Stochastic Olie Schedulig with Precedece Costraits Nicole Megow Tark Vredeveld July 15, 2008 Abstract We cosider the preemptive ad o-preemptive problems of schedulig obs with precedece costraits o parallel machies with the obective to miimize the sum of (weighted) completio times. We ivestigate a olie model i which the scheduler lears about a ob whe all its predecessors have completed. For schedulig o a sigle machie, we show matchig lower ad upper bouds of Θ() ad Θ( ) for obs with geeral ad equal weights, respectively. We also derive correspodig results o parallel machies. Our result for arbitrary ob weights holds eve i the more geeral stochastic olie schedulig model where, i additio to the limited iformatio about the ob set, processig times are ucertai. For a large class of processig time distributios, we derive also a improved performace guaratee if weights are equal. 1 Itroductio Oe of the classical schedulig problems that has attracted research for decades is the problem of processig obs with precedece costraits o parallel machies with the obective to miimize the sum of (weighted) completio times. We cosider a stochastic olie versio of this problem where processig times are modelled as radom variables ad the obs become kow to the scheduler olie. I traditioal olie paradigms, i. e., the olie-time ad the olie-list model [21, 24, it is assumed that all data about a request are revealed as soo as the request becomes kow. Iterpreted for a olie schedulig problem with precedece costraits, this meas that wheever a ob arrives, a scheduler lears about its weight ad processig time ad most importatly about ob depedecies. However, these depedecies occur betwee two obs ad it is ot clear which ob gets assiged the iformatio about such a bilateral relatio. Certaily, there are various optios to specify the iformatio that should be revealed at ob arrival. However, we cosider a model i which the momet of uveilig obs ad all their data is desigated by other ob completios: a scheduler lears about the existece of a ob whe all its predecessors have completed their processig. The, its weight, (expected) processig time ad all precedece relatios to predecessors become kow. This model has bee used earlier for olie schedulig to miimize makespa [8, 2. 1.1 Problem defiitio Let J = {1, 2,..., } be a set of obs which must be scheduled o m idetical, parallel machies. Each of the machies ca process at most oe ob at the time, ad the obs ca be executed by ay of the machies. All obs must be scheduled i compliace with the give precedece costraits. Max Plack Istitute for Iformatics, Campus E1 4, 66123 Saarbrücke, Germay. Email: megow@mpiif.mpg.de. Research partially supported by the DFG Research Ceter Matheo Mathematics for key techologies i Berli. Maastricht Uiversity, Departmet of Quatitative Ecoomics, P.O. Box 616, 6200 MD Maastricht, The Netherlads. Email: t.vredeveld@ke.uimaas.l. Research partially supported by METEOR, the Maastricht research school of Ecoomics of Techology ad Orgaizatios. 1
These costraits defie a partial order (J, ) o the set of obs J, where k implies that ob k must ot start processig before has completed. If o precedece costraits are give, the we call the obs idepedet. Each ob must be processed for P uits of time, where P is a o-egative radom variable. By E [ P we deote the expected value of the processig time of ob ad by p a particular realizatio of P. We assume that all radom variables of processig times are stochastically idepedet. We may or may ot allow preemptio. I the preemptive settig, a ob ca be iterrupted at ay time ad resume processig o the same or aother machie at ay time later. I the o-preemptive settig, each ob must ru util its completio oce it has started. Additioally, each ob has associated a o-egative weight w. The goal is to fid a o-aticipatory schedulig policy so as to miimize the total weighted completio time of the obs, w C, i expectatio, where C deotes the completio time of ob. For details o stochastic schedulig policies we refer to Möhrig, Radermacher, ad Weiss [19. I this paper we cosider the olie versio of these stochastic problems i which a ob becomes kow to the scheduler whe all its predecessors k have completed