A Neural Network for the Travelling Salesman Problem with a Well Behaved Energy Function

Transcription

1 A Neual Netwok fo the Tavelling Saleman Poblem with a Well Behaved Enegy Function Maco Budinich and Babaa Roaio Dipatimento di Fiica & INFN, Via Valeio, 347 Tiete, Italy mbh@tiete.infn.it (Contibuted pape to MANNA Confeence, Oxfod, July 3-7, 995) Abtact - We peent and analyze a Self Oganizing Featue Map (SOFM) fo the NP-complete poblem of the tavelling aleman (TSP): finding the hotet cloed path joining N citie. Since the SOFM ha dicete input patten (the citie of the TSP) one can examine it dynamic analytically. We how that, with a paticula choice of the ditance function fo the net, the enegy aociated to the SOFM ha it abolute minimum at the hotet TSP path. Numeical imulation confim that thi ditance augment pefomance. It i cuiou that the ditance function having thi popety combine the ditance of the neuon and of the weight pace. - Intoduction Solving difficult poblem i a natual aena fo a would-be new calculu paadigm like that of neual netwok. One can delineate a hape image of thei potential with epect to the blued image obtained in imple poblem. Hee we tackle the Tavelling Saleman Poblem (TSP, ee [Lawle 985], [Johnon 990]) with a Self Oganizing Featue Map (SOFM). Thi appoach, popoed by [Angéniol 988] and [Favata 99], tated to poduce epectable pefomance with the elimination of the non-injective output poduced by the SOFM [Budinich 995]. In thi pape we futhe impove it pefomance by chooing a uitable ditance function fo the SOFM. An inteeting featue i that thi net i open to analytical inpection down to a level that i not uually eachable [Ritte 99]. Thi happen becaue the input patten of the SOFM, namely the citie of the TSP, ae dicete. A a conequence we can how that the enegy function, aociated with SOFM leaning, ha it abolute minimum in coepondence to the hotet TSP path. In what follow we tat with a bief peentation of the woking pinciple of thi net and of it baic theoetical analyi (ection ). In ection 3 we popoe a new ditance function fo the netwok and how it theoetical advantage while ection 4 contain numeical eult. The appendix contain the detailed deciption of paamete needed to epoduce thee eult.

2 - Solving the TSP with elf-oganizing map The baic idea come fom the obevation that in one dimenion the exact olution to the TSP i tivial: alway tavel to the neaet unviited city. Conequently, let u uppoe we have a mat map of the TSP citie onto a et of citie ditibuted on a cicle, we will eaily find the hotet tou fo thee image citie that will give alo a path fo the oiginal citie. It i eaonable to conjectue that the bette the ditance elation ae peeved, the bette will be the appoximate olution found. In thi way, the oiginal TSP i educed to a each of a good neighbohood-peeving map: hee we build it via unupevied leaning of a SOFM. The TSP we conide i contituted of N citie andomly ditibuted in the plane (actually in the (0,) quae). The net i fomed by N neuon logically oganized in a ing. The citie ae the input patten of the netwok and the (0,) quae it input pace. Each neuon eceive the (x, y) = q coodinate of the citie and ha thu two weight: (w x,w y ) = w. In thi view both patten and neuon can be thought a point in two dimenional pace. In epone to input q, the -th neuon poduce output o = q. w. Figue give a chematic view of the net while figue epeent both patten and neuon a point in the plane. q i x y Figue Schematic net: not all connection fom input neuon ae dawn. Leaning follow the tandad Kohonen algoithm [Kohonen 984]: a city q i i elected at D w - D w w w - w + D w + Figue Weight modification in a leaning tep: neuon (mall gay cicle) ae moved towad the patten q i (black cicle) by an amount given by (). The olid line epeent the neuon ing. The hape of the defomation of the ing i given by the elative magnitude of the D w that i in tun given by the ditance function h. andom and popoed to the net; let be the mot eponding neuon (i.e. the neuon neaet to q i ) then all neuon weight ae updated with the ule: Dw = e h qi ( - w ) () whee 0 < e < i the leaning contant and h i the ditance function.

