Asymptotically Optimal Algorithms for Job Shop Scheduling and Packet Routing

Transcription

1 Journal of Algorthms 33, Ž Artcle ID agm , avalable onlne at on Asymptotcally Optmal Algorthms for Job Shop Schedulng and Packet Routng Dmtrs Bertsmas Sloan School of Management and Operatons Research Center, Massachusetts Insttute of Technology, E53-363, Cambrdge, Massachusetts and Davd Gamarnk T.J. Watson Research Center, IBM, Yorktown Heghts, New York We propose asymptotcally optmal algorthms for the ob shop schedulng and packet routng problems. We propose a flud relaxaton for the ob shop schedulng problem n whch we replace dscrete obs wth the flow of a contnuous flud. We compute an optmal soluton of the flud relaxaton n closed form, obtan a lower bound C to the ob shop schedulng problem, and construct a feasble schedule from the flud relaxaton wth obectve value at most C OŽ C. ', where the constant n the OŽ. notaton s ndependent of the number of obs, but t depends on the processng tme of the obs, thus producng an asymptotcally optmal schedule as the total number of obs tends to nfnty. If the ntally present obs ncrease proportonally, then our algorthm produces a schedule wth value at most C OŽ. 1. For the packet routng problem wth fxed paths the prevous algo- rthm apples drectly. For the general packet routng problem we propose a lnear programmng relaxaton that provdes a lower bound C and an asymptotcally optmal algorthm that uses the optmal soluton of the relaxaton wth obectve value at most C OŽ C. '. Unlke asymptotcally optmal algorthms that rely on probablstc assumptons, our proposed algorthms make no probablstc assumptons and they are asymptotcally optmal for all nstances wth a large number of obs Ž packets.. In computatonal experments our algorthms produce schedules whch are wthn 1% of optmalty even for moderately szed problems Academc Press * The research of ths author was partally supported by NSF Grant DMI $30.00 Copyrght 1999 by Academc Press All rghts of reproducton n any form reserved. 296

2 ASYMPTOTIC SCHEDULINGROUTING ALGORITHMS INTRODUCTION The ob shop schedulng and the packet routng problems are fundamental problems n operatons research and computer scence. The ob shop schedulng problem s the problem of schedulng a set of I ob types on J machnes. Job type conssts of J stages, each of whch must be completed on a partcular machne. The par Ž,. represents the th stage of the th ob and has processng tme p,. The completon tme of ob s the completon tme of the last stage J of ob type. Assumng that we have n obs of type, the obectve s to fnd a schedule that mnmzes the mum completon tme, called the makespan, subect to the followng restrctons: 1. The schedule must be nonpreemptve. That s, once a machne begns processng a stage of a ob, t must complete that stage before dong anythng else. 2. Each machne may work on at most one task at any gven tme. 3. The stages of each ob must be completed n order. The classcal ob shop schedulng problem nvolves exactly one ob from each type,.e., the ntal vector of ob types s Ž 1, 1,..., 1.. The ob shop schedulng problem s a classcal NP-hard problem, notorously dffcult to solve even n relatvely small nstances. As an example, a specfc nstance nvolvng 10 machnes and 10 obs posed n a book by Muth and Thompson 11 n 1963 remaned unsolved for over 20 years untl solved by Carler and Pnson 2 n The packet routng problem n a communcaton network Ž V, A. s the problem of routng a collecton of packets from a source node to a destnaton node. It takes one tme unt for a packet to traverse an edge n A, and only one packet can traverse a gven edge at a tme. As n the ob shop schedulng problem, the obectve s to fnd a schedule that mnmzes the tme, called the makespan, that all packets are routed to ther destnatons. For the case that the paths along whch packets need to be routed are gven, the problem can be modeled exactly as a ob shop schedulng problem. However, when we can select the paths along whch to route packets, the problem s more complcated as t nvolves both routng Žpath selecton. and sequencng Ž whch packet each edge process. decsons. Our overall approach for these problems reles on two deas from two dstnct communtes. Frst, we consder a relaxaton for the ob shop schedulng problem called the flud control problem, n whch we replace dscrete obs wth the flow of a contnuous flud. The motvaton for ths approach comes from optmal control of multclass queueng networks. Multclass queueng networks are stochastc and dynamc versons of ob