their processig; at this poit i time the weight w ad the probability distributio of the processig time P are revealed. The solutio of such a stochastic olie schedulig problem is a o-aticipatory, olie schedulig policy; for more details see [17. We aim for approximative policies, ad as suggested i [17, we use a geeralized defiitio of approximatio guaratees from traditioal stochastic offlie schedulig by [20. The, a (olie) stochastic policy Π is a ρ-approximatio, for some ρ 1, if for all problem istaces I, E [ Π(I) ρ E [ Opt(I), where E [ Π(I) ad E [ Opt(I) deote the expected values that the policy Π ad a optimal o-aticipatory offlie policy, respectively, achieve o a give istace I. The value ρ is called performace guaratee of policy Π. 1.2 Previous Work The determiistic offlie problem of schedulig obs with precedece costraits to miimize the sum of (weighted) completio times has bee show to be N P-hard [12, 13 eve if there is oly a sigle processor. The preemptive problem is N P-hard already o two machies whe precedece costraits form chais [5 ad i the weighted settig eve whe additioally all obs have uit processig times [5, 27. The determiistic sigle machie problem has attracted research for more tha thirty years ad a vast amout of results has bee obtaied o this problem. Several classes of schedulig algorithms based o differet LP-formulatios are kow that achieve a approximatio ratio of 2 i polyomial time whereas special cases are eve solvable optimally; we refer to [4, 1 for a recet comprehesive overview. For the parallel machie variat of this problem, the curretly best kow approximatio algorithm is by Queyrae, ad Schulz [22 ad yields a approximatio guaratee of 4 2/m. Whe preemptio is allowed, Hall et al. [9 propose a LP-based algorithm that yields a 3 1/m-approximate solutio. Despite the obvious research iterest i the schedulig problem uder cosideratio, literature is very limited whe assumig ucertaity i the problem data. As far as we kow, the oly work that deals with precedece costraits i schedulig uder ucertaity is by Skutella ad Uetz [25, who cosider the stochastic offlie schedulig model. The performace guaratees they prove are fuctios of a parameter that bouds the squared coefficiet of variatio of processig times. Their policies require to solve a liear programmig relaxatio i which all obs must be kow i advace. This approach is ot directly applicable i our olie settig. To the best of our kowledge, we preset the first results o schedulig with precedece costraits to miimize the sum of completio times whe the problem istace is revealed olie. O the other had, research has bee doe o the determiistic olie problem whe the goal is to miimize the makespa. Probably, oe of the earliest publicatios usig the olie paradigm 2
itroduced above is by Feldma et al. [8. They cosider a differet schedulig model, i which parallel obs are processed by more tha oe machie at the same time. Cosiderig obs that are processed by at most oe machie at the time, o algorithm ca achieve a costat competitive ratio. Azar ad Epstei [2 derived a lower boud of Ω( m) for the competitive ratio of ay determiistic or radom olie algorithm that schedules obs, preemptively or ot, o m related machies. This boud matches a upper boud give earlier by Jaffe [11. Fially, there exists relevat work o the determiistic offlie problem of schedulig obs with geeralized precedece costraits, so called Ad/Or-precedece relatios. While ordiary precedece costraits force a ob to wait for the completio of all its predecessors (represeted as a Ad-ode i the correspodig precedece graph), there is a additioal relaxed waitig coditio that allows a ob to start after at least oe of its predecessors has completed (Or-ode). Clearly, ordiary precedece costraits are cotaied as a special case i Ad/Or-precedece costraits. Erlebach, Kääb, ad Möhrig [7 aalyze the performace of a Shortest Processig Time (Spt) Algorithm for the determiistic offlie problem of schedulig to miimize the sum of weighted completio times subect to Ad/Or-precedece costraits o a sigle machie. The classic Spt algorithm schedules obs i o-decreasig order of their processig times. Erlebach et al. s variat cosiders at ay time oly obs that