Scheduling parallel jobs to minimize the makespan

Transcription

1 J Sched (006) 9: DOI /s Schedling parallel jobs to minimize the makespan Berit Johannes C Springer Science + Bsiness Media, LLC 006 Abstract We consider the NP-hard problem of schedling parallel jobs with release dates on identical parallel machines to minimize the makespan. A parallel job reqires simltaneosly a prespecified, job-dependent nmber of machines when being processed. We prove that the makespan of any nonpreemptive list-schedle is within a factor of of the optimal preemptive makespan. This gives the best-known approximation algorithms for both the preemptive and the nonpreemptive variant of the problem. We also show that no list-schedling algorithm can achieve a better performance garantee than for the nonpreemptive problem, no matter which priority list is chosen. List-schedling also works in the online setting where jobs arrive over time and the length of a job becomes known only when it completes; it therefore yields a deterministic online algorithm with competitive ratio as well. In addition, we consider a different online model in which jobs arrive one by one and need to be schedled before the next job becomes known. We show that no list-schedling algorithm has a constant competitive ratio. Still, we present the first online algorithm for schedling parallel jobs with a constant competitive ratio in this context. We also prove a new information-theoretic lower bond of.5 for the competitive ratio of any deterministic online algorithm for this model. Moreover, we show that 6/5 is a lower bond for the competitive ratio of any deterministic online algorithm of the preemptive version of the model jobs arriving over time. Keywords mltiprocessor schedling, parallel jobs, approximation algorithms, online algorithms, list schedling, release dates 1. Introdction Schedling parallel jobs has recently gained considerable attention. The papers (Amora et al., 1997; B lażewicz, Drabowski, and Wȩglarz, 1986; Chen and Miranda, 1999; D and Leng, 1989; Feldmann, Sgall, and Teng, 1994; Jansen and Porkolab, 1999, 000; Ldwig and Tiwari, 1994; M alem and Feitelson, 1999; Trek, Wolf, and Y, 199) are jst a small sample of work in this B. Johannes ( ) Institte of Theoretical Compter Science, ETH Zrich, 809 Zrich, Switzerland beritj@inf.ethz.ch

2 434 J Sched (006) 9: area. In fact, the stdy of compter architectres with parallel processors has prompted the design and analysis of good algorithms for schedling parallel jobs. Parallel jobs are often alternatively called mltiprocessor tasks. We refer to Feitelson et al. (1997) for a comprehensive introdction to the schedling aspect of parallel compting, and to Drozdowski (1996) for an overview of reslts on comptational complexity and approximation algorithms; (Feitelson, 1997) contains a collection of frther references. We discss the following class of schedling problems. We are given m identical parallel machines and a set of n independent parallel jobs j = 1,...,n. Each job j has a positive integer processing time p j, which we also call its length. Job j simltaneosly reqires m j m machines at each point in time it is in process. The positive integer m j is also called the width of job j. Note that we assme that m j is part of the inpt; in particlar, jobs are nonmalleable. Moreover, each job j has a nonnegative integer release date r j at which it becomes available for processing. Every machine can process at most one job at a time. The objective is to find a feasible schedle of minimal completion time; that is, the makespan is to be minimized. Whenever a machine falls idle or a job is released, a list-schedling algorithm schedles from a given priority list the first job that is already released and that does not reqire more machines than are available. We consider both the preemptive and the nonpreemptive variants of this problem. If preemptions are allowed, a job may be interrpted at any point in time and contined later, possibly on a different set of machines. Hence, in preemptive list-schedling, jobs with lower priority can be preempted by jobs with higher priority. We denote the preemptive problem by P m j, r j, pmtn C max and the nonpreemptive problem by P m j, r j C max, following the three-field notation introdced in Graham et al. (1979). We note that some athors se size j instead of m j to refer to parallel jobs of the natre described before; see, e.g., Drozdowski (1996). Occasionally, we shall also refer to problems featring nonparallel jobs only; they arise as the special case in which m j = 1 for all j = 1,...,n. The associated short-hand notation is P r j, pmtn C max and P r j C max, respectively, and, in the absence of nontrivial release dates, P pmtn C max and P C max. Since both the parallel job-schedling problems considered here are NP-hard, we are interested in approximation algorithms. An α-approximation algorithm for a minimization problem is a polynomial-time algorithm that constrcts for any instance a soltion of vale at most α times the optimal vale; α is also called the performance garantee of the algorithm. Motivated by the application context of parallel job schedling, we also stdy (deterministic) online algorithms. While we shall consider different online scenarios, we always measre the qality of an online algorithm in terms of its competitive ratio. An online algorithm is α-competitive if it prodces for any instance a soltion of vale at most α times the vale of an offline optimm. While preemptive schedling of parallel jobs is NP-hard, even in the absence of release dates (Drozdowski, 1994), preemptive schedling of nonparallel jobs (that is, m j = 1) can be solved in polynomial time (McNaghton, 1959; Horn, 1974). For the strongly NP-hard problem of schedling nonparallel jobs withot release dates, P C max, Graham (1966) showed that every list-schedling algorithm is a ( 1/m)-approximation algorithm. Gsfield (1984) and Hall and Shmoys (1989) observed that Graham s reslt holds for nonparallel jobs with release dates, P r j C max, as well. If nonparallel jobs are schedled in nonincreasing order of their lengths, listschedling is a ((4m 1)/3m)-approximation algorithm for P C max, see (Graham, 1969), and a 3/-approximation algorithm for P r j C max, see Chen and Vestjens (1997). In contrast thereto, we provide an instance of parallel jobs with release dates that is simltaneosly bad for all possible priority lists; that is, no variant of list-schedling can achieve a better performance garantee than for P m j, r j C max. Hochbam and Shmoys (1987) and Hall and Shmoys (1989) gave polynomial-time approximation schemes for the problems P C max and P r j C max, respectively. In contrast, there is no approximation algorithm with a performance garantee better than 3/ for P m j C max, nless P=NP,aswepoint ot in Section.