3 Thi function detemine the local defomation along the chain and contol the numbe of neuon affected by the adaptation tep (); thu it i cucial fo the evolution of the netwok and fo the whole leaning poce (ee figue ). Step () i epeated eveal time while e and the width of the ditance function ae being educed at the ame time. A common choice fo h i a Gauian-like function like h = e - Ê d ˆ Ë whee d i the ditance between neuon and (the numbe of tep between and ) and i a paamete which detemine the numbe of neuon uch that Dw π 0; duing leaning e, Æ 0 o that Dw Æ 0 and h Æ d. Afte leaning, the netwok map the two dimenional input pace onto the one dimenional pace given by the ing of neuon and neighboing citie ae mapped onto neighboing neuon. Fo each city it image i given by the neaet neuon. Fom the tou on the neuon ing one obtain the path fo the oiginal TSP. The tandad theoetical appoach to thee net conide the expectation value E[ Dw w ] [Ritte 99]. In geneal E[ Dw w ] cannot be teated analytically except when the input patten have a dicete pobability ditibution a it happen fo the TSP. In thi cae, E[ Dw w ] can be expeed a the gadient of an enegy function, i.e. E[ Dw w ]=-e w VW ( ), with VW ( )= N h q i ŒF q i - w ( ) () whee W = { w } and the econd um i ove the et of all the citie q i having w a neaet neuon, i.e. the citie contained in F, the Voonoi cell of neuon w. On aveage, VW ( ) deceae duing leaning ince E[ DV W]=-e w V. Subtantially, in thi cae, thee exit an enegy function which decibe the dynamic of the SOFM and which i minimized duing leaning; fomally, the leaning poce i the decent along the gadient of VW ( ). Unfotunately VW ( ) ecape analytical teatment until the end of the leaning poce when ome implification ae applicable. Since at the end of leaning h Æ d, we can uppoe h i ignificantly diffeent fom zeo only fo =, ±; in thi cae () become VW ( )@ N h qi -, - w ( - ) + h qi - w ( ) + h qi +, ( - w + ) q i ŒF [ ] (3) The main weakne of thi algoithm i that, in about half of the cae, the map poduced i not injective. The definition of a continuou coodinate along the neuon ing olve thi poblem yielding a competitive algoithm [Budinich 995]. 3

4 In addition, imulation uppot that, at the end of leaning, mot neuon ae elected by jut one city to which they get neae and neae. Thi mean that F contain jut one city, let' call it q i( ), and that w Æ q i(), conequently VW ( N h qi -, ( ( ) - q i ( - ) [ + h ) qi +, ( ( ) - q i ( + ) ] ) (4) and auming h ymmetic i.e. h -, = h +, = h, we get VW ( ) = = [ ( ) ] ( ) h ( N q i ( ) - q i ( - ) + q ) i( ) - q i + h N L TSP whee L TSP i the length of the tou of TSP conideing the quae of the ditance between citie. Thu the Kohonen algoithm fo TSP minimize an enegy function which, at the end of the leaning poce, i popotional to the um of the quae of the ditance. Numeical imulation confim thi eult. 3 - A new ditance function Ou hypothei i that we can obtain bette eult fo the TSP uing a ditance function h uch that, at the end of the poce, VW ( ) i popotional to the imple length of the tou L TSP, namely VW ( )µ q i ( ) - q i( -) = L TSP ince, in geneal, ( ) minimizing L TSP i not equivalent to minimizing L TSP. We thu conide a function h depending both on the ditance d and on anothe ditance D defined in weight pace: D = w j - w j - j = + w 3 3 w 4 w D 3 D 4 D w If we define Figue 3 Ditance between neuon =4 and =: D,4 = D + D + D 3 and d,4 =3. h = Ê + D ˆ Ë -d (5) 4

5 D h d Figue 4 Set of h given by (5) fo 0 < D < and d = 0,,...,4. when Æ 0, we get fo h ±, h ±, = + D - Ê ±, ˆ D ±, = w - w q i( ) - q i( ±) and ubtituting thi expeion in (4) we obtain 5

