An Efficient Genetic Algorithm Based Approach for the Minimum Graph Bisection Problem

Transcription

1 118 An Effcent Genetc Algorthm Based Approach for the Mnmum Graph Bsecton Problem Zh-Qang Chen, Rong-Long WAG and Kozo OKAZAKI Faculty of Engneerng, Unversty of Fuku, Bunkyo 3-9-1,Fuku-sh, Japan Summary The goal of the mnmum graph bsecton problem s to dvde the vertces set V of the gven undrected graph G = ( V, E) nto two equal-sze subsets V1 and V2 such that the number of edges connectng vertces n V 1 to vertces n V 2 s mnmzed In ths paper, we propose an effcent method based on genetc algorthm wth condtonal genetc operators In the proposed approach, the genetc operators are performed condtonally nstead of probably to make sure the algorthm has good global search and local search ablty; furthermore, a selecton method combnng roulette selecton wth tournament selecton s presented to renforce the local search ablty The proposed approach s tested on a large number of nstances and s compared wth other optmzaton methods The expermental results show that the proposed approach s superor to ts compettors Key words: Mnmum graph bsecton problem, Genetc algorthm, combnatoral optmzaton problems, P-complete problem 1 Introducton The graph bsecton problem s an mportant problem n prnted crcut board layout and communcaton networks [1] [2] It was proved to be a P-complete problem [3] It s well known that there are no tractable algorthms to solve P-complete problem, whch motvates to fnd better algorthms that yeld better approxmate solutons Several approxmate algorthms have been proposed [4] [5] [6] for the mnmum graph bsecton problem The benchmark algorthm for graph bsecton problem s due to Kernghan and Ln [7] By combnng Kernghan-Ln algorthm [7] wth the parallel hll clmbng [8], and the seed-growth algorthm [9], Marks et al proposed a new heurstc algorthm called PHC/SG+KL [10] We have proposed an artfcal neural network [11] based algorthm for the problem [12] For solvng such combnatoral optmal problems, the Genetc algorthm (GA) [13] also consttutes an mportant avenue GA s an adaptve search technque based on the prncples and mechansms of natural selecton from natural evoluton In GA, there s no rule of thumb to desgn the GA operators and select GA parameters [14] Instead, tral-and-error has to be appled for every problem Besdes, due to the poor local search ablty [15], the soluton qualty of GA s not very good Some modfed versons of genetc algorthm [16] [17] [18] have been proposed to mprove the GA s performance on combnatoral optmzaton problems In ths paper, we propose an effcent approach based on our prevously proposed genetc algorthm wth condtonal genetc operators [18] for the mnmum graph bsecton problem In the proposed approach, the genetc operators are performed condtonally nstead of probably and roulette selecton and tournament selecton s combned to renforce the search ablty of the algorthm We test proposed method by smulatng a large number of randomly generated graphs We also analyze the nfluence of the man parameters We compare the smulaton results wth other knds of optmzaton method ncludng the PHC/SG+KL [10], the neural network based method [12] The expermental results show that the proposed genetc algorthm approach works remarkably well and s superor to ts compettors to solve the mnmum graph bsecton problem 2 Problem Formulaton Gven an undrected graph G = ( V, E), where V s the set of vertces and E s the set of edges The edge from vertex to vertex j s represented by e j The mnmum graph bsecton problem s to fnd a partton of V nto two nonempty, dsjon sets V 1 and V2, such that V1 I V 2 = φ, V 1 U V2 = V, V 1 = V2, and the number of the edges connectng vertces n V 1 to vertces n V2 (The sze of cut set, notated cut ( V1, V 2 )) s mnmzed For solvng such problem, a typcal genetc algorthm requres two thngs to be defned frstly: one s a genetc representaton of the soluton doman; another one s a ftness functon to evaluate the soluton doman For a gven graph wth vertces and M edges, as a graph bsecton problem, ts Manuscrpt receved June 5, 2008 Manuscrpt revsed June 20, 2008