3 J Sched (006) 9: It follows from the work of Garey and Graham (1975) on project schedling with resorce constraints that list-schedling has performance garantee for schedling parallel jobs withot release dates, P m j C max. Trek, Wolf, and Y (199) presented a direct, simplified proof of this reslt. Feldmann, Sgall, and Teng (1994) observed that the length of a nonpreemptive listschedle is actally at most 1/m times the optimal preemptive makespan, if no release dates are present. This implies that there is a (3 1/m)-approximation algorithm for both, P m j, r j, pmtn C max and P m j, r j C max, since a delay of the start times of all jobs by the maximal release date increases the performance garantee by at most 1 (see also M alem and Feitelson, 1999). In Section 4, we show that any nonpreemptive list-schedling algorithm prodces a schedle with makespan at most twice the makespan of an optimal preemptive schedle. This leads at the same time to a -approximation algorithm for both P m j, r j C max and P m j, r j, pmtn C max. We show that in contrast to schedling with nonparallel jobs, this approximation ratio cannot be improved pon by applying different priority strategies. Independently of or work, first presented in Johannes (001), Naroska and Schwiegelshohn (00) also showed that list-schedling is a -approximation algorithm for these problems. While Naroska and Schwiegelshohn prove this reslt by indction over the nmber of different release dates, we directly compare the strctre of a list-schedle with that of an optimal preemptive schedle, which reslts in a somewhat more perspicos proof. As a matter of fact, we se or techniqes to obtain additional reslts. For P m j, r j, pmtn C max, we show that preemptive m j -list-schedling, that is, preemptive list-schedling where jobs are in order of nonincreasing widths, generates a schedle that is at most 1/m times as long as an optimal preemptive schedle. Althogh this reslt is essentially otperformed by the main reslt, we will present it anyway in Section 3, becase its proof is short, and it provides a first idea of the techniqes sed in the main proof. While it is not difficlt to see that list-schedling algorithms that do not depend on job characteristics (e.g., the length) also work in the online settings nknown rnning times and jobs arriving over time with corresponding competitive ratios, no list-schedling algorithm has a constant competitive ratio in the context of schedling jobs one by one. In Section 5, we present the first online algorithm with a constant competitive ratio for schedling parallel jobs one by one. We also show that no deterministic online algorithm has a competitive ratio smaller than.5. For the preemptive version of the online model jobs arriving over time, we show that no deterministic online algorithm can achieve a better competitive ratio than 6/5. So far, no lower bond is known for this problem. A problem closely related to P m j C max is strip packing, sometimes also called orthogonal packing in two dimensions. In contrast to the model considered here, machines assigned to a job need to be contigos in a soltion to the strip-packing problem. Trek, Wolf, and Y (199) pointed ot that there is indeed an advantage to sing noncontigos machine assignments. From a parallel compter architectre perspective, P m j C max corresponds to schedling on a PRAM, while strip packing is eqivalent to schedling on a linear array of processors (Ldwig and Tiwari, 1994). The strip-packing problem was first posed by Baker, Coffman, and Rivest (1980). Varios athors proposed approximation algorithms with performance garantees 3 (Baker, Coffman, and Rivest, 1980; Coffman et al., 1980; Golan, 1981),.7 (Coffman et al., 1980),.5 (Sleator, 1980), and (Steinberg, 1997), respectively. Kenyon and Remila (1996) gave an asymptotic flly polynomial-time approximation scheme when m is fixed. If no network topology is specified (PRAM) and the nmber m of machines is fixed, Pm m j, pmtn C max can be solved as a linear programming problem in polynomial time (B lażewicz, Drabowski, and Wȩglarz, 1986). Jansen and Porkolab (000) presented an algorithm with rnning time O(n) + poly(m) ifm is not fixed, thereby showing that it cannot be