3 298 BERTSIMAS AND GAMARNIK shops. In recent years there has been consderable progress n solvng the flud control problem n multclass queueng networks. Focusng on obectve functons that mnmze a weghted combnaton of the number of obs at the varous machnes, as opposed to makespan, Avram, Bertsmas, and Rcard 1 show that by usng the Pontryagn mum prncple, we can fnd the optmal control explctly. However, the descrpton of the optmal control, whle nsghtful for the orgnal problem, nvolves the enumeraton of an exponental number of cases. Luo and Bertsmas 10, buldng upon the work of Pullan 12, use the theory of contnuous lnear programmng to propose a convergent numercal algorthm for the problem that can solve effcently problems nvolvng hundreds of machnes and ob types. For the obectve we consder Žmnmze the length of the schedule,.e., the mum completon tme. the optmal soluton of the flud control problem can be computed n closed form and provdes a lower bound C to the ob shop schedulng problem. Wess 17 has consdered and solved the makespan obectve for a flud control problem wth arrvals. Our proof of the flud control problem wthout arrvals follows along smlar lnes. The second dea of the paper s motvated by the consderable progress n the determnstc schedulng communty n provdng approxmaton algorthms for schedulng problems that rounds the soluton of a lnear programmng relaxaton of the schedulng problem. Shmoys, Sten, and Wen 15, Goldberg et al. 4, and Fege and Schedeler 3 provde algorthms that are wthn a multplcatve logarthmc guarantee from the optmal soluton value. Very recently Jansen, Sols-Oba, and Svrdenko 6 provded a polynomal tme approxmaton scheme. For a revew of ths approach see Hall 5 and Karger, Sten, and Wen 7. However, the paper closest n sprt to the current work s a schedulng algorthm for ob shop problems constructed by Sevast anov 13 Žsee also 14.. Sevast anov s algorthm s based on an nterestng geometrc method, unrelated to the methods of the current paper, and produces a schedule wth length C OŽ. 1, and as a result, s asymptotcally optmal as the number of obs tends to nfnty. Our algorthm s sgnfcantly smpler than Sevast anov s and produces superor bounds for a varety of nstances. For example, for the 10 by 10 nstance defned n Muth and Thompson 11 wth the same number n of obs for every ob type, then the bound for our algorthm s always stronger for all n. We compare the bounds gven by our methods and those by Sevast anov n Secton 4.3. We use the optmal soluton of the flud control problem to construct a feasble schedule wth the obectve value C OŽ C. '. If the ntally present obs ncrease proportonally, then our algorthm produces a schedule wth a value of at most C OŽ. 1. Smlarly, for the packet routng problem we propose a lnear programmng relaxaton that provdes a lower

4 ASYMPTOTIC SCHEDULINGROUTING ALGORITHMS 299 bound C and use ts soluton to construct a feasble schedule wth obectve value C OŽ ' C.. We note that the constant n the OŽ. notaton s ndependent of the number of obs, but t does depend on the processng tmes of the obs. Ths mples that as the total number of obs Ž packets, respectvely. tends to nfnty, the proposed algorthm s asymptotcally optmal. Unlke asymptotcally optmal algorthms that rely on probablstc assumptons, the above algorthm makes no probablstc assumptons, and t s asymptotcally optmal for all nstances wth a large number of obs Ž packets, respectvely.. The classcal result n ths area s the work of Karp 8, who provded an asymptotcally optmal algorthm for the travelng salesman problem when the ponts are randomly and unformly dstrbuted n the unt square n the Eucldean plane. The combnatoral structure of the ob schedulng problem makes the problem very complcated to solve when there s a small number of obs n the system. Interestngly, the results of the paper ndcate that as the number of obs ncreases, the combnatoral structure of the problem s ncreasngly less mportant, and as a result, a flud approxmaton of the problem becomes ncreasngly exact. Smlarly, the packet routng problem has an even rcher combnatoral structure. The results of the paper also mply that a contnuous approxmaton to the problem s asymptotcally exact. The paper s structured as follows. In Secton 2, we formulate the ob shop schedulng problem and descrbe the notaton. In Secton 3, we ntroduce the flud control problem for the ob shop schedulng problem and solve t n closed form. In Secton 4, we present and analyze the roundng algorthm, called the synchronzaton algorthm. We also provde some computatonal results and contrast our bounds wth those by Sevast anov 13. In Secton 5, we address packet routng n communcaton networks wth fxed paths as an applcaton of ob shop schedulng. In Secton 6, we propose an asymptotcally optmal algorthm for the general packet routng problem n communcaton networks. Secton 7 contans some concludng remarks. 2. PROBLEM FORMULATION AND NOTATION In the ob shop schedulng problem there are J machnes, 1, 2,..., J, whch process I dfferent types of obs. Each ob type s specfed by the sequence of machnes to be processed on and the processng tme on each machne. In partcular, obs of type, 1, 2,..., I, are processed on J machnes 1, 2,...,J n that order. Let J J. The tme to

