Application of Genetic Algorithms in Graph Theory and Optimization. Qiaoyan Yang, Qinghong Zeng

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1 3rd Internatonal Conference on Materals Engneerng, Manufacturng Technology and Control (ICMEMTC 206) Alcaton of Genetc Algorthms n Grah Theory and Otmzaton Qaoyan Yang, Qnghong Zeng College of Mathematcs, Baoshan Unversty, Baoshan Yunnan, , Chna Keywords: Grah theory lannng model, Otmzaton, Genetc algorthm, Peak coverage roblems, Codng, Alcaton Abstract. There wll be a lot of NP- comlete roblems n grah theory and otmzaton rocess, as the most mortant roblem n scentfc engneerng comutng, now t s generally used genetc algorthm to solve. Therefore, ths artcle wll manly study the basc theory of genetc algorthms and grah theory, and ut forward the otmzaton algorthm n grah theory vertex cover roblem n the secfc alcaton. Introducton Genetc Algorthm (GA) s a knd of bonc otmzaton algorthm based on macro-scentfc ersectve, whch reles on the rncle of evolutonary changes n nature to selecton, crossover, mutaton and genetc smulated oerator, and by mathematcal calculatons to acheve grou oeratng globally adatve otmzaton, and access to the most otmal soluton. In modern scence, genetc algorthm n grah theory and otmzaton rocess s most common. Genetc algorthm and grah theory Genetc algorthm Genetc algorthm based on two factors: the robablty of natural selecton and genetc search. They calculate the target oulaton through the mlementaton of genetc manulaton to acheve the nternal restructurng of ndvdual grous and teratve, and therefore covers the followng four man concets n the genetc algorthm. Chromosome Chromosome s also called ndvdual, t comes from bology concets. In the genetc algorthm, chromosomes are usng "strng" to reresent, t reresents a vector, n a bologcal seces "strng" and the vector can be consdered equvalent. For examle, the vector X = x, x2... x. These x, x2... x and the lke are known to be a strng of basc unt vector X, also known as a gene (gene), and ndcates the total number of genes, also called the length of the strng, so that the entre chromosome s also known as gene strng. Poulaton Poulaton reresents a collecton of ndvduals. When usng genetc algorthms to solve any roblems, we should frst randomly select a number of ndvduals to form a comlete whole. But for chromosome geometry, the grou can only be regarded as a collecton of ndvdual whole chromosomes subset. Selecton oerator There wll be a hgh ftness ndvduals n the oulaton, they wll consttute a new Poulaton, select these grous must rely selecton oerator. It s assumed that f every ndvdual robablty of beng selected as (X), t deends on the ndvdual factor s X on the envronment nclusve ftness f (X) = f, therefore based on the selected oerator to select the most commonly used method s to ftness rato method, also known as the roulette wheel selecton, t exressed the followng formula: S = ( x ) = 206. The authors - Publshed by Atlants Press 24 f f + f f n

2 Crossover oerator Crossover can act smultaneously on two gene strngs, t s characterzed by a artcular gene strng nterchangeable arts, resultng n a new entty, whle the cross secton of the exchange s comletely random selecton, so the crossover of genes strngs resence robablty swa mechansm. For examle, there are two genes strng X = xx2... x, Y = yy2... y, use them () that f meet the (l n) ths condton, you can acheve mutual exchange oston robablty. Grah theory and related lannng model The man urose s to lan grah theory model, t s caable of ractcal roblems descrbed for the knowledge of grah theory to answer wth scorng, s lannng a model formula t uses. In real lfe, grah theory lannng model s more common, such as factory roducton work arrangements, schedules and so on. These ssues also have a common feature, that s to be scored wth the grah theory roblems, are often those who want to otmze NP-C roblem solvng. Therefore, lannng models based on grah theory to solve the roblem of otmzng the key ont s to set a reasonable and effectve roblem solvng aroach, ts urose s to reduce the sze of the sace to solve otmzaton to mrove local condtons for the global convergence seed and search accuracy, to enhance work effcency. It can be seen based on the theory and dscuss related ssues n the roblem of genetc algorthm to grah theory rogrammng model s the effectveness of havng a reasonable; t can realze the majorty of roblem solvng and otmzaton. Genetc Programmng Model for Solvng grah theory-based aroach Usng genetc algorthm roblems n grah theory and otmzaton solver s dvded nto fve stes Ste one: frst of all to the rogrammng model to otmze the form of grah theory to solve encode comled, that s, ndvdual codng genetc algorthm, whch s to enable the otmal soluton and the ndvdual would form a more ntutve transformng relatonsh. Ste Two: After the codng constrants to consder the full range of the model, and the constrants of the desgn and constructon of the grou's total generaton algorthm. Ste three: the algorthm constrants and objectve functon combned wth each other n order to adjust and desgn conforms to the corresondng ftness functon roblems, so as to ensure the rad convergence of the genetc algorthm. Ste Four: Select the ratonal and effectve genetc arameters and genetc oerators to artcate n the lannng grah theory model otmzaton, genetc algorthm to ensure the smooth unfoldng whle also achevng stablty rogram executon. Ste fve: after the end of the rogram can make the arorate nterretaton and verfcaton of ts otmzaton results []. Otmzaton solutons based on genetc algorthm grah theory vertex cover roblem Grah theory vertex cover roblem Overvew Vertces Coverng Problems(VCP) dscusses n grah theory gven grah G n a subset of vertces (set S), when the vertex G each edge has at least when one endont n S, and to ensure that there s no cover G S ', reached S' < S, then S can be consdered the mnmum coverage of G. Vertex cover roblem n bologcal sequence, nformaton retreval, the assembly lne balancng, schedulng roblems, and so the feld of theory and ractce n both are wdely used. Takng nto account t belongs to the NP- comlete roblem, so many aroxmaton algorthms, based on genetc algorthm to determne the mnmum coverage ssues, and to acheve vertex cover roblem grah otmzaton algorthms, t has smle calculaton takes tme and fewer restrctons Shaoshao Features. Nearby Fnd method Also, n order for the otmzaton roblem n grah theory, the aer also secfcally genetc algorthms and strateges wth local amendments combne to form a more targeted, recson Hybrd Cenetc Algorthm(HGA), by the mnmum vertex method to solve and otmze the grah coverng roblem. 25