4 436 J Sched (006) 9: strongly NP-hard, nless P = NP. They also gave a polynomial-time approximation scheme for the nonpreemptive problem with a fixed nmber of machines, Pm m j C max, see (Jansen and Porkolab, 1999). D and Leng (1989) showed that this problem is strongly NP-hard for any fixed m 5.. Preliminaries In this section, we briefly discss the limits of approximability for some parallel job-schedling problems as well as the rnning time of list-schedling algorithms. The problem of nonpreemptively schedling parallel jobs of length 1 to minimize the makespan, P m j, p j = 1 C max, is eqivalent to the strongly NP-hard BIN PACKING problem, as was observed by B lażewicz, Drabowski, and Wȩglarz (1986). It follows that there is no approximation algorithm for schedling parallel jobs to minimize the makespan with a performance garantee better than 3/, nless P = NP. Moreover, in contrast to BIN PACKING, item sizes (i.e., lengths of jobs) can be scaled. Therefore, we can state the following theorem. Theorem.1. There is no polynomial-time algorithm that prodces for every instance of P m j C max a schedle with makespan at most αcmax + β with α<3/ and β being a polynomial in n, nless P = NP. Here, Cmax denotes the optimal makespan. Proof: Let s consider the NP-complete decision version of the problem P C max : Is it possible to schedle the n nonparallel jobs of a given instance I 1 on two identical parallel machines sch that the makespan is at most T? We transform I 1 into an instance I of the problem P m j, p j = p C max by introdcing T machines and n jobs. Each job i with length p i in I 1 will be transformed into a job j in I with width m j = p i and length p for some p > 0. Hence, a feasible schedle for the instance I has a makespan of at most p if and only if I 1 is a yes-instance. If there was a polynomial-time algorithm A for the problem P m j, p j = p C max with Cmax A αc max + β with α<3/ and β being a polynomial in n, we wold have the means to recognize any yes-instance of P C max in polynomial time by choosing p > β 3 α. Li (1999) gave a polynomial-time algorithm with asymptotic performance garantee 31/18 for P m j C max. Drozdowski (1994) observed that if all jobs have length 1, then the existence of a preemptive schedle of length implies the existence of a nonpreemptive schedle of length as well. Ths, preemptive schedling of parallel jobs with length 1 and therefore P m j, pmtn C max do not have a better than 3/-approximation algorithm either, nless P = NP. It is well known that list-schedling algorithms for classic (i.e., nonparallel) job-schedling problems can easily be implemented in polynomial time. However, one needs to be more carefl for parallel job-schedling problems becase we may not assme that the nmber m of machines is at most the nmber n of jobs. Therefore, any polynomial-time schedling algorithm that otpts job-machine assignments has to find a compact way of encoding this otpt since not m, bt log m is part of the inpt size (as machines are identical). The following lemma ensres that there is always an assignment of the jobs to the machines sch that no job is split over too many machines, and we may therefore safely restrict orselves to algorithms that specify the starting times of jobs. Lemma.. Let S be a feasible schedle for an instance of the problem P m j, r j C max, given by job starting times. Then, there is a polynomial-time algorithm that comptes a feasible assignment

5 J Sched (006) 9: of jobs to machines (withot changing the starting times of jobs). In particlar, the job-machine assignment can be represented with polynomial size. Proof: Let t 1 < t < < t z be the different starting times of the jobs in S. Let i 1 < i < < i mt be the idle machines in S at time t and let j 1, j,..., j nt be the jobs that start at time t, t = t 1,...,t z. We assign the jobs j 1,..., j nt to the machines i 1,...,i mt in the following way. Job j 1 is assigned to the first m j1 machines in {i 1,...,i mt }, job j is assigned to the next m j machines in {i 1,...,i mt }, and so on. Let f t be the nmber of machine-intervals that exist or are created at time t, for t = 1,...,z. A machine-interval is a set of consective machines (in the order 1,,...,m) of maximal cardinality sch that all machines in this set are processing the same job. In particlar, a machine-interval is completely specified by (the index of) its first and its last machine. Hence, we will otpt for every t {t 1,...,t z } the set of machine-intervals with the corresponding jobs. At time t 1 = 0, n t1 jobs start. Therefore, f t1 n t For every additional split of a machineinterval at time t k+1, we need a new job that starts at time t k+1. Hence, for all 1 k z we have f tz 1 + t z t=t 1 n t = n + 1. We conclde that for every point in time t = t 1,...,t z there are no more than n + 1 machine-intervals and ths the size of the otpt is polynomial in the nmber of jobs. In particlar, the nonpreemptive list-schedling algorithm described in Section 1 comptes a feasible schedle (inclding job-machine assignments) in polynomial time. For preemptive list-schedling, it is not necessary to invoke Lemma.. At any event (release date or completion time of a job,) one can simply interrpt the processing of all jobs and then newly assign each job (highest priorities first) to consective machines. Hence, the total nmber of preemptions is bonded from above by n. Ths, the job-machine assignments can again be compactly described. Note also that preemptive list-schedling only preempts at integer points in time, whereas an optimal preemptive schedle may preempt at any time. 3. Preemptive list-schedling of parallel jobs with release dates We now present a -approximation algorithm for schedling parallel jobs with release dates when preemptions are allowed. More specifically, we prove that preemptive m j -list-schedling delivers for all instances of P m j, r j, pmtn C max a schedle that is at most 1/m times as long as an optimal preemptive schedle. The algorithm works as follows. At every decision point (i.e., release date or completion time) all crrently rnning jobs are preempted. Then, the already released, bt not yet completed jobs are considered in order of nonincreasing widths m j, and as many of them are greedily assigned to the machines as feasibly possible. Theorem 3.1. Consider an instance of P m j, r j, pmtn C max. The length of the schedle constrcted by the preemptive m j -list-schedling algorithm is at most 1/m times the optimal makespan C max. Proof: Let e be the job that determines the makespan in the preemptive list-schedle, and let C e be its completion time. We divide the time horizon (0, C e ] into intervals (t, t + 1] of length one (t = 0, 1,...,C e 1). We call sch an interval time-slot. We distingish two cases.