5 300 BERTSIMAS AND GAMARNIK process a type ob on machne k s denoted by p, k. Throughout the paper we assume that p are ntegers., k The obs of type that have been processed on machnes,..., 1 k1 but not on machne k are queued at machne k and are called type obs n stage k. The set of obs n stages k 2 n any specfc machne s called a nonntal queue n machne. In partcular, at tme zero all nonntal queues are empty. We wll also thnk of each machne as a collecton of all types and stage pars that t processes. Namely, for each 1, 2,..., J 4 Ž, k. :,1I, 1 k J. k There are n obs for each type ntally present at ther correspondng frst stage. Our obectve s to mnmze the makespan,.e., to process all the n n1 n2 ni obs on machnes 1,..., J, so that the tme t takes to process all the obs s mnmzed. Each machne has a certan processng tme requred to process obs that eventually come to ths machne. Specfcally, for machne ths tme s C p n., k Ž, k. The quantty C s called the congeston of machne. We denote the mum congeston by C C. 1J The followng proposton s mmedate. PROPOSITION 1. The mnmum makespan C of the ob shop schedulng problem satsfes C C. In the next secton we consder a flud Ž fractonal. verson of ths problem, n whch the number of obs n of type can take arbtrary postve real values, and machnes are allowed to work smultaneously on several types of obs Žthe formal descrpton of the flud ob shop schedulng problem s provded n the next secton.. For the flud control problem we show that a smple algorthm leads to a makespan equal to C and therefore s optmal.

6 ASYMPTOTIC SCHEDULINGROUTING ALGORITHMS THE FLUID JOB SHOP SCHEDULING PROBLEM In ths secton, we descrbe a flud verson of the ob schedulng problem. The nput data for the flud ob shop schedulng problem are the same as for the orgnal problem. There are J processng machnes 1, 2,..., J, I ob types, each specfed by the sequence of machnes k, k 1, 2,..., J, and the sequence of processng tmes p, k for type obs n stage k. We ntroduce the notaton, k 1p, k whch represents the rate of machne k on a type ob. The number of type obs ntally present, denoted by x, takes nonnegatve real values. In order to specfy the flud control problem we ntroduce some notaton. We let x Ž. t be the total Ž fractonal n general., k number of type obs n stage k at tme t. We call ths quantty the flud level of type n Ž. stage k at tme t. We denote by T, k t the total tme the machne k works on type obs n stage k durng the tme nterval 0, t. Fnally 1A4 denotes the ndcator functon for the set A. The flud control problem of mnmzng makespan can be formulated as follows: H ½, k 5 mnmze 1 x Ž t. 0 dt Ž I,1kJ subect to x,1ž t. x,1t,1ž t., 1,2,..., I, t 0, Ž 2. x, kž t., k1t, k1ž t., kt, kž t., Ž. k 2,..., J, 1,2,..., I, t 0, Ž 3. Ž T, kž t2. T, kž t1.. t2 t 1,, k t 2, t 1, t 1, t2 0, 1,2,..., J. Ž 4. x, kž t. 0, T, kž t. 0. Ž 5. The obectve functon Ž. 1 represents the total tme that at least one of the flud levels s postve. It corresponds to the mnmum makespan schedule n the dscrete problem. Equatons Ž.Ž. 2, 3 represent the dynamcs of the system. The flud level of type n stage k at tme t s the ntal number of type obs n stage k Ž x for k 1, zero for k 1. plus the number of type obs processed n stage k 1 durng 0, t Žgven by T Ž.., k1, k1 t, mnus the number of type obs processed n stage k durng 0, t Žgven by T Ž t... Constrant Ž 4., k, k s ust the aggregate feasblty constrant for machne.