2 119 vertex set can be represented usng vector X = ( x1, x2,, x ), where x expresses # vertex s parttoned nto ( = 1, ) the subset V1 or V2 Thus, x has only two possble values and a soluton of the graph bsecton problem can be represented as a bnary strng A value of 1 for x mples that # vertex s parttoned nto subset V1, and a value of 0 denotes that t s parttoned nto subset V 2 Usng the above representaton, the graph bsecton problem can be mathematcally transformed nto the followng optmzaton problem: Mnmze x x j ej (1) j 2 Subject ( ) = 0 2 x (2) = 1 Where s the number of the vertexes, e j s 1 f there s an edge from vertex # to vertex # j otherwse 0 Then the ftness functon can be expressed as followng: F( x) = Edgeum A x x j ej j (3) 2 + B( ) 2 x = 1 where Edgeum s the number of edge of the gven graph, A and B s parameters v 3 Proposed genetc algorthm based approach 31 Genetc algorthm wth condtonal genetc operators In our genetc algorthm approach, the crossover and mutaton behavor s based on our prevously proposed genetc algorthm wth condtonal genetc operators [18] The detal s as depcted n Fg1, where p denotes the populaton sze and c denotes the current number of chldren generated Frst c s set to zero, then ( c - p )/2 pars of parent chromosomes are randomly selected, and the dfference-degree d for every par of parent chromosomes are calculated The dfference-degree d s defned as follows: d d = (4) g Where s the sze of chromosome, and s the g number of the dfferent genes between two parent chromosomes As an example, we consder the followng two parent chromosomes for a graph wth 6 vertces: d Fg1 Outlne of crossover and mutaton behavor of the proposed genetc algorthm v x v = (1,0,0,1,1,1) y = (0,1,01,0,1) It s evdent that d = 3 and thus the dfference-degree ( d ) of the parent par s 05 To control the crossover and mutaton behavor, a new parameter called settng dfference-degree D s s ntroduced As shown n Fg1, for every parent par, f the dfference degree ( d for # parent par) s larger than the settng dfference-degree D s, then the crossover s appled wth 100% probablty on the parent par to generate two chldren After crossover, we calculate the total number of chldren generated, and f the total number of chldren ( c ) s smaller than the populaton sze p, the mutaton s performed wth 100% probablty on chromosomes of parent pars whose dfference-degree are smaller than the settng dfference- degree D s The mutaton operators used n ths work ndrectly produce chldren to next generaton (5)