are available for processig accordig to the precedece costraits ad schedules oe with miimal processig time. Thus, it coicides with the olie Spt algorithm that kows of obs oly after all predecessors have fiished. Therefore, the approximatio results traslate ito competitiveess results i our olie settig if all processig times are determiistic. Theorem 1 (Erlebach, Kääb, ad Möhrig [7). The olie versio of the Spt algorithm has a competitive ratio of for the determiistic, o-preemptive problem o a sigle machie. If all obs have equal weights, the Spt is 2 -competitive. I their paper, Erlebach et al. [7 state (without proof) that a parallel machie versio of Spt also yields a approximatio guaratee of for the determiistic parallel machie schedulig problem. This result would also hold i our model restricted to determiistic istaces. 1.3 Our results We provide the first results o olie schedulig with precedece costraits to miimize the (weighted) sum of completio times. We complemet the results i Theorem 1 by Erlebach et al. [7 with matchig lower bouds o the competitive ratio of ay olie algorithm for the sigle machie problem. It follows that a olie Spt algorithm achieves the best possible performace for this problem which is a competitive ratio of. If all weights are equal, the we improve this boud to 2. O the other had, we show that o olie algorithm ca have a competitive ratio less tha 2/3 1. For the correspodig schedulig problem o idetical parallel machies we also provide lower ad upper bouds o the competitive ratio. Here, we leave a gap growig with the umber of machies m. Table 1 gives a summary of the lower ad upper bouds we derived. Notice, that o a sigle machie preemptio does ot lead to a improved schedule sice olie iformatio is revealed oly at the completio of obs. Therefore, the results trasfer immediately to the preemptive settig. This is also true for the bouds o parallel machie schedulig. The worst-case istaces for the lower bouds are costructed o chais i which case preemptio is redudat agai [5 whereas i the aalysis of the algorithm (for the upper boud) we use lower bouds o the optimal offlie value which hold i the preemptive as well as i the o-preemptive settig. We derive most of the performace guaratees above i a much more geeral model for schedulig uder ucertaity, i which additioally to the lack of iformatio about ob arrivals, also processig times of kow obs are ucertai. We show that a olie variat of the Shortest Expected Processig Time (Sept) policy yields the best possible approximatio guaratees,, for the stochastic sigle machie problem, idepedetly of the probability distributio of processig times. If the weights of all obs are equal, the we improve this boud for processig time 3
lower boud upper boud Ad/Or-prec [7 sigle machie, w = 1 2 3 1 2 2 sigle machie, arbitrary w 1 2 parallel machies, w = 1 3 m 1 2m 1 parallel machies, arbitrary w m Table 1: Bouds o the performace guaratee of ay preemptive or o-preemptive olie algorithm [lower boud ad o the performace guaratee of a olie versio of S(e)pt [upper boud for opreemptive stochastic olie schedulig of obs with precedece costraits. Upper bouds for the uweighted settig are give for the special case where processig time distributios obey Var [P E [ P 2. The approximatio results for arbitrary weights also hold true whe ob preemptio is allowed. For determiistic problem istaces this is true also for the improved results for trivial weights. Offlie cosideratios i [7 for problems with Ad/Or-precedece relatios trasfer to our determiistic olie settig ad ispire our ew results; their approximatio guaratees are give i the third colum. distributios with Var[P /E [ P 2 /(m 1) 1. For so called NBUE distributios, that is, distributios for which holds that E [ P t P > t E [ P, for all t 0, ad thus satisfyig Var[P E [ P 2 [10, the competitive ratio is o more tha 2m. Obviously, this holds also for determiistic processig times i which case the Sept policy coicides with the Spt algorithm. Fially, the lower bouds o the competitive ratio for determiistic olie schedulig traslate directly ito lower bouds o our more geeral model, sice olie schedulig istaces with determiistic processig times ca be see