6 438 J Sched (006) 9: Case 1. m e > m/. Let r z be the minimal point in time after which every time-slot contains a wide job (i.e., a job j with m j > m/) in the m j -list-schedle. It follows from the definition of m j -list-schedling that r z is the minimal release date of all jobs with m j > m/ that are schedled in this last contigos block of time-slots with wide jobs. Hence, r z pls the total processing time of wide jobs in this block is a lower bond for Cmax. Ths, C e = r z + (C e r z ) Cmax, and the list-schedle is optimal in this case. Case. m e m/. Sppose C e > ( 1/m)C max. Let r z be the minimal point in time after which every time-slot contains a job j with m j m e in the list-schedle. By definition, r z is the minimal release date of all jobs with m j m e that are schedled in this last contigos block of time-slots containing jobs at least as wide as e. Conseqently, all these jobs have to be schedled after r z in the optimal schedle as well. Ths, the total load of jobs to be processed in a space of (C max r z)m is strictly more than (r e r z )m e + (C max ( 1/m) r e p e )(m m e + 1) + p e m e. The first term in the smmation reslts from jobs j with m j m e schedled prior to the time at which e is released, and from the definition of r z. The second term acconts for those time-slots after the release date of job e in which e is not schedled. Finally, the third term is the load prodced by e itself. A simple calclation shows that this load is too mch to fit into the space provided by an optimal schedle between r z and C max. It is not hard to show that this approximation bond of 1/m for (preemptive) list-schedling of (parallel) jobs to minimize makespan is tight. The following instance has already been proposed by Graham (1966) to show that list-schedling for nonparallel jobs withot release dates has no better performance garantee than 1/m. Example 3.. Consider an instance with m m + 1 nit-width jobs, each one with length 1, except for the last job in the list, which has length m. The reslting schedle has no preemptions and is of length m 1 + m = m 1, while the optimal schedle has makespan m. 4. Nonpreemptive list-schedling of parallel jobs with release dates The following reslt gives a niversal performance garantee of for all nonpreemptive listschedling algorithms, regardless of which priority list is sed. It holds for both the preemptive and the nonpreemptive version of the problem. Theorem 4.1. For every instance of the problems P m j, r j, (pmtn)/c max, the length of the schedle constrcted by any nonpreemptive list-schedling algorithm is less than twice the optimal preemptive makespan. Proof: To prove the reslt, we directly compare the nonpreemptive list-schedle LS ofagiven instance with a preemptive optimal schedle OPT. Let Cmax LS be the makespan of LS and let COPT max be the makespan of OPT. We partition the time horizon into periods of the same length, sch that all jobs are started, preempted, restarted, and completed at the beginning or the end of a period. By scaling, we may assme that sch a period, which we call time-slot s, starts at time s 1 and ends at time s (for some nonnegative integer s). The load of a time-slot is the nmber of machines that are bsy dring this period. We define the load of a set of time-slots as the sm

7 J Sched (006) 9: of the loads in the individal time-slots. We refer to the part of a job that is schedled in one time-slot as a slice. Let r 0 < r 1 < < r z be the different release dates of the given instance. The release date of a slice is the release date of its job. A slice of job j is wide if m j > m/, otherwise, it is called small. The proof will refer to slices of jobs instead of jobs since this is the level of granlarity needed to prove the reslt. We will combine time-slots in LS to sets and then match sets to add p their contained load in a way that enables s to compare the load in LS to the load in OPT. Two disjoint sets of time-slots E 1 and E in LS of eqal size ( E 1 = E ) can be matched if the load of any time-slot in E 1 pls the load of any time-slot in E exceeds m, and hence the sm of the load of all slices in E 1 E exceeds E 1 m. For instance, if a job j is released at time r j bt started only at time s j > r j in or list-schedle, then the sm of the load in any time-slot between r j and s j and the load in any time-slot between s j and the completion time C j of j, is greater than m. Hence, any sch two time-slots can be matched and therefore two sets of same size, consisting of time-slots between r j and s j, and time-slots between s j and C j, respectively, can also be matched. Also, if we know that every time-slot in two disjoint sets of the same size contains a wide job, then these two sets can be matched as well. De to the fact that or sets of time-slots defined dring the proof are not necessarily of the same size, we have occasionally to distingish several cases. For instance, if we have two disjoint sets of time-slots E 1 and E, and any time-slot in E 1 can be matched with any time-slot in E, then depending on which set is greater, we might have a total load of more than E 1 m pls some remaining nmatched time-slots in E, or we have a total load of more than E m and some remaining nmatched time-slots in E 1.Fork = 0, 1,...,z, let D(r k ) be the set of time-slots in LS after r k that contain wide slices, which are completed in OPT before r k. In particlar, D(r 0 ) =. These sets have the properties described in Observations 1,, and 3. We denote by S(r k ) the set of time-slots {1,...,r k } D(r k ), for each k = 1,...,z. Let S(r 0 ) = and let S(C LS max of all time-slots in LS; ths, S(C LS max ) ={1,...,CLS max }. Observation 1. D(r h ) {r k + 1,...,C LS max } D(r k) for 0 h < k z. ) be the set Observation. For every set D D(r k ) we define the bipartite graph B(D) in the following way. For every time-slot h {(r k D +1),...,r k } we introdce one node on the left side of the graph B(D). For every time-slot d D we introdce one node on the right side of the graph B(D). A node h on the left side of B(D) and a node d on the right side of B(D) are adjacent if and only if the wide slice in the time-slot d is released before h, that is, r d h 1. Then B(D) contains a perfect matching. Observation 3. The ineqality D(r k ) r k holds for all k = 0,...,z. Observation 4. For all 0 h < k z we have (S(Cmax LS )\S(r k)) (S(r k )\S(r h )) =. Moreover, (S(Cmax LS )\S(r k)) (S(r k )\S(r h )) = S(Cmax LS )\S(r h). Observation 5. If there exists a k z with S(C LS max )\S(r k) =, then the claim of Theorem 4.1 is tre. Reason.IfS(C SL max )\S(r k) = for some k z, then all time-slots between the time r k and C LS part of D(r k ), therefore Cmax LS = r k + D(r k ). On the other hand, Cmax OPT This implies Cmax OPT max. Therefore, we assme henceforth that S(Cmax LS )\S(r k) for all k z. > rk+ D(rk) = C LS > r k and C OPT max max are D(r k).