7 302 BERTSIMAS AND GAMARNIK Smlar to the defnton for the dscrete problem, we defne congeston n staton as C p x, Ž 6. and the mal congeston as, k Ž, k. C C. Ž 7. 1J We next show that the flud control problem can be solved n closed form. PROPOSITION 2. The flud control problem Ž. 1 has an optmal alue equal to the mum congeston C. Proof. We frst show that the mum congeston C s a lower bound on the optmal value of the control problem. For any postve tme t and for each I, k J, we have from Ž.Ž. 2, 3 : k, l, k, k l1 x Ž t. x T Ž t.. For each staton we obtan k p x Ž t. p x T Ž t. C t,, k, l, k, k Ž, k. l1 Ž, k. Ž, k. where the last nequalty follows from the defnton of C and constrant Ž. 4 appled to t 0, t t. It follows then, that the flud levels are 1 2 postve for all tmes t smaller than C. Therefore, the obectve value of the optmal control problem s at least C C. We now construct a feasble soluton that acheves ths value. For each I, k J and each t C p, kx T, kž t. t, C we let x Ž t. x T Ž t. x t, 1,...,I,,1,1,1 x C x, kž t. 0, k 2,3,..., J, 1,...,I. For all t C we set T Ž. t p x, x Ž., k, k, k t 0. Clearly, ths soluton has an obectve value equal to C. We now show that ths soluton s feasble. It s nonnegatve by constructon. Also by constructon, Eq. Ž. 2 s

8 ASYMPTOTIC SCHEDULINGROUTING ALGORITHMS 303 satsfed for all t C. In partcular, x Ž C.,1 0, 1, 2,..., I. Moreover, for all, k 2, 3,..., J and t C we have, k1t, k1ž t., kt, kž t. p, 1 k1t, k1ž t. p, 1 kt, kž t. x x t t 0 x, kž t., C C and Eq. Ž. 3 s satsfed. Fnally, for any t t C, we have 1 2 and for any machne p, kx p, kx T Ž t. T Ž t. t t C C Ž. ž /, k 2, k Ž, k. Ž, k. C Ž t t. t t, C and constrant Ž. 4 s satsfed. Note that for the constructed soluton x Ž C., k 0 for all I, k J. Therefore, the feasblty for tmes t C follows trvally. The constructed soluton has a structure resemblng a processor sharng polcy. It calculates the mal congeston C and allocates a propor- tonal effort to dfferent ob types wthn each machne to acheve the target value C. Such an optmal polcy s possble, snce we relaxed the ntegralty constrant on the number of obs and allowed machnes to work smultaneously on several ob types. In the followng secton, we use the flud soluton to construct an asymptotcally optmal soluton for the orgnal dscrete ob shop schedulng problem. 4. AN ALGORITHM FOR THE JOB SHOP SCHEDULING PROBLEM In ths secton, we consder the orgnal ob shop schedulng problem, descrbed n Secton 2. Recall that we are ntally gven n obs for each type, 1, 2,..., I, where n s some nonnegatve nteger. Staton has congeston C gven by Ž, k. p, kn. Agan, let C denote the - mal congeston. Let be a certan postve real value. An exact value for wll be specfed later. For each ob type we let n a. Ž 8. C

9 304 BERTSIMAS AND GAMARNIK For each staton let U p. Ž, k., k Namely, U s the workload of staton when only one ob per type s present. Fnally, let U U. Ž 9. 1J The proposed algorthm revsts the schedule n tme ntervals of length U. THE SYNCHRONIZATION ALGORITHM. For each nterval mž U., Ž m 1.Ž U., m 0, 1,..., of length U, each machne, and each par Ž, k., machne processes exactly a obs of type Ž whch takes p a tme unts. and dles, k U p a, k Ž, k. tme unts. If for some, at tme mž U., the number of type obs n machne s less than a, then machne processes all the avalable type obs and dles for the remanng tme. Note that the synchronzaton algorthm produces a feasble schedule, snce for each machne and each Ž, k., t takes p, ka tme unts to process a obs of type. Snce n p a 1 p U U ž /, k, k Ž, k. Ž. C, k t follows that the schedule s ndeed feasble. In the flud relaxaton, each ob Ž, k. receves Ž p nc., k % of the effort from the correspondng machne. The synchronzaton algorthm over each nterval of length U allocates tme ap on obs Ž, k., k. Thus, ob Ž, k. receves ap, k p, kn % U C of the effort from the correspondng machne. We used approxmaton n the last equalty, snce n the synchronzaton algorthm we deal wth dscrete obs. Intutvely, the synchronzaton algorthm gves essentally the same amount of effort as n the flud relaxaton.