as a special case. I stochastic (olie) schedulig preemptio is a powerful tool for dealig with the ucertaity of processig times eve o a sigle machie; see e.g. [28, 18. However, the approximatio guaratee of for schedulig obs with arbitrary weights by Sept holds also i the preemptive schedulig settig. This is ot true for the improved result for obs with equal ob weights because we use a lower boud o the optimum value which does ot hold whe preemptio is allowed. Actually, simple examples show that Sept as well as ay other policy that does ot utilize preemptio may perform arbitrarily bad. Our results are still valid whe cosiderig the more geeral Ad/Or-precedece costraits eve though this is ot focus of our work. Thereby we give a full proof for the approximatio guaratee of for parallel machie schedulig with Ad/Or-precedece costraits which was metioed i [7. Moreover, we improve this result for schedulig istaces i which all obs have equal weights ad give a approximatio guaratee of 2m. 2 Schedulig obs with arbitrary weights For schedulig idepedet obs, good performace guaratees have bee obtaied for olie versios of Smith s classic rule [26, also kow as Weighted Shortest Processig Time Rule (Wspt), its stochastic couterpart, ad various extesios [23, 16, 17, 15. If obs must obey precedece relatios which are revealed after all predecessors have completed, o such variat yields a bouded performace. We give a simple sigle machie example where all obs have eve determiistic processig times. Note, that o a sigle machie o waitig time will reveal ew iformatio o the olie sequece ad o preemptio will improve the schedule. Example 1. Cosider a istace that cosists of the followig three obs. The first ob has processig time p 1 = k 3 ad weight w 1 = 1. Jobs 2 ad 3 must obey the precedece costrait 2 3; they have processig times p 2 = 1 ad p 3 = ε, respectively, ad their weights are w 2 = ε ad w 3 = k. Let k ad ε be such that the ratios of weight over processig time of the two idepedet obs 1 ad 2 fulfill w 1 /p 1 > w 2 /p 2, that meas ε < 1/k. The the olie versio of the Wspt algorithm schedules the obs i icreasig order of their 4
idices, 1, 2, ad 3 achievig a obective value of k 2 + 2k + ε(2k + 1). I cotrast, a optimal schedule has ob 2 beig processed first, followed by 3 ad 1, ad yields thus a value of 2k + 1 + ε(k + 2). For k 3, the ratio of values of the Wspt schedule ad a optimal schedule is larger tha k 2 which is ubouded for icreasig k. Cosider the olie variat of the Shortest Processig Time (Spt) policy that schedules at ay poit i time the ob with the shortest processig time amog the available obs. Eve though it seems couter ituitive to igore kow ob weights, Theorem 1 states that this algorithm yields a competitive ratio that matches the lower boud o the performace guaratee for ay olie algorithm o a sigle machie as we prove later i Theorems 5 ad 6. We exted this result to the more geeral parallel machie settig i which all obs have stochastic processig times, without loosig i the performace guaratee. Cosider the stochastic olie preemptive ad o-preemptive schedulig problems o parallel machies ad the stochastic olie policy that rus the olie variat of the Shortest Expected Processig Time (Sept) policy o oly oe out of m machies. This o-preemptive policy simply igores the m 1 remaiig available machies. We deote this policy by 1-Sept. Lemma 2. The order of obs i a schedule obtaied by 1-Sept is idepedet of the realizatio of processig times. Proof. We claim that for ay two realizatios of processig times, at the completio of ob the same set of obs is available, for ay J. This implies the lemma, as 1-Sept chooses the ob to process oly based o the set of available obs ad the expected processig times of these obs. To see the claim, cosider two realizatios of processig times. We assume that obs are idexed i order i which they are processed i the first realizatio. First ote that whe o ob has bee processed, obviously the same set of obs is available to 1-Sept i both realizatios. Suppose the claim is true up to ob. As 1-Sept chooses ob + 1 to be processed after ob i the first realizatio ad i the secod realizatio the same set of obs is available to 1-Sept, the policy