8 440 J Sched (006) 9: We can also exclde for every release date r k, k = 1,...,z, the case that the time-slot r k is empty in LS; in this case all sbseqent jobs wold not be released before time r k, and we cold consider the remaining part of LS separately. We consider the following two cases for Cmax LS : Either there is a release date r h, h {0,...,z}, LS S(Cmax sch that there is more load than )\S(rh) m in S(C LS max )\S(r h), or there is no sch release date. Case 1. There is no release date r h, h {0,...,z}, sch that the load in S(Cmax LS )\S(r h) is more LS S(Cmax than )\S(rh) m. It follows that the load in S(Cmax LS )\S(r LS S(Cmax z) cannot be more than )\S(rz) m. Together with Observation 5, this implies that there is a time-slot in S(Cmax LS )\S(r z) {r z + 1,...,Cmax LS } containing a small job. Let e be a small job in S(Cmax LS )\S(r z) that completes last. Let r e be the release date, s e the start time, and C e the completion time of e. The load in S(Cmax LS )\S(r e) is at most S(Cmax LS )\S(re) m. We partition the set S(Cmax LS )\S(r e) into the disjoint sets Ē, E := E 1 E, and Ẽ. Let E be the set of time-slots in S(Cmax LS )\S(r e) containing a slice of e. Let E 1 be the set of time-slots in E before the date r z, and let E be the set of time-slots in E after r z. Notice that E. Let Ē be the set of time-slots in S(Cmax LS )\S(r e) containing no slice of e and which are before r z. Let Ẽ be the set of time-slots in S(Cmax LS )\S(r e), which do not contain a slice of e and which come after r z. Finally, Dre e is the set of all time-slots between r e and r z that contain a slice of e, bt which are not elements of the set S(Cmax LS )\S(r e), and hence not part of the set E. Observation 6. More than m machines are bsy in every time-slot in Ē. Moreover, any two time-slots s 1 E and s Ẽ can be matched. Reason. Job e with m e m is released at date r e, bt is started at time s e only. Therefore, at least m m e + 1 machines are bsy at any time between r e and s e. Observation 7. In every time-slot s 1 Ẽ more than m s 1 Ẽ and s E can be matched. machines are bsy. Any two time-slots Reason. Since job e with m e m is released at time r e, bt started only at time s e, the statement is clearly tre for all time-slots s 1 before s e. Each time-slot s 1 Ẽ after C e contains a wide slice. We consider now a time-slot s E. Either s contains a slice of the wide job from s 1 or the wide job cold not be processed in time-slot s althogh it was already released, becase too many machines are bsy in s. In both cases the total nmber of bsy machines in both time-slots exceeds m. Let s consider the set S(Cmax LS )\S(r z). We partition the time-slots between date r z and Cmax LS that are part of the set S(Cmax LS )\S(r e), bt not part of the set S(Cmax LS )\S(r z), and which are therefore part of the set D(r z ), into two disjoint sets. The ones that contain a slice of e form the set Dr E z, and those withot a slice of e form the set D Ẽr z. Ths, we have Dr E z = E D(r z ) and D Ẽr z = Ẽ D(r z ). Each time-slot in Dr E z D Ẽr z contains a wide slice. We have therefore partitioned the set S(Cmax LS )\S(r z) into the disjoint sets E \Dr E z and Ẽ\D Ẽr z. All jobs in S(Cmax LS )\S(r z) are released not later than r z. Observation 7 implies the following insight. Observation 8. E \D E r z > Ẽ\D Ẽr z.

9 J Sched (006) 9: We distingish for frther cases with respect to the relationship between the size of the sets Ē and E 1 and the relationship between the size of the sets E and Ẽ. Case 1.A. Ẽ < E. Case 1.A.1. Ē E Ẽ. Observation 7 implies that the set Ẽ and any set of Ẽ many time-slots in E can be matched. Becase of Observation 6, the set of the remaining E Ẽ many nmatched time-slots in E, together with the time-slots in E 1, can be matched with any set of E Ẽ + E 1 = E Ẽ many time-slots in Ē. The nmber of bsy machines in any remaining nmatched time-slot in Ē exceeds m LS. Ths, the set S(C max )\S(r e) contains in total more load than ( Ẽ + E Ẽ )m + ( Ē E +Ẽ)m = ( Ē + E +Ẽ)m LS S(Cmax = )\S(re) m, which contradicts the assmption of Case 1. Case 1.A.. Ē < E Ẽ. LS S(Cmax In this case, we have E > )\S(re). From C OPT max r e + p e and Observation 3 follows Cmax OPT r LS S(Cmax e + E > r e + )\S(re) = r e + C max LS re D(re) = C max LS + re D(re) C max LS, and Theorem 4.1 is proved in this case. Case 1.B. Ẽ E. Case 1.B.1. Ē E 1. Observation 7 shows that we can match any set of E many time-slots in Ẽ with the set E. Using Observation 6, we can match any set of E 1 many time-slots in Ē with the time-slots in E 1. In each of the remaining nmatched time-slots in Ẽ and Ē, more than m machines are bsy. Ths, the load in S(Cmax LS )\S(r ( Ē E e) exceeds ( E 1 + E )m + 1 )m ( Ẽ E + )m = LS ( Ē + E + Ẽ )m S(Cmax = )\S(re) m. This is in contradiction to the assmption of Case 1. Case 1.B.. Ē < E 1. We denote the set of the first Ē time-slots in E 1 by E1 1. Observation 6 implies that E 1 1 can be matched with Ē. From Observations 7 and 8 follows that any set of Ẽ\D Ẽr z many time-slots in E \Dr E z can be matched with the time-slots in Ẽ\D Ẽr z. Frthermore, Dr E z < D Ẽr z becase of Ẽ E and Observation 8. Therefore, the set Dr E z can be matched with any set of Dr E z many time-slots in D Ẽr z. Since E Ẽ, we obtain the following ineqality: D Ẽr z Dr E z E \Dr E z Ẽ\D Ẽr z. Hence, there are E \Dr E z Ẽ\D Ẽr z many time-slots in the remaining D Ẽr z Dr E z many time-slots in D Ẽr z that can be matched with the set of the remaining E \Dr E z Ẽ\D Ẽr z many time-slots in E \Dr E z. Let D rest be the set of the D Ẽr z Dr E z ( E \Dr E z Ẽ\D Ẽr z ) many presently nmatched time-slots in D Ẽr z. By Observation, there is a set Drest match of time-slots in {r z ( D Ẽr z Dr E z ( E \Dr E z Ẽ\D Ẽr z )) + 1,...,r z } that can be matched with D rest. We combine the time-slots in D Ẽr z that are matched with a timeslot in D(r e ) D match rest, ths with a time-slot in D e r e, which is therefore not part of the set S(Cmax LS )\S(r e), in the set D spare. The load in S(Cmax LS )\S(r e) is at most time-slot in D spare contains more load than m S(C LS max )\S(re) m. Since every LS, it follows that the load in (S(Cmax )\S(r e))\d spare LS (S(Cmax is bonded from above by )\S(re))\Dspare m. If every time-slot in E 1 \E1 1 has been matched with a time-slot in D rest, then every timeslot in E has been matched with another time-slot in (S(Cmax LS )\S(r e))\d spare )\E. Each of the remaining time-slots in (S(Cmax LS )\S(r e))\d spare )\E contains more than m load. Ths, (S(Cmax LS )\S(r LS (S(Cmax e))\d spare contains more load than )\S(re))\Dspare m, which is a contradiction.