3 Examles vertex cover roblem analyss In grah theory gven grah G, assumng ts vertces and edges, resectvely (V, E), and the formaton of an undrected grah, where V t means a collecton of vertces n the grah, and E ndcates the set of edges. In addton, V and E denote the number of vertces and set the fgure edges. Vertex cover s to fnd a subset of V and satsfy, and and j belong to the subset S, they are to meet the (, j) E, ts mathematcal form exressed as: Gven vertex cover roblem based on the mathematcal exresson for the constrant: = 2 x s = V When the length of vector x [ x x ] comly wth the followng condtons. P P = j= + mn S( x S) α ( x )( x j t j ) = 0 (, then ( a j ) x s the adjacent matrx, so x shall, comlywthv S x = 0, non comlywthv S On the current vertex cover relevant academc feld of grah theory roblems aroxmaton algorthm manly for greedy algorthm. It set n accordance wth the degree of the vertex unt Fgure descendng order, from whch select them, then added to cover the maxmum vertex set S were, and contnue to remove the vertces of G and all edges and vertces of a gven grah s connected, untl t s comletely deleted. Theoretcally reach the greedy algorthm grah vertces ssue of mnmum coverage, such as t s f you choose the vertex A, then ts vertex cover otmum result s B / C / D / E / F, A ont wll not be covered but becomes redundant vertces. And greedy algorthm also can not get rd of coverage and redundancy n a gven grah G vertces A, so ths wll select targeted and hgher recson hybrd genetc algorthm HGA [2]. Grah Theory vertex cover roblem hybrd genetc algorthm otmzaton desgn Hybrd genetc algorthm n the calculaton urose s to be acheved by selectng the best chromosome, and they crossover, mutaton and genetc combnaton oerator and have an effect, and ultmately the formaton of a whole new generaton of chromosome grous. Whle the ftness values of chromosomes n terms of a relatvely large, after teratve genetc algorthm converges Oerators wll be classfed as to the best chromosome, and the formaton of the otmal soluton of the roblem. Desgn and Analyss of ftness functon and chromosome reresentaton Generally seakng for vertex cover roblem n grah theory gven grah t s based on natural codng methods 0/ codng exlaned. and 0, resectvely, n or out of coverage, the followng encodng methods for the natural vertex cover roblem of grahs to exlan. Suose there s a subset of the set of x, x = {x, x2... xn}, then the ossblty that t can cover the vertex x = {x, x2... xn} = { }. Ths ndcates that a gven vertex cover fgure s comosed by the vertces,3,4, t forms a grahc nfeasble. That s the soluton to the gven grah does not cover all edges. So based on ths case, we use the method of enalty functon of the genetc algorthm for solvng constraned otmzaton roblems nfeasble solutons to solve the roblem, t s the rncle of otmzaton enalty gven by some fgure to be nfeasble solutons wth constrants no roblem nto a constraned otmzaton roblem, and at the tme dd not meet the constrants mosed on the arorate enalty for the ftness functon. In the enalty functon, the constant can be resent as the value of the enalty functon, ths value s determned accordng to the degree of ftness functon s volated, the greater the degree of volatons, the hgher the enalty. For the vertex cover roblem, the objectve functon reresents the mnmum cost functon of the form, t needs to be based on the target maed condton functon becomes the ftness functon form, 26