10 ASYMPTOTIC SCHEDULINGROUTING ALGORITHMS 305 The next theorem shows that the algorthm does a good ob of synchronzng and ppelnng the total workload n the system and acheves asymptotc optmalty. THEOREM 1. Consder a ob shop schedulng problem wth I ob types and J machnes 1, 2,..., J. Gen ntally n obs of type 1, 2,..., I, the synchronzaton algorthm wth ' C U J produces a schedule wth makespan tme C such that H C C C C 2 C U J U J, Ž 10. H ' where U s defned by Ž. 9.In partcular, C H C CH 1, Ž 11. C as I 1 n, where C s the optmal makespan. In addton, all the nonntal queue lengths at each staton are at most ( UC J U. Ž 12. Proof. For each I, k J and each nteger tme t, let N Ž., k t denote the number of type obs n stage k Žwatng to be processed on the. Ž. machne k. Note that N, k 0 0 for all k 2. For each, machne 1 wll process exactly a obs durng the nterval 0, U. Therefore, N,1Ž U. n a, N,2Ž U. a, N, kž U. 0, k 3. Also, for each machnes 1, 2 wll process exactly a obs durng the nterval U,2Ž U.. Therefore, Ž. Ž. N 2Ž U. n 2 a, N 2Ž U. a,,1,2 Ž. Ž. N 2Ž U. a, N 2Ž U. 0, k 3.,3, k

11 306 BERTSIMAS AND GAMARNIK Smlarly, we observe that for each Ž, k., machne does not receve type obs untl tme t Ž k 1.Ž U., and durng each subsequent nterval mž U., Ž m 1.Ž U., m k 1 t re- ceves and processes exactly a obs of type, untl no new obs of type arrve. Let Then C T Ž U.. Ž 13. C n C N,1Ž T. n a n 0 Ž 14. C for all. Therefore, all obs leave the ntal stage durng the tme nterval 0, T. We next estmate the tme to process all 1 I n obs ntally present n the system. We tag a gven ob of type. Ths ob, as shown n Ž 14., leaves the frst stage Ž, 1. at some tme not bgger than T. Gven any k J, suppose ob arrves at the kth stage at some tme nterval Ž m 1.Ž U., mž U.. All the type obs that ar- rved at stage k before tme Ž m 1.Ž U. have been processed, and ob s one of the a obs of type that arrved at stage k Žmachne. k durng the tme nterval Ž m 1.Ž U., mž U., from the pre- vous stage. Durng the tme nterval mž U., Ž m 1.Ž U. machne k processes a obs of type, so ob s n stage k 1 at tme Ž m 1.. We conclude that the delay for ob between leavng the frst stage and beng processed at the last machne s at most Ž J 1.Ž J U. Ž J 1.Ž U.. Combnng ths wth Ž 14., we conclude that the total delay for the ob s at most T Ž J 1.Ž U. C Ž U. Ž J 1.Ž U. C ž / 1 Ž U. Ž J 1.Ž U. UC C JU J. Recall that we have not specfed the value of. We now select to be the mnmzer of the expresson above;.e.,

12 ASYMPTOTIC SCHEDULINGROUTING ALGORITHMS 307 ( J UC. Therefore, the total makespan tme under the algorthm s at most CH C 2' CU J U J. Ths proves the bound Ž 10.. Notce that the mum congeston C tends to nfnty, as the ntal total number of obs I n tends to nfnty. Therefore, 1 C H 1 C as 1 I n. From Proposton 1, C s a lower bound on the optmal makespan tme C. Therefore, Ž 11. follows,.e., the synchronzaton algorthm s asymptotcally optmal. Fnally, as we have seen, the number of type obs n machne k s never more than a for k 2. As a result, the nonntal queue length n machne s at most Ž, k., k2 From the ntegralty of processng tmes, t follows that the nonntal queue length n machne s at most n UC ap p p C J a. ž ( /, k, k, k Ž, k., k2 Ž, k., k2 UC ( U. Ths proves Ž 12.. Note that the makespan C H J of the synchronzaton algorthm satsfes ž ' / C C O C. H 4.1. The Proportonal Case In ths secton, we address the case wth n bn, and b are ntegers. In ths case, we let Ž, k. p, kb. Then C n. We modfy slghtly the synchronzaton algorthm n ths case: For each nterval