will also choose ob + 1 to be processed for the secod realizatio. Hece, at the completio of ob + 1 the same set of obs will be set free to 1-Sept i the first as i the secod realizatio. Let E [ C Π deote the expected completio time of a ob i the schedule obtaied by policy Π. Ispired by [7, we defie a stochastic versio of the threshold ξ Π of a ob for policy Π as the maximum expected processig time of a ob that fiishes i expectatio o later tha. More formally, ξ Π Thresholds have a useful property. = max k J { E [ P k E [ C Π k E [ C Π }. Lemma 3 (Threshold-Lemma). Let Π be a feasible policy for the stochastic preemptive or opreemptive schedulig problem o parallel machies. The for ay ob J with threshold ξ Π holds ξ ξ Π. Proof. For ay o-aticipatory policy Π, we have that ξ Π E [ P sice E [ P E [ C Π. If ξ = E [ P, the the lemma holds. Suppose that ξ > E [ P. The there exists a ob k that was completed before ob by 1-Sept, ad that has expected processig time E [ P k = ξ > E [ P. As 1-Sept chooses the ob with smallest expected processig time, we kow by Lemma 2 that i ay realizatio of processig times, ob caot be available to 1-Sept whe ob k is started to be processed. Hece, k must be a predecessor of. Thus, ay policy processes ob k before, from which follows ξ Π E [ P k = ξ. Now, we ca establish a performace guaratee for 1-Sept. Theorem 4. The 1-Sept policy that utilizes oly oe machie is a -approximatio for the stochastic olie schedulig problem o parallel machies with ad without preemptio. 5
Proof. Let obs be idexed i their order i the 1-Sept schedule. Recall from Lemma 2 that the order of obs i the 1-Sept schedule is idepedet of the realizatio of processig times. The k < implies E [ C [ k < E C. Sice there is o idle time, the expected completio time E [ C of a ob i the 1-Sept schedule is E [ C = k=1 E [ P k ξ. (1) With the Threshold-Lemma 3 ad the fact that ξ Π E [ C Π holds by defiitio for ay policy Π thus, also for a optimal policy Opt we coclude from iequality (1) E [ C Sept ξ Opt E [ C Opt. Weighted summatio over all obs J proves the theorem. I the followig we show that o olie algorithm usig m machies ca have a competitive ratio of less tha ( 1)/m. Thus the aalysis of the simple 1-Sept policy usig oe machie is tight if there is ust oe machie available, whereas i geeral it leaves a gap i the order of m. The lower boud is achieved eve whe the processig times are give determiistically. By defiitio of the model, these bouds carry over to the more geeral stochastic olie schedulig model. Theorem 5. No preemptive or o-preemptive determiistic olie algorithm ca achieve a competitive ratio less tha ( 1)/m for schedulig with precedece costraits o ay umber of machies m. Proof. Cosider the followig istace that cosists of obs ad assume, w.l.o.g., that 1 is a multiple of the umber of machies m. We have 1 idepedet obs 1, 2,..., 1 with weights w = 0 ad uit processig time. Suppose, that the olie algorithm chooses the ob l to be scheduled such that it completes as the last ob. The, we have oe fial ob i the istace with l as its predecessor ad with processig time zero ad weight 1. Clearly, the olie algorithm ca schedule the highly weighted last ob oly as the fial ob, achievig a schedule with value ( 1)/m. I cotrast, a offlie algorithm would choose ob l as oe of the m first obs to be processed, followed by the highly weighted ob. This yields value of 1. Thus, the ratio betwee both value is ( 1)/m. Addig a radomizig igrediet to the istace above, we exted the result to a lower boud for ay radomized olie algorithm. Here, we make use of Yao s priciple [29 which states that a lower boud o the expected competitive ratio of ay determiistic olie algorithm o a appropriate iput distributio also lower bouds the competitive ratio of ay radomized olie algorithm agaist a oblivious adversary. Sice this is the weakest type of adversaries, these lower bouds hold agaist ay other adversary with more power as well; see, e.g., Be-David et al. [3. Theorem 6. No preemptive or o-preemptive radomized olie algorithm ca achieve a competitive ratio less tha ( 1)/(m + 1) for schedulig with precedece costraits o ay umber of machies m. Proof. Cosider the istace i the previous proof. Whe playig agaist a radomized algorithm, a oblivious adversary does ot kow which will be the last completig ob l amog the idepedet obs. Therefore, we modify the istace by addig m radom precedece relatios to each ob 1,..., 1 with the same probability. Let k := ( 1)/m be a itegral umber. The, ay set of m obs (excludig ) has probability 1/ ( ) km m for beig the set of predecessors of. 6