10 44 J Sched (006) 9: We may therefore assme that there is at least one time-slot in E 1 \E1 1 that has not been matched with a time-slot in D rest. Conseqently, D spare Dr e e. Frthermore, E > Ē + Ẽ\D Ẽr z + Dr E z + E \Dr E z Ẽ\D Ẽr z +( D Ẽr z Dr E z ( E \Dr E z Ẽ\ D Ẽr z )) D spare = Ē + Ẽ D Ẽr z + Dr E z + E Dr E z Ẽ + D Ẽr z + D Ẽr z Dr E z E + Dr E z + Ẽ D Ẽr z D spare = Ē + Ẽ D spare. It follows that E > (S(Cmax LS )\S(re))\Dspare, and becase C OPT max r e + pe we conclde that Cmax OPT (S(C LS max )\S(re))\Dspare r e + E + Dr e e > r e + Dr e e + = r e + Dr e e + C max LS re D(re) D spare C max LS + re D(re) + De re Dspare C max LS, and herewith the correctness of Theorem 4.1 in this case. A schematic illstration of the relevant part of the list-schedle LS for this case is given in Fig. 1. Case. There is a release date r h < Cmax LS LS sch that S(Cmax )\S(r h) contains more load than S(Cmax LS )\S(rh) m. Let r k be the smallest release date of this kind. If r k = 0, the theorem follows immediately. Let s therefore assme that r k > 0. Observation 9. S(r k )\S(r h ) for every release date r h < r k. Reason. If S(r k )\S(r h ) = for r h < r k, Observation 4 implies S(Cmax LS )\S(r h) = (S(Cmax LS )\S(r k)) (S(r k )\S(r h )) = S(Cmax LS )\S(r k). Since S(Cmax LS )\S(r k) contains more load Fig. 1 Illstration of Case 1.B.. The bars nderneath the pictre nderline the time-slots that belong to the sets S(r e ), S(Cmax LS ), or (S(C LS max )\S(r e))\d spare, respectively. The horizontal bar within LS, going from s e to C e, corresponds to the job e. Sets of time-slots with the same pattern have been matched with each other.