13 308 BERTSIMAS AND GAMARNIK m, Ž m 1., m 0, 1,..., of length, each machne, and each par Ž, k., machne processes exactly b obs of type Ž whch takes p b tme unts. and dles, k, k Ž., k p b tme unts. By followng the same analyss as n Theorem 1, we show that the modfed synchronzaton algorthm produces a schedule wth makespan at most C Ž J 1. C OŽ Computatonal Results We have mplemented three algorthms for the ob schedulng problem. The frst algorthm s the synchronzaton algorthm Žcalled orgnal flud trackng heurstc n Fgs. 1 and 2., the second s the modfed synchronzaton algorthm outlned n the proportonal case Žcalled new heurstc for unform N n Fgs. 1 and 2., and the thrd s a fnal modfcaton of the synchronzaton algorthm, n whch the machnes do not dle f they do not have work to do Ž called varable omega heurstc n Fgs. 1 and 2.. We run these algorthms on the 10 by 10 nstances n Muth and Thompson 11 wth n N obs present and vared N. Fgure 1 shows the performance of the synchronzaton algorthm and ts modfcaton for N 50, whle Fg. 2 shows the performance of all three algorthms for N It s nterestng that for N 500, the varable heurstc produces solutons wthn 1% from the lower bound Comparson wth Seast ano s Algorthm Sevast anov 13has constructed a schedulng algorthm wth makespan Ž.Ž 2 tme not exceedng C J 1 JJ 2 J 1. p, where p p 4, k s the mal processng tme of a sngle ob. We now compare ths performance bound wth the one gven by Theorem 1. PROPOSITION 3. Ž. a If the total number of obs n satsfes n n ŽŽ 0 J.Ž 2 1 JJ 2 J J, then ' ' C 2 C U J U J Ž. C Ž J 1. JJ 2 2 J 1 p ; Ž 15..e., the upper bound on the makespan tme C H correspondng to the synchro-

14 ASYMPTOTIC SCHEDULINGROUTING ALGORITHMS 309 FIG. 1. N 50. The performance of the synchronzaton algorthm and ts modfcaton for nzaton algorthm s superor to the upper bound on the Seast ano schedulng algorthm. Ž b. If all ob types hae the same number n of obs ntally present, there are at most J ob types, and each ob type s processed by any gen machne at most once, then the bound gen by the synchronzaton algorthm s always stronger than the bound gen by Seast ano s schedulng algorthm. Proof. Ž. a Trvally, U C np. Then, the left hand sde of Ž 15. s smaller than C 2np' J np J. Therefore, the left hand sde of Ž 15. s smaller than C 2np J, whch s less than or equal to f n n. 0 C Ž J 1.Ž JJ 2 2 J 1. p,

15 310 BERTSIMAS AND GAMARNIK FIG. 2. The performance of all three algorthms for N Ž b. If all ob types have the same number n of obs ntally present,.e., n n, the performance bound of the synchronzaton algorthm s C Ž J 1. U. If there are at most J ob types, and each ob type s processed by any gven machne at most once, then U J p, and thus the bound gven by the synchronzaton algorthm s always stronger. For example, for the 10 by 10 nstance defned n Muth and Thompson 11 wth the same number n of obs for every ob type, then the bound for the synchronzaton algorthm s always stronger for all n. 5. THE PACKET ROUTING PROBLEM WITH FIXED PATHS In ths secton, we apply our results on the ob shop schedulng problem to the problem of packet routng n communcaton networks. Gven a drected graph Ž V, A. that represents a communcaton network, there s a

16 ASYMPTOTIC SCHEDULINGROUTING ALGORITHMS 311 collecton of packets that needs to be sent from a source node to a destnaton node along gven paths. Each packet s gven wth a prespecfed smple path Ž each node vsted at most once., connectng the source to the destnaton,.e., each packet s represented by the trplet Ž s, t, P. k k k, where sk s the source node, tk s the destnaton node, and Pk s the prespecfed smple path. It takes one tme unt for a packet to traverse an edge n A, and only one packet can traverse a gven edge at a tme. We wll consder two versons of the packet routng problem: n ths secton, we address the problem where the paths are gven, and n the next secton, we address the problem where we need to select the paths. For the packet routng problem wth fxed paths there are np packets that need to be sent along path P, for each smple path P. Let P denote the collecton of all smple paths, for whch np 0. A scheduler decdes whch packets traverse any gven edge and whch packets wat n queue. The goal s to fnd a schedule whch routes all packets from ther sources to ther destnatons n mnmal possble Ž makespan. tme, gven the ntal number of packets np for each smple path P. Extensve research has been conducted on ths problem Žsee Leghton. 9. It s easy to see that the packet routng problem wth fxed paths s a specal case of the ob shop schedulng problem. Each edge can be seen as a processng machne. Each path s a sequence of machnes Ž edges. that obs Ž packets. need to follow. All the processng Ž traversng. tmes are equal to one. The ob types correspond to paths n P, and the stages correspond to edges wthn the path. Also, the quantty Ž, k. p, k n the ob shop schedulng problem corresponds smply to the number of paths crossng any gven edge e. The number s at most P. The congeston Ce of a gven edge e A s smply the number of packets that eventually cross e Ž the correspondng paths contan e.: C n. e P : ep P The mal congeston C s ece and s clearly a lower bound on the optmal makespan tme. Applyng the synchronzaton algorthm for the ob shop schedulng problem we obtan the followng result.