6 VW ( = N N È Í q i( ) - q ÎÍ i - q i( ) - q i( +) ( ( ) - q i( -) ) + ( ) qi q i( ) - q i + ( ( ) - q i( +) ) ( ) qi = N L TSP With thi choice of h the minimization of the enegy VW ( ) i equivalent to the minimization of the TSP path. We emak that the intoduction of the ditance D between weight i a lightly unuual hypothei fo thi kind of net that uually keep well epaated neuon and weight pace in the ene that the ditance function h depend only on the ditance d. 4 - Numeical eult Since the pefomance of thi kind of TSP algoithm ae good fo poblem with moe than 500 citie and moe citical in malle poblem [Budinich 995], we began teting the pefomance poduced by the new ditance function (5) in poblem with 50 citie. We compaed the quality of TSP olution obtained with thi net to thoe of two othe algoithm both deiving fom the idea of a topology peeving map and that both actually minimize L TSP : the elatic net of Dubin and Willhaw [Dubin 987] and thi ame algoithm with a tandad ditance choice. A a tet et, we conideed the vey ame 5 et of 50 andomly ditibuted citie ued fo the elatic net. Table contain a compaion of the bet TSP path obtained in eveal un of the diffeent algoithm expeed a pecentual incement ove the bet known olution fo the given poblem. 6

7 City et Min. length [Dubin 987] [Budinich 995] Thi algoithm %.65 % 0.96 % %.66 % 0.3 % %.06 %.05 % %.37 % 0.70 % % 5.5 % 0.43 % Aveage.68 %.0 % 0.69 % Table Compaion of the bet TSP olution obtained in 0 un of the vaiou algoithm. Row efe to the 5 diffeent poblem each of 50 citie andomly ditibuted in the (0,) quae. Column epot the length of the bet known olution fo the given poblem. Column 3 to 5 contain the bet length obtained by the thee algoithm unde tudy expeed a pecentual incement fom the minimal length; the numbe of un of the algoithm i epectively: unknown, 5 and 0. Lat ow give the incement aveaged ove the 5 city et. Anothe meaue of the quality of the olution i the mean length obtained in the 0 un. The pecentual incement of thee mean length, aveaged ove the 5 et, wa fo thi algoithm.49%, howing that even the aveage found with the new ditance function ae bette than the minima found with the elatic net. Thee eult clealy how that ditance choice (5) give bette olution in thi SOFM application, thu uppoting the gue that an enegy VW ( ) diectly popotional to the length of the tou L TSP, i bette tuned to thi poblem. One could wonde if adding weight pace infomation to the ditance function could give inteeting eult alo in othe SOFM application. Appendix Hee we decibe the netwok etting that poduce the quoted numeical eult. Apat fom the ditance definition (5) we apply a tandad Kohonen algoithm and exponentially deceae paamete e and with leaning epoch n e (a leaning epoch coepond to N weight update with ule ()) e = e 0 a n e = 0 b n e and leaning top when e eache Numeical imulation clealy indicate that bet eult ae obtained when the final value of i vey mall (ª ) and when e and deceae togethe eaching thei final value at the ame time. Conequently given value fo a and 0 one eaily find b. In othe wod thee ae jut thee fee paamete to play with to optimize eult, namely e 0 and 0 and a. Afte ome invetigation we obtained the following value that poduce the quoted eult: e 0 = 0.8, 0 = 4 and a =

8 Refeence: [Angéniol 988] [Budinich 995] [Dubin 987] [Favata 99] [Johnon 990] B. Angéniol, de G. La Coix Vauboi and J.-Y. Le Texie, Self Oganiing Featue Map and the Tavelling Saleman Poblem, Neual Netwok 988 pp ; M. Budinich, A Self-Oganiing Neual Netwok fo the Tavelling Saleman Poblem that i Competitive with Simulated Annealing, to appea in: Neual Computation; R. Dubin and D. Willhaw, An Analogue Appoach to the Tavelling Saleman Poblem uing an Elatic Net Method, Natue pp ; F. Favata and R. Walke, A Study of the Application of Kohonen-type Neual Netwok to the Tavelling Saleman Poblem, Biological Cybenetic pp ; D. S. Johnon, Local Optimization and the Taveling Saleman Poblem, in: Poceeding of the 7 th Colloquium on Automata, Language and Pogamming, 990 Spinge-Velag New Yok, pp ; [Kohonen 984] T. Kohonen, Self-Oganiation and Aociative Memoy, 984 (3 d Ed. 989) Spinge-Velag Belin Heidelbeg; [Lawle 985] E. L. Lawle, J. K. Lenta, A. G. Rinnoy Kan and D. B. Shmoy (edito), The Taveling Saleman Poblem - A Guided Tou of Combinatoial Optimization, John Wiley & Son, New Yok 990, IV Repint, pp. x 474; [Ritte 99] H. Ritte, T. Matinetz and K. Schulten, Neual Computation and Self-Oganizing Map, Addion-Weley Publihing Company, Reading Maachuett 99, pp