11 J Sched (006) 9: S(C LS max )\S(rk) m than, it follows that the set S(C LS max )\S(r h) contains more load than a contradiction to the definition of r k. S(C LS max )\S(rh) m, Observation 10. The load of the set S(r k )\S(r h ) is at most S(rk)\S(rh) m, for every release date r h < r k. Reason. If the load in S(r k )\S(r h ) was more than S(rk)\S(rh) m, we wold be able to merge the loads in S(r k )\S(r h ) and S(Cmax LS )\S(r k), with the help of Observation 4. The cmlated load in S(Cmax LS )\S(r LS S(Cmax h) wold then exceed )\S(rh) m, a contradiction to the definition of r k. Observation 11. Every wide slice in S(C LS max )\S(r k) is processed in OPT after r k. Reason. The wide slices, which are processed in OPT before r k, bt in LS after r k, are inclded in D(r k ) and therefore not part of the set S(Cmax LS )\S(r k). If all small slices in S(Cmax LS )\S(r k) are not released before r k, Observation 11 and the assmption of Case imply Cmax OPT > r LS S(Cmax k + )\S(rk) C max LS, and we are done. We henceforth assme that there is at least one small job in S(Cmax LS )\S(r k) which has been released before r k. We call sch small jobs, which are released before r k bt completed by the list-schedling algorithm after r k ot-jtting jobs. Let j be the minimm of the nmber of slices after r k of an ot-jtting job j and the nmber of time-slots between r j and r k that contain no slice of j. Ths, j is an pper bond of the nmber of slices of job j, which are processed in LS after r k, bt in OPT possibly before r k. Let be an ot-jtting job for which this bond is maximal among all ot-jtting jobs. Let D r before be the set of time-slots in D(r ) {r + 1,...,r k } that contain no slice of. Let Dr before be the set of time-slots in D(r ) {r + 1,...,r k } that contain a slice of. Let Dr after := D(r ) {r k + 1,...,Cmax LS }. We partition the time-slots between date r and r k into the disjoint sets U, Ū, D r before, and Dr before. Let U be the set of time-slots in {r + 1,...,r k }\Dr before that contain a slice of the job. Let Ū be the set of time-slots in {r + 1,...,r k }\ D r before that do not contain a slice of the job. Hence, the set {r + 1,...r k } (D(r k )\Dr after ) is composed of the disjoint sets of time-slots U, Ū, D r before, Dr before, and D(r k )\Dr after. Observation 1. The term Ū + D before r is an pper bond for ũ. Observation 13. We have ũ + D(r k ) < r k. Reason. Becase of Observation 10, the set S(r k )\S(r ) contains at most S(rk)\S(r) m load. Together, more than m machines are bsy in each pair of time-slots s 1 Ū and s U since there is no slice of job in s 1 althogh job was already released. Moreover, more than m m m machines are bsy in each time-slot in Ū and D r before. Since each time-slot in D(r k )\D after r contains a wide slice, more than m machines are bsy in each of them. Becase of Observation, for every time-slot s 1 D(r k )\Dr after there exists another time-slot s {r k D(r k )\Dr after +1,...,r k } sch that s 1 and s can be matched. This way at most Dr before many time-slots from D(r k )\Dr after are matched with a time-slot in Dr before. We combine those time-slots to the set D spare.wehave D spare Dr before. Since S(r k )\S(r ) contains at most S(r k)\s(r ) m load, it follows that (S(r k )\S(r ))\D spare contains at most (S(rk)\S(r))\Dspare m load, becase every time-slot in D spare is loaded by more than m. The set (S(r k)\s(r ))\D spare decomposes

12 444 J Sched (006) 9: into the disjoint sets Ū, U and (D(r k )\Dr after )\D spare.if U Ū + (D(r k )\Dr after )\D spare,we cold match every time-slot in U with another time-slot in ((S(r k )\S(r ))\D spare )\U. The remaining time-slots in ((S(r k )\S(r ))\D spare )\U each contain more than m load. Ths, the load in (S(r k )\S(r ))\D spare wold exceed (S(rk)\S(r))\Dspare m, in contradiction to or earlier observations regarding this set. We conclde that U > Ū + D(r k )\Dr after Dr before. Combining this ineqality with Observations 1 and 3 reslts in ũ + D(r k ) < U + D r before + Dr before + D after r U + D(r ) U +r r k. Observation 14. The sm of the widths of all ot-jtting jobs is at most m. Reason. If this wold not be the case, the sm of the widths of all small jobs between r k 1 and r k wold exceed m. Bt then there wold be no time-slot between the release dates r k 1 and r k that is filled with at most m load. Using Observation 9, this is in contradiction to Observation 10 since it wold follow that the load in S(r k )\S(r k 1 ) exceeds S(rk)\S(rk 1) m. Let s consider the set S(Cmax LS )\S(r k). The load between r k and Cmax LS, exclding the load in LS (Cmax the time-slots in D(r k ), is more than rk D(rk) )m. It follows from Observations 11 and 14 that at most an amont of ũm of this load can be processed in OPT before r k. Therefore, the load LS (Cmax in LS that has to be processed in OPT after r k exceeds rk D(rk) )m ũm. Conseqently, by sing the ineqality from Observation 13, we obtain Cmax OPT > r k + C max LS rk D(rk) ũ > C max LS. A schematic illstration of the relevant part of the list-schedle LS for this case is given in Fig.. Therefore, the length of LS is less than two times the length of the optimal preemptive schedle OPT. An immediate implication on the power of preemption is stated in the following corollary. Fig. Illstration of Case. The bars nderneath the pictre nderline the time-slots that belong to the sets S(r ), S(r k ), or (S(r k )\S(r ))\D spare, respectively. Sets of time-slots with the same pattern have been matched with each other.

13 J Sched (006) 9: Fig. 3 Worst-case example for any list-schedling with release dates Corollary 4.. An optimal nonpreemptive schedle is less than twice as long as an optimal preemptive schedle for all instances of P m j, r j, pmtn C max. This bond is tight. Chen and Vestjens (1997) proved that the p j -list-schedling algorithm (also called the LPT rle) is a 3/-approximation algorithm for P r j C max. In contrast, the following example shows that for nonpreemptive schedling with parallel jobs and release dates, no list-schedling algorithm has a performance garantee smaller than, no matter which priority list is chosen. Example 4.3. Let m be the nmber of machines. Let m 3 jobs d = 1,...,m 3 of length p d = 1 and width m d = m be given. We call these jobs big jobs. The release date of each big job d = 1,...,m 3isr d := d. Let there also be m small jobs k = 1,...,m with length and width 1. The release date of a small job is r k := k 1, k = 1,...,m. An optimal schedle has length m, by schedling the big jobs from time 1 to time m and sing the last two time-slots for the small jobs. The first time-slot remains empty. Each list-schedling algorithm receives at time 0 only one job, namely the first small job to be schedled. This job therefore starts at time 0. At time 1 the list-schedling algorithm receives the second small job and the first big job. Since there are not enogh idle machines to schedle the big job, the list-schedling algorithm assigns the second small job to start at time 1. This pattern reoccrs at each point in time thereafter; one small job is still rnning, ths preventing the start of big jobs and casing the next small job to be started. At time m + 1 all small jobs are completed and the big jobs can be started. The reslting schedle has length m m 3 = m. Therefore, the ratio between the makespan of any list-schedling algorithm and the optimal makespan is /m. Figre 3 illstrates Example 4.3 with eight machines. Note that the variant of list-schedling in which jobs are assigned to machines in order of their priorities (sometimes called serial or job-based list-schedling) does not lead to improved performance garantees either, even if jobs are in order of nonincreasing widths (Example 3.) or nonincreasing processing times (Example 4.3). 5. Online-schedling In this section, we discss online-schedling of parallel jobs. In an online environment, parts of the inpt data are not known in advance. In schedling, one typically distingishes between three basic online models, each characterized by a different dynamics of the sitation (see, e.g., Sgall, 1998). In the first online model, schedling jobs one by one, jobs arrive one after the