17 312 BERTSIMAS AND GAMARNIK THEOREM 2. Gen a drected graph Ž V, A., suppose for each P P there are np packets that need to follow path P. There exsts a schedule that brngs all the packets to ther destnatons n tme C at most ' C 2 C PL PL, where P s the cardnalty of the set P and L s the sze of the longest smple path n the graph. In partcular, H as C H 1 C PP n, P and C s the optmal makespan tme. 6. THE PACKET ROUTING PROBLEM WITH PATH SELECTION In ths secton, we consder a more general verson of the packet routng problem n whch the collecton of paths s not gven a pror, but needs to be determned. Gven a drected network Ž V, A., for each par of nodes k, l V there s a number of packets n Žcalled packets of type Ž k, l.. kl that need to be routed from source k to destnaton l va some path n the network. Let P denote the collecton of all types n the network P Ž k, l. : nkl 0. 4 The scheduler s free to choose a path for each packet. The obectve s to construct a schedule Žwhch selects paths and chooses packets to traverse any gven edge. so as to mnmze the total tme t takes to route all the packets to ther destnatons. Srnvasan and Teo 16 provde an algorthm that uses a lnear programmng problem that fnds a schedule wthn a constant factor from the mnmum makespan. In ths secton, we construct a schedule that has makespan C C H ' OŽ C., where C denotes the mnmum makespan tme. Therefore, our algorthm s asymptotcally optmal as the total number of packets ncreases to nfnty. In the prevous sectons we used a dynamc flud relaxaton to construct a schedule. We wll now use a statc multcommodty flow relaxaton of the problem. For any feasble schedule, we defne decson varables x kl to be

18 ASYMPTOTIC SCHEDULINGROUTING ALGORITHMS 313 the total number of type Ž k, l. packets that traverse the edge Ž,.. We can assume wthout loss of generalty that orgn and destnaton nodes are not revsted by any packet. In partcular, xk kl xl kl 0 for all k, l,, V. For each edge Ž,. A the value C x kl Ž 16., Ž k, l. P represents the total tme that edge Ž,. s processng packets. Clearly, the optmal makespan tme C s at least Ž,. A C,. Therefore, the followng multcommodty flow problem provdes a lower bound on C : mnmze C Ž 17. k kl : Ž k,. A subect to x kl n, Ž k, l. P, Ž 18. l kl : Ž, l. A x kl n, Ž k, l. P, Ž 19. x kl xr kl, Ž k, l. P, k, l, : Ž,. A r : Ž, r. A Ž 20. C x kl, Ž,. A, Ž 21., Ž k, l. P C, C, Ž,. A, Ž 22. x kl, C 0, Ž,. A, Ž k, l. P. Ž 23. Eqs. Ž 18. Ž 20. represent conservaton of flow. The obectve functon value of ths lnear programmng problem, denoted also by C, s clearly a lower bound on the optmal makespan tme C. The lnear programmng problem has A P varables and V P A constrants. Thus, t can be solved n polynomal tme even f the nkl are large. We next propose an algorthm that constructs a schedule wth performance close to C when the number of packets s large. PACKET ROUTING SYNCHRONIZATION ALGORITHM 1. Calculate the optmal value of C of the lnear programmng problem Ž Let be a postve real value, to be specfed later. For each nterval m, Ž m 1., m 0, 1,..., C 1, each edge Ž,.

19 314 BERTSIMAS AND GAMARNIK processes kl x kl a Ž 24. C packets of type Ž k, l. P and dles the remanng tme. If there are less kl than a obs of type Ž k, l. avalable, then all avalable packets are processed and the rest of the dedcated tme the edge dles,.e., the edge does not process other packets. 3. At tme C T C, Ž 25. we process all remanng M packets sequentally, takng MP tme unts, where P s the length of the mal smple path n the network. Note that the last step s neffcent, but as we wll see the number of packets left n the network after tme T s small. Let us frst show that the algorthm s feasble. For each edge Ž,. A we have x kl C, kl a. C C Ž k, l. P Ž k, l. P We wll show frst that the number of packets left n the network at tme T s OŽ ' C., by selectng approprately. For each node, let d Ž. denote the outdegree of node PROPOSITION 4. Let ' C P. Then the total number of packets present n the network at tme T defned n Ž. 25 s at most 2 A' C P A P. Proof. We frst show that for each Ž k, l. P, the total number of packets of type Ž k, l. present n node k at tme T s not bgger than dk Ž.Ž C.. Durng each nterval m, Ž m 1., m 0, 1, 2,..., C 1, the number of type Ž k, l. packets processed from node k s