Clearly, the optimal offlie solutio has still value 1. Now, cosider ay determiistic olie algorithm Alg. E [ Alg = E [ C Alg = k k 1 1 Pr [ C Alg i Pr [ i C Alg < i = k k < i + 1 + k Pr [ k C Alg = k Pr [ C Alg i Pr [ C Alg < i. (2) Moreover, we give Alg the advatage that it schedules ob as soo as all its predecessors have completed. The umber of obs that ca be completed strictly before a fixed (itegral) poit i time i + 1 is at most im. Therefore, the probability that all m radom predecessors of ob complete before a fixed time 2 i + 1 k is Pr [ C Alg < i + 1 ad Pr [ C Alg boud combied with (2) ad (3) yields ( im m ) ( km m ) = < 1 = 0. Now we use the boud k 1 (im)! ((i 1)m)! (km)! ((k 1)m)! ( i ) m k k m+1 k 1 ( ) m ( i E [ Alg k k 1 1 ) = 1 k m + 1 m + 1. ( ) m i, (3) k which we prove below. This By Yao s priciple [29 this gives the desired lower boud o the competitive ratio of ay olie algorithm. It is left to show k 1 ( i ) m k k m+1. We prove the boud by iductio o k; it is certaily true for k = 2. k ( ) m k 1 i ( ) m ( ) m ( ) m ( ) [ m k 1 i k k k ( ) m i = + = + 1 k + 1 k k + 1 k + 1 k + 1 k ( ) m [ k k k + 1 m + 1 + 1 = km+1 + (m + 1)k m (k + 1) m (4) (m + 1) (k + 1) m+1 (k + 1) m (m + 1) = k + 1 m + 1. (5) Iequality (4) follows from the iductio hypothesis. The secod iequality (5) follows from (k + 1) = which cocludes the iductive step. i=0 2.1 Schedulig Jobs with equal Weights ( ) k i > k + k 1, i If all ob weights are equal the, ituitively, the 1-Sept policy should perform better tha i the geeral settig. We show a performace boud for o-preemptive schedulig which improves the -approximatio i Theorem 4 for a large class of problem istaces; i particular, for istaces with NBUE distributed processig times, which iclude determiistic istaces, it is 2m. We achieve this boud by extedig ideas of Erlebach et al. [7 to the stochastic olie parallelmachie settig. Moreover, we apply the followig lower boud o the expected optimal value for the relaxed problem without precedece costraits give by Möhrig, Schulz, ad Uetz [20. 7
Lemma 7 (Möhrig et al. [20). Cosider the stochastic schedulig problem o parallel machies to miimize the expected total weighted completio time i which all obs are available for processig from the begiig. Assume that the obs are idexed i o-decreasig order of expected processig times E [ P. The, a optimal policy Opt yields a value E [ C Opt k=1 E [ P k m (m 1)( 1) 2m E [ P, where bouds the squared coefficiet of variatio of the processig times, that is, Var[P E [ P 2 for all obs = 1,..., ad some 0. Theorem 8. The 1-Sept policy achieves a approximatio guaratee of ρ = 1 2 (m 1)( 1) + 1 2 [(m 1)( 1)2 + 8m, with Var[P E [ P 2 for ay istace of the o-preemptive stochastic olie problem o parallel machies. Proof. Cosider a 1-Sept schedule. Let α > (m 1)( 1)/ be a parameter that will be specified later. For otatioal coveiece, we defie for each ob the set of obs completed o later tha ob i the 1-Sept schedule as B() = {k J E [ C [ k E C }. Let x be the last ob i the 1-Sept schedule such that all obs scheduled before this ob have a expected processig time of at most E [ /(α ), that is, { x := arg max E [ C J ad E [ P k E [ C } α for all k B(). This desigated ob x is used to partitio the set of obs ito two disuctive subsets: J deotes the set of obs that complete before x i the 1-Sept schedule, that is, J = { J E [ C E [ }, ad J > cosists of the remaiig obs J \J. Obviously, the expected completio time of ob x is E [ be expressed as J = J E [ P. Now, the expected value of the Sept schedule ca E [ C = E J [ C + E J > [ C. We boud the expected completio times of obs of both ob sets separately. To boud the cotributio of the obs i J, assume that J. Let Opt be a optimal policy for all obs J ad Opt a optimal policy that schedules oly the obs i J. Clearly, [ [ E C Opt E C Opt. (6) J J By igorig the release dates, we ca use Lemma 7. Assumig that the obs i J are idexed, 1,..., J, i o-decreasig order of their expected processig times, we obtai: J E [ C Opt J k=1 E [ P k m (m 1)( 1) 2m J E [ P, (7) where Var[P E [ P 2 for all obs J ad some 0. We claim that J k=1 E [ P k is bouded from below by α 2 E [. 8