3 120 The above procedure s performed n a loop untl the total number of chldren s equal to the populaton sze p The settng dfference-degree D s s decreased n every generaton and s defned as follows: Ds ( t + 1) = μds ( t) (6) Where t denotes the t# generaton and μ s a constant varaton called coolng rato, 0< μ <1 From the Eq6, t can be seen that D s decreases slowly to a near zero wth the evoluton of generaton From the above crossover and mutaton behavor, we can know that n the early stage of optmal process D s has a relatvely large value, thus many parent pars have to undergo dverse mutaton to produce the enough offsprng by crossover As a result, t makes sure the dversty of the soluton and the effcent global search ablty of the algorthm On the other hand, n the late stage of optmal process, D s become small and even equal approxmately to zero, only few parent pars perform mutaton and most of parent pars produce drectly offsprng by crossover As a result, t makes sure the algorthm has good local search ablty n the late stage of optmal process Thus, the crossover and mutaton behavor s controlled by settng dfference-degree D s dynamcally nstead of statcally so that the ablty of local and global search of the GA s balanced effcently 32 Selecton method For further renforcng the local search ablty of the proposed approach, we desgn a selecton method whch nherts the advantage of roulette selecton and tournament selecton The proposed selecton has two phases The frst phase s to create the canddate populaton whose ndvdual s stochastcally selected from the current generaton usng roulette selecton, where the current generaton s expressed as P cur = { a1, a2, an}, and a ( = 1,2,, n) expresses the # ndvdual of the current generaton, n s the sze of populaton The canddate populaton selected n the frst phase s expressed as P And express the # cd = { b1, b2,, b } b n ( = 1,2,, n) ndvdual (chromosome) of the canddate populaton The second phase decdes f the ndvdual (n the canddate populaton) wth low ftness survves n the next generaton or not The detal of the second phase s as follows: (1) Randomly par the ndvdual between Pcur and Pcd We use < a, b > to express the # par (2) Compare the ftness of ndvdual for each par We use f a ) to expresses the ftness of a If f b ) > f ( a ) then ( b survve n the next generaton Else then ( a replaces b and survves n the next generaton by probablty P The probablty P s defned by the followng equaton: t P = 1 μ (7) Where t denotes the t# generaton and μ s the same one as that n Eq6 In other words, the ndvdual (n the current generaton) wth better than ts compettor survve n the next generaton by probablty P The proposed selecton method nherts the advantage of roulette selecton and tournament selecton In addton, t does not requre global reorderng Selecton pressure can be easly adjusted by changng probablty P Wth the evoluton of populaton, P s gradually ncreasng and selecton pressure also s gradually ncreasng wth P In the early stage, because P has rather smaller value, the dversty of the populaton sn t destroyed and as evoluton of generaton the local search ablty of the proposed approach s renforced 33 Crossover and mutaton method In GA, besdes the behavor of crossover and mutaton, the methods of crossover and mutaton are also very mportant for effcent search In ths secton, we desgn the crossover and mutaton methods Our fundamental rule of desgnng crossover and mutaton methods s that chldren generated are vald solutons One pont-crossover or ts extenson (multpont crossover) [19] s well-used and effcent methods for the problem wth bt strng encodngs because of ts smplfcaton However, for the graph bsecton problem, the nvald solutons are also produced f performng the tradtonal one pont-crossover or multpont crossover We now present a modfed two-pont crossover whch can always produce vald soluton The detal s as follows: (1)Randomly select two crossover pont CP1 and CP2 ( CP 1 < CP2 ) (2)Use S 1 to express the number of the gene that value s 1 n parent 1, and S 2 n parent 2 Adjust the crossover pont CP1 and CP1 accordng to the followng procedure: Whle ( S1 S 2 ) (1) CP1 = CP1 1 or CP2 = CP2 +1 (2) Calculate S1 and S 2 End (3) Interchange the segment between CP1 and CP1 to generate two vald chldren Besdes the crossover method, we also modfy the smplenvert mutaton [13] method to guarantee the feasblty and vald of the soluton generated by mutaton operator as follow: (1)Randomly select two mutaton pont and Mp 1

4 121 Coolng raton μ Table 1: Smulaton results on the graph wth 150 vertexes, 558 edges usng dfferent parameters p = 50 p =80 p =100 p =150 best Average best Average Best Average Best Average Mp2 for a chromosome (2) Interchange the gene of # Mp1 and # Mp2 It s evdent that the above crossover and mutaton method can always produce vald chld Thus, the ftness of a soluton can be evaluated accordng to the followng equaton smplfed from Eq3 F( x) = Edgeum x x e (8) 4 Algorthm j The followng procedure descrbes the proposed approach for the bsecton problem of -vertex graph: 1: Set the parameter p, Ds and μ, and generate p vald ntal chromosomes (solutons) 2: Produce p chld chromosomes accordng to followng sub-procedure: (1) Set c =0 (2) Randomly select ( p - c )/2 pars of parent chromosomes (3) Calculate the dfference-degree d of every parent par accordng to Eq4 (4) Crossover s appled on parent pars whose dfference-degree d s larger than Ds (5) Calculate the total chld chromosomes c, f = p termnate ths sub-procedure c j j (6) Mutaton s appled on parent pars whose dfference-degree d s smaller than Ds (7) Go to step (2) 3 Calculate the ftness of each chromosome accordng to Eq8 4: Decrease the settng dfference-degree D s usng Eq6 5: Termnate ths procedure f D s <001 6: Create the chromosomes of next generaton usng our mproved selecton explaned n the last secton 7: Go to step 2 5 Smulaton results and analyss The proposed approach has three man parameters: populaton sze p, ntal value of Ds and ts coolng rato μ To analyze the nfluence of the man parameters, we smulated an undrected graph wth 150 vertces 558 edges usng dfferent parameters To measure the performance of the proposed approach, a total 18 graphs was smulated As mentoned, the coolng rato μ s used to control the decreased speed of D s n Eq6 and the ncreased speed of P n Eq7 In other word, t s used to balance the ablty of global search and local search of the algorthm If μ s too