4 and based on the dualty theorem to determne the mnmum cost functon for maxmum roft and the roft functon equvalents. It wll functon to convert the mnmum roblem to the maxmum roblem under normal crcumstances and multled wth the cost functon, and then multled by -, and always ensure the ftness functon value s ostve, the followng converson forms: f ( x x 2 Emax g( xx2 ), 当 g( xx2 ) = 0, 当 g( xx2 ) >Emax ) <E max Selecton of Max there are many ways, whch tself can be used as nut arameters, whch change wth the arameter grous vares. Desgn and Analyss of Genetc Oerators Frst, the genetc algorthm s based on Darwn's theory of natural selecton, whch n terms of the strength of the sze of the selecton has a key role, and therefore the ntal selecton n genetc algorthms; usually choose a small amount, so that you can fully reflect the dversty of sace. Ths artcle s selected Dejon excluson genetc algorthm, whch can be used to exclude new generaton of offsrng and relace the old arent chromosome, thereby ncreasng the dversty of the oulaton. Fgure s a samle sace crowdng genetc algorthm based on the rule of the selecton rocess. Fg.. Based on the samle sace selecton rules flowchart based on crowed out genetc algorthm Snce the samle sace by randomly selectng chromosomes manly samlng selecton s comleted, we must frst determne the chromosome exectatons, and exectatons to transform accordng to the number of offsrng. In other words, the basc dea of the algorthm s based on the robablty of each chromosome to select and mlement adataton value rato between the robablty assumtons select, then t s calculated as follows: f = o, sze When the robablty of chromosome reflected n a certan ercentage of the total accounted for chromosome grou n ftness, t should calculate the exected value of the chromosome, namely: e = o_sze x, then t should genetc algorthm s descrbed as follows: sum 0; tr rand(); fork to o _ szedo j= f j 27

5 sum sum + e whle( sum>tr) do endwhle endfor end [ ] Selectchrom k tr tr + Accordng roortonal selecton method for selectng grous of two chromosomes n the chromosome and chromosome randomly select a ont n unform. Then the swtch neutron chromosome strng, and generatng two offsrng chromosomes. Call the scan of chromosome grous - correcton mechansm, combned wth the resultng offsrng chromosomes. Therefore, the fnal mutaton oerator should be defned as: When X Gene n X=X X 2 X transforms to ~X wth robablty of Pm, t reresents negaton of X, namely: x { 0,} Mxed genetc algorthm Hybrd genetc algorthm for randomly generated frst of length n chromosome vertex cover roblem n a gven grah otmzaton, where reresents the number of vertces n the fgure, and the use of standard genetc algorthm, scannng - Fxed and local mrovement strategy to mrove the chromosome grous. The followng s a hybrd genetc algorthm gven n Fgure G vertex cover roblem of otmzng the alcaton rocess: t 0; Intalzaton oulaton P (t); Scan - the amendment rocess Evaluaton oulaton P (t); Whle (termnaton condton s not satsfed); The comlete genetc oerators actng on oulaton P (t) generated by the grou C (t); and A scannng - the revson rocess; Imlementaton of Local-Imrovement-Search Process; Evaluaton of grou C (t); t t + End Such as the mxed genetc algorthm dervaton, n the outut of the vertex cover roblem grah otmal coverage otmzaton, we can acheve the best exlanaton for the otmzaton of grah theory [3]. Conclusons Based on the genetc algorthm to exlore n grah theory and otmzaton, resents the best soluton on the settlement of the mnmum vertex cover roblem gven grah, namely hybrd genetc algorthm, whch cover ssues for the mlementaton of the scan - correcton algorthm, and the realzaton of the local otmzaton technques to generate new rogeny grous, and fnally generate the otmal soluton. All ths shows the suerorty of the genetc algorthm, confrmng the robustness of ts desgn can be aled to the vertex cover roblem and more dfferent toologes of grah theory gven grah. Acknowledgments Ths artcle s a general roject n Yunnan Provnce Deartment of Educaton, Project Name: Study on the Classfcaton 2q ^ 2 rank 5 degree arc transtve grahs, NO: 204Y486; Yunnan k 28

6 Provncal Deartment of Educaton s also the general roject, Project Name: K tmes short-crcut and short-crcut roblem algorthm, NO: 204Y485. References [] Kong Dejan. Multobjectve otmzaton roblem research based on mroved genetc algorthm. Comuter smulaton,202,29(2): [2] Lu Xang. The alcaton of Genetc Algorthm and Its grah theory n lannng model. The PLA Informaton Engneerng Unversty,2007,-5. [3] Hong Hongwe Alcaton of genetc algorthm n grah theory and otmzaton. X'an Unversty of Electronc Scence and Technology,