To see this claim, ote that J k=1 E [ P k = k J ( J k + 1) E [ P k. This value ca ot be less tha the miimum of this value over all possible expected processig times for obs i J satisfyig J E [ P = E [ ad E P E [ /(α ). This miimum is obviously obtaied by settig E [ P = E [ /(α ) for = J b + 1,..., J, E [ P = (1 b/(α ))E [, for = J b, ad E [ P = 0 for all other, where b = α. This proves the claim ad with J E [ P = E [ we have J E [ C Opt α 2m E [ (m 1)( 1) 2m = α (m 1)( 1) 2m E [. E [ With this estimate of the relevat portio of the expected optimal value we ca boud the value achieved by the 1-Sept policy J E [ C if ad oly if α > (m 1)( 1). [ E C 2 m x α (m 1)( 1) J E [ C Opt, (8) Cosider ow obs i the remaiig ob set J > ; by defiitio, there exists for each ob J > a ob k that completes i 1-Sept earlier tha ad has processig time E [ P k > E [ C /(α ). We coclude from this fact ad the Threshold-Lemma 3 (icludig the otio of the threshold ξ S) that for all J > holds E [ C Opt ξ Opt ξ E [ P k > E [ C α. Summatio over all obs J > yields a boud o the completio times i the 1-Sept schedule, [ E C [ α E C Opt. J > J > Fially, combiatio with Equality (8) yields the boud E [ { C 2 m max α (m 1)( 1), α J } J E [ C Opt. The performace boud is miimized whe choosig the parameter α := ((m 1)( 1) + [(m 1)( 1)2 + 8m)/(2 ) which gives the desired approximatio guaratee ρ = 1 2 (m 1)( 1) + 1 2 [(m 1)( 1)2 + 8m. Observe that the optimal choice of α fulfills the coditio α > (m 1)( 1) i equality (8). I cotrast to the previous, more geeral approximatio guaratee of value i Theorem 4, this result depeds o the variace of processig times. I particular, the performace guaratee ρ grows with the parameter. However, for istaces with distributios of small relative variace, 9
this boud improves o the -approximatio for the geeral weighted problem i Theorem 4. More precisely, for istaces with a upper boud o the squared coefficiet of processig times m 1 1 the performace guaratee ρ i Theorem 8 is at most. Moreover, this theorem leads immediately to the followig result for a restricted class of probability distributios the NBUE distributios, which imply 1 [10. Corollary 9. If all obs have processig times that follow a NBUE distributio, that is, 1, the 1-Sept policy that utilizes oly oe machie is a 2m-approximatio for the o-preemptive stochastic olie schedulig problem o parallel machies with equal ob weights. This icludes determiistic istaces. It follows from the aalysis that the result holds also if Ad/Or-precedece costraits are preset. Thus, we improve the approximatio factor of for the offlie schedulig problem where obs have equal weights ad processig times are determiistic i [7 eve though we cosider a more geeral model. Corollary 10. The 1-Spt algorithm that utilizes oly oe machie is a 2m-approximatio for the schedulig problem o parallel machies with Ad/Or-precedece costraits with equal ob weights. For preemptive schedulig of obs with trivial weights, the aalysis of 1-Sept above does ot improve the geeral -approximatio of Theorem 4. The reaso is, that the lower boud o the expected optimum value i Lemma 7 does ot hold whe obs are allowed to be preempted. I fact, the Sept policy which does ot preempt a ob performs arbitrarily bad i the preemptive stochastic schedulig eviromet as simple examples show. However, for determiistic istaces without precedece costraits preemptio is redudat; see McNaughto [14. Therefore, the determiistic versio of the lower boud o the optimum value i Lemma 7 holds also for preemptive schedulig. I fact, i that case it coicides with the classical result by Eastma, Eve, ad Isaacs [6. Now, the aalysis of 1-Spt as i the proof of Theorem 8 holds true also for determiistic olie schedulig with precedece costraits. Choosig α = 2m leads to the corollary. Corollary 11. The 1-Spt algorithm is 2m-competitive