied. In the right picture the second bit from the right is xed due to recombination. Obviously, the reduced neighbourhood contains two points which are not included in the full neighbourhood To investigate both eects of recombination a population of 50 hill climbers was implemented in the scheme of the abstract GA. Table 8-11 display the results for a typical run of ve generations. Convergence and diversity of the population can be measured by the size of the seach space averaged over the population and the deviation in the cost values. Furthermore, the average number of moves is given which is culmunative over the generation. Thus it can be considered as measure for how ecient the local optima encountered at the end of one generation are circumvented in the next generation. For the given set of test problems the following observations may be made. Owing to neighbourhood size reduction all the operators are speeded up. Thus ve generations require less than double the time of the rst generation. As predicted, recombination does not have any eect on the 1-bit Hamming hill climber. The increasing average performance is merely due to selection. Clearly, for 1CX and LPK, recombination leads the search away from the local optima found in the rst generation. The increasing average number of moves indicates the 8

Operator max min average deviation min cut average moves average size time in s LPK 572 562 569 3 26 2.280 50.00 1.0 572 566 571 1 26 2.660 12.96 2.0 572 572 572 0 26 2.860 5.72 2.0 572 572 572 0 26 2.860 1.92 3.0 572 572 572 0 26 2.860 2.08 3.0 1CX 532 488 511 8 46 1.720 50.00 0.0 536 510 520 6 44 2.460 21.60 0.0 538 520 529 5 43 3.160 21.44 1.0 544 526 535 4 40 3.620 15.36 1.0 550 536 539 3 37 4.100 11.12 1.0 2CX 530 498 515 8 47 6.340 50.00 9.0 538 512 524 6 43 7.540 22.12 12.0 544 518 530 6 40 8.680 19.84 14.0 548 526 535 5 38 9.500 20.44 16.0 548 530 538 5 38 10.040 18.12 18.0 MUT 552 498 531 13 36 13.060 50.00 6.0 552 514 539 10 36 14.380 20.04 7.0 556 532 547 6 34 15.340 20.40 8.0 558 534 551 5 33 15.860 18.72 8.0 558 544 553 3 33 16.540 17.64 9.0 HC 544 482 515 13 40 2.920 50.00 1.0 544 494 522 12 40 2.920 21.32 1.0 544 512 527 9 40 2.920 21.28 1.0 544 518 533 8 40 2.920 19.92 1.0 544 522 537 7 40 2.920 18.28 2.0 Table 8: Typical run of the GA when applied to R.50.5

9


10

For mutation and 2CX the local search stops after l trials have been failed. However, in the

next generation the selected points will be evaluated again which essentially exceeds the stopping criterion and accounts for the progress of one move per generation. Obviously, this eect is an artefact of the selection scheme and cannot be attributed to recombination. However, both operators bene t from the speed up.

Operator max min average deviation min cut average moves average size time in s LPK 600 572 597 5 76 2.220 50.00 1.0 600 598 600 0 76 2.460 8.64 1.0 600 600 600 0 76 2.460 3.20 1.0 600 600 600 0 76 2.460 2.72 2.0 600 600 600 0 76 2.460 3.32 2.0 1CX 540 478 512 13 106 1.980 50.00 0.0 546 508 526 9 103 2.980 21.28 0.0 556 522 538 7 98 3.640 18.44 1.0 556 534 546 6 98 4.300 15.40 1.0 566 542 554 5 93 5.040 14.32 1.0 2CX 542 498 518 11 105 6.020 50.00 5.0 548 516 532 9 102 7.340 22.08 7.0 572 520 540 10 90 8.120 20.28 9.0 588 528 553 12 82 9.220 18.24 10.0 586 540 562 9 83 10.020 16.76 11.0 MUT 580 490 539 18 86 12.360 50.00 5.0 580 518 550 15 86 13.300 20.68 6.0 592 540 561 13 80 14.180 20.04 6.0 592 548 569 10 80 14.900 17.12 7.0 594 552 576 10 79 15.300 16.12 7.0 HC 558 478 519 17 97 2.920 50.00 0.0 558 494 528 15 97 2.920 21.80 0.0 558 510 536 13 97 2.920 20.40 1.0 558 520 544 10 97 2.920 17.12 1.0 558 522 548 9 97 2.920 18.64 1.0 Table 10: Typical run of the GA when applied to R.50.10 However, since the LPK-heuristic found the best values always within the rst generation, it is not clear whether recombination is more eective than simply trying dierent initial points. To obtain a preliminary answer on this question a hill climber was compared to the GA when applied to a more ambitious random graph with 500 vertices and 1224 edges corresponding to a mean degree of 5. In 30 runs the CPU time has been xed for both algorithms and the best values found per run are compared. The results are shown in Table 12. It appears that LPK coupled with recombination performs better than LPK alone. In particular, the worst value found in 30 runs with recombination exceeds the average performance of LPK on its own. The last column display the eective population size, i.e. 80 restarts require the same amount of time as 50 individuals evolved for 6 generations.

11


Operator max min average deviation popsize LPK 5778 5754 5762 6 80 GA + LPK 5788 5764 5774 5 50 Table 12: The LPK heuristic applied alone and coupled with recombination.

12

We have shown elsewhere [15] that GAs are closer to NS than is sometimes believed, in that a GA works by rst nding a reduced neighbourhood and then searching it in a random fashion. Adding local optimization as an extra `operator' (as suggested by previous work) has been found to improve the GA's performance, albeit at a cost in terms of the computational requirements. However, integrating local optimization more rmly into the GA can help to maintain the bene ts of recombination, and if implemented eciently can speed up the time to reach a good solution. In this paper an abstract genetic algorithn has been presented and tested on a special case of graph partitioning. The proposed GA couples a local hill climbing techniques with recombination and selection in a general way, so that any local search technique may be accommodated within the proposed scheme. In this paper the investigation has focused on several operators and the eects of recombination. Initially the operators have been studied individually when implemented in the framework of a simple local hill climber. Such preliminary investigations may form a rst step in the design of more sophisticated algorithms. Secondly, we have separated the eect of the operator used from that of the recombinative eect of a GA, and shown that, at least for larger problem instances, recombination has been proved to save computational time as well as oering a mechanism to circumvent local optima for certain operators. Both eects provide a reasonable explanation why GAs can improve traditional hill climbing techniques. Furthermore, both characteristics may be used to guide the design of application-speci c genetic algorithms. However, the eciency of recombination certainly depends on the population size as well as the selection scheme applied. Both problems deserve further investigation. In future work the performance of the GA will be investigated when more sophisticated hill climbers are incorporated.

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