5 122 Table 2 Smulaton results on the graph wth 150 vertexes, 558 edges usng dfferent populaton sze Populaton Average sze Best value Error% p value than 0999, the optmzaton process s to converge quckly and no good large, Ds decreases too slowly and P ncreases too slowly, the algorthm would spend too long computaton tme If μ s too small, the algorthm would converge quckly and lose the ablty of global search for global optmal soluton To study the reasonable range of the value of the coolng rato μ, we smulated an undrected graph wth 150 vertexes and 558 edges The experments are classfed nto four groups by populaton sze p =50, p =80, p =100, p =150 The value of the coolng rato μ s 0999 to The ntal value of D s s set to tmes smulatons wth dfferent ntal chromosomes have been performed for every parameters group The best soluton and average soluton n 100 smulatons are recorded The smulaton results are descrbed n the Table 1 As table 1 shown, when the coolng rato μ s smaller Fg2 Generatng procedure of soluton usng proposed approach soluton was found When the coolng rato μ s or larger than 09999, the algorthm balances effcently the ablty of global search and local search and can fnd good solutons We have also performed a large number of smulaton to study the ntal value of D s, we found that the reasonable ntal value of D s s from 01 to 06 In the later smulatons, the coolng rato μ and the ntal value of D s were fxed at and 04, respectvely Populaton sze p s also an mportant parameter whch affects the performance of genetc algorthm To see the nfluence of p n the proposed approach, we performed smulatons usng dfferent populaton sze ote that the ntal value of Ds and the coolng rato μ are 04 and respectvely The experment results are shown n the table 2 The Error% column corresponds to the average percentage error of solutons obtaned wth dfferent runs as defned n the followng equaton ( Soluton( ) Best) Error% = (9) 1 100% Best Where Soluton() s the soluton found n the # run, Best s the best soluton of the problem As the table 2 shown, when the populaton sze p s larger than 60, good soluton can be found In the later smulatons, the populaton sze p was fxed at 100 To verfy the ablty of global search and local converge of the proposed approach, we present the evoluton procedure of the soluton on the graph wth 150 vertces 558 edges

6 123 usng the proposed approach Fg 2 shows the varaton n the best, worst and average solutons durng the evoluton procedure of the soluton From Fg 2, we can know the Graph Vertex Probablty Edges Table 3: The comparsons of smulaton results produced by dfferent algorthms SGA PHC/SG+KL[10] Wang etc[12] Proposed algorthm characterstc of the proposed approach In the early stage, there s large dfference between the best and worst solutons, whch means that n ths stage the proposed approach has good dversfcaton In the fnal stage, the dfference between best and worst solutons become very small, whch means that the proposed approach has good local search ablty n ths stage In order to wdely verfy the proposed approach, we have also tested the algorthm wth a large number of randomly generated graphs [20] defned n terms of two parameters, n and p The parameter n specfes the number of the vertces n the graph; the parameter p, 0<p<1, specfes the probablty that any gven par of vertces consttutes an edge To evaluate our results, we compared our results wth the other knds of optmzaton method ncludng the PHC/SG+KL proposed by JMarkset al[10], the mproved neural network by Wang et al [12], and the smple genetc algorthm(sga, the crossover rato s 07, the chromosome mutaton rato s 004, the crossover and mutaton method s the same as the proposed ones n the paper) Informaton on the test graphs as well as all results s shown n Table 3 From Table 3 we can know that the proposed approach has more excellent ablty than other exstng algorthm for solvng the mnmum graph bsecton problem 6 Conclusons We have proposed an effcent GA based approach for the graph bsecton problem and showed ts effectveness by smulaton experments In the proposed method, the genetc operators are performed condtonally nstead of probably to renforce the ntensfcaton and dversfcaton n dfferent stages We have analyzed the approach by performng a large number of smulatons The smulaton results showed that the proposed approach has good dversfcaton and local search ablty We have also compared the proposed approach wth the exstng algorthms and found that the proposed approach works remarkable well and s superor to ts compettors to solve the mnm-um graph bsecton problem

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