20 ASYMPTOTIC SCHEDULINGROUTING ALGORITHMS 315 equal to / xk kl nkl kl ak 1 dž k., ž C C : Ž k, l. A : Ž k,. A where the second nequalty follows from Ž 18.. Therefore, the number of type Ž k, l. packets present n node k at tme T C s at most / nkl C C nkl dž k. dž k.. ž C We next consder any node V and any type Ž k, l. such that k. Intally, there are no type Ž k, l. packets at node. Let m0 be the largest nteger such that there are no type Ž k, l. packets at node at tme m0. The total number of type Ž k, l. packets that arrve nto durng m, Ž 0 m0 1. s at most x kl kl a. Ž 26. : Ž,. A : Ž,. A C Durng each subsequent nterval m, Ž m 1., m m0 the total number of type Ž k, l. packets that arrve nto durng m, Ž m 1. s also at most : Ž,. A x kl. C The schedule wll allocate at least / x kl x kl kl a 1 dž. ž C C : Ž,. A : Ž,. A : Ž,. A tme unts to type Ž k, l. durng each nterval m, Ž m 1., m m0 1. Thus, durng each subsequent nterval m, Ž m 1., m m0 1, m0 2,... the number of type Ž k, l. packets n node ncreases by at most x kl x kl dž. dž., C C : Ž,. A : Ž,. A where the equalty follows from Ž 20.. Combnng wth Ž 26., the total number of type Ž k, l. packets at node at tme T s at most

21 316 BERTSIMAS AND GAMARNIK kl kl C x C x dž. dž. dž.. C C : Ž,. A : Ž,. A By summng over all nodes V and types Ž k, l. P, we obtan that the total number of packets n the network at tme T s at most ž C x kl d Ž. d Ž. / V Ž k, l. P : Ž,. A C ž / ž / C C, 1 A P Ž,. A C 1 A P A. C We now select to mnmze the quantty above. Namely, set ' C P. Then the total number of packets n the network at tme T s at most 2 A' C P A P. We next apply Proposton 4 to obtan an upper bound on the makespan tme realzed by the algorthm. THEOREM 3. The packet routng synchronzaton algorthm routes all n kl packets wth orgn k and destnaton l n tme C satsfyng ' C C C C C P 2 A C P V A P V, H where C s the optmal makespan tme. In partcular, and H ' ž ' / C C O C, rmh C H C CH 1 C as Ž k, l. P n kl.

22 ASYMPTOTIC SCHEDULINGROUTING ALGORITHMS 317 Proof. From Proposton 4, at tme T we have at most 2 A' C P A P packets n the network. These packets can be routed to ther destnaton Ž. by any trval algorthm as n Step 3 of the algorthm wthn tme at most ž ' / 2 A C P A P P, where P s the length of the mal smple path n the network. Snce P V, we obtan that the total makespan tme of the algorthm s, usng the bound Ž 25., at most ž ' / ' T 2 A C P A P V C C P 2 A C P V A P V. ' 7. CONCLUDING REMARKS We presented algorthms for the ob shop schedulng and packet routng problems that are asymptotcally optmal as the number of obs Žpackets, respectvely. n the system approaches nfnty. Unlke asymptotcally optmal algorthms that rely on probablstc assumptons, the proposed algorthms make no probablstc assumptons, and they are asymptotcally optmal for all nstances wth a large number of obs Ž packets, respectvely.. The algorthm for ob shop schedulng and ts analyss underscores the mportance of the flud control problem and t shows that for nstances of the problem wth a large number of obs, t s the dynamc and not the combnatoral character of the problem that domnates. Interestngly, the dynamc character of the problem that can be captured by the flud control problem has a very smple structure. The algorthm for packet routng underscores the mportance of the dea already observed n other dscrete optmzaton problems that contnuous relaxatons carry nformaton about the dscrete optmzaton problem that can be used to construct near optmal solutons. Fnally, the results of the paper mply that n the lmt of a large number of obs Ž packets. the combnatoral structure of the problems, whch s the essental dffculty of the problems, becomes ncreasngly unmportant as both problems are well approxmated by contnuous relaxatons that are effcently solvable.

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