for the determiistic preemptive olie schedulig problem o parallel machies, with the equal ob weights Fially, we complemet the ew improved performace guaratees by a lower boud which leaves a gap i the order of m. Theorem 12. The competitive ratio of ay preemptive ad o-preemptive, determiistic olie algorithm for schedulig obs with precedece costraits has a lower boud of 2 3 /m 1/3 ad ay radomized olie algorithm has a lower boud of 2 3 m/(m + 1)2 1/3 o the competitive ratio. Proof. We have mk idepedet obs with processig times p = 1 for all = 1,..., mk where k >> m. Moreover, there are mk 2 mk obs that have legth 0 ad which must obey precedece costraits that form oe log chai mk + 1 mk + 2.... Let l {1,..., mk} be the ob to be scheduled last by the olie algorithm amog the idepedet obs. This ob l is a predecessor of mk + 1, the first ob of the chai. Note, that a olie algorithm Alg caot start the ob chai with mk 2 mk obs earlier tha time k. Alg yields a schedule of value Alg m k i + (mk 2 mk)k = 1 2 mk (1 + (2k 1) k) 1 2 mk2 (2k 1). (9) 10
I cotrast, a optimal offlie algorithm Opt kows the sequece i advace. By processig ob l at time 0 ad startig the chai of zero legth obs at time 1, it ca achieve a obective value Opt m k i + mk 2 mk = 1 2 mk (3k 1) 3 2 mk2. (10) The ratio of the bouds i (9) ad (10) combied with the umber of obs, = mk 2, gives the lower boud o the competitive ratio of ay determiistic olie algorithm. Alg Opt 2k 1 3 2 3 m 1 3. We achieve almost the same boud for radomized olie algorithms by radomizig the determiistic istace i the same way as i the proof of Theorem 6. We cosider the istace above ad replace the precedece costrait l mk + 1 by m radom precedece costraits. That meas, ay set of m obs precedes ob mk + 1, the first ob of the ob chai, with the same probability. The with the same argumets as i the Theorem 6, the expected completio time of the obs i the chai is k(1 1/(m + 1)) which gives the result. 3 Coclusio We preseted first results for (stochastic) olie schedulig with precedece costraits to miimize the (expected) sum of weighted completio times. The bouds for the sigle machie settig are tight whereas i the parallel machie settig we leave a gap of O(m). For closig this gap, it is ot sufficiet to ru simply a parallel versio of the Sept or Spt algorithm. The followig determiistic istace shows that the competitive ratio of Spt is also at least i the order of for the problem with arbitrary ob weights. Example 2. Cosider a istace with obs ad m machies. Job 1 has processig time ε ad weight 0. Jobs 2,..., m have uit processig time ad weight 0, ad they ca start oly after ob 1 has completed. Job m + 1 has m obs out of 2,..., m as direct predecessors ad w m+1 = 1 ad p m+1 = 0. The adversary chooses the m precedece relatios such that these are the latest fiishig obs. Fially there are m 1 large idepedet obs with processig times m ad weight 0. Whe ε teds to 0, the parallel olie versio of Spt has value m whereas a optimal solutio has value 1. A slightly modified istace i which the ob with high weight is substituted by a log chai of obs with zero processig time shows that i the schedulig settig where all obs have equal weights, the parallel Spt algorithm yields a competitive ratio Ω( ). Our fial remark cocers preemptive stochastic olie schedulig. We metioed that simple examples show that the Sept policy ad actually ay other policy that does ot utilize preemptio ca perform arbitrarily bad whe obs are allowed to be iterrupted. Therefore other policies must be cosidered. But still Lemma 7, which provided oe of the lower bouds that were used i the aalysis, does ot do so i the preemptive schedulig eviromet. However, [18 provides a geeral lower boud for the preemptive stochastic olie schedulig problem o parallel machies. Applied to a variat of the preemptive olie policy Geeralized Gittis Idex Policy also proposed i [18 with a modificatio such that it respects precedece costraits, this could lead to a improved approximatio result. Ackowledgemets. We thak Jiří Sgall for poitig out that the radomized lower bouds i previous versios of Theorems 6 ad 12 could be stregtheed. 11
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