14 446 J Sched (006) 9: other, and the crrent job needs to be (irrevocably) schedled before the next job and all its characteristics become known. In the model jobs arriving over time, the characteristics of a job become known when the job becomes known, which happens at its release date. In contrast, in the model nknown rnning times, the processing time of a job remains nknown ntil it is completed. Since list-schedling (withot special ordering of the list) complies with the reqirements of the online model nknown rnning times, it follows from the work of Garey and Graham (1975) that list-schedling is -competitive in the context of the online model nknown rnning times for parallel jobs withot release dates. For the same model with release dates and for the online model jobs arriving over time, Theorem 4.1 or Theorem 4 in Naroska and Schwiegelshohn (00) implies that list-schedling is -competitive for both the preemptive and the nonpreemptive version. Shmoys, Wein, and Williamson (1995) showed for both versions of the online model nknown rnning times that there is no deterministic online algorithm with a better competitive ratio than 1/m, even if all jobs are nonparallel. Therefore, list-schedling algorithms achieve the best possible competitive ratio within the class of deterministic online algorithms for this model. Chen and Vestjens (1997) showed that is a lower bond for any deterministic nonpreemptive online algorithm for the model jobs arriving over time, even if every job needs only one machine for its processing. This bond does not hold if preemptions are allowed. In fact, Gonzalez and Johnson (1980) presented for the preemptive case of the online model jobs arriving over time with nonparallel jobs an exact algorithm. In comparison, Example 5.1 shows that there is no deterministic online algorithm for the preemptive version of online model jobs arriving over time with a competitive ratio smaller than 6/5, when dealing with parallel jobs. Example 5.1. There are for identical parallel machines. At time 0 we release three jobs, job a and job b, each with length 1 and width, and job l with length and width 1. If the online algorithm decides to dedicate at most 3/5 of the first time-slot to the processing of job l, wedo not release any more jobs. In this case, the makespan of the schedle of the online algorithm is at least 3 3/5, whereas the length of the optimal schedle is. If the online algorithm dedicates more than 3/5 of the first time-slot to job l, we release at time 1 a job d with length and width 3. In this case the online algorithm prodces a schedle of length at least 3 + 3/5, whereas the optimal makespan is 3. Both cases give s a ratio between the length of the online schedle and the optimal makespan of 6/5. For the online model schedling jobs one by one, list-schedling achieves a competitive ratio of 1/m for nonparallel jobs (Graham, 1966). In contrast, list-schedling of parallel jobs does not have a constant competitive ratio, which is revealed by Example 5. below. In fact, for parallel jobs and the online model schedling jobs one by one no algorithm with constant competitive ratio was known heretofore, to the best of the athor s knowledge. We present the first sch algorithm below (Algorithm 5.4) and show that it is 1-competitive (Theorem 5.5). Theorem 5.3 provides a lower bond of.5 for the competitive ratio of any deterministic online algorithm for the model schedling jobs one by one with parallel jobs. The latter reslt shows again that schedling parallel jobs is significantly harder than schedling nonparallel jobs. For nonparallel job schedling, deterministic online algorithms are known with competitive ratio smaller than (Albers, 1999; Fleischer and Wahl, 000). Example 5.. Let I be an instance of the online-schedling model schedling jobs one by one with m machines and m jobs. Half of the jobs in I are wide; they have length 1 and width m. The other jobs j = 1,...,m are small with width 1 and length p j = j. The list alternates between

15 J Sched (006) 9: Fig. 4 Worst-case example for list-schedling in the online model schedling jobs one by one small and wide jobs, and shorter small jobs precede longer small jobs. The optimal schedle has length m, whereas the list-schedle has a makepan of m + m j=1 j = m +3m. The two schedles of an instance with for machines are depicted in Fig. 4. Theorem 5.3. No deterministic online algorithm for the online model schedling jobs one by one of the schedling problem P m j C max has a competitive ratio smaller than or eqal to.5. Proof: Let A be any deterministic online algorithm for the model schedling jobs one by one of P m j C max with competitive ratio.5. We constrct an instance with a list in which jobs of width 1 and jobs of width m alternate. The length of each job is set in a way sch that it can start only after the completion time of the previos job in the list. Let the nmber of machines be m := 10. The total nmber of jobs is at most 0. More specifically, let k i be the ith job of width 1 in the list, and let d i be the ith job of width m, i = 1,...,m. Let z ki be the delay introdced by A prior to starting job k i. Similarly, let z di be the delay before d i. Then, the job lengths are defined as follows: p k1 := 1, p ki+1 := z ki + p ki + z di + 1, and p di := max j<i {z k j, z d j, z ki }+1. The length of the schedle prodced by Algorithm A after the completion of job k i is therefore z ki + p ki + i 1 j=1 (z k j + p k j + z d j + p d j ), and it is i j=1 (z k j + p k j + z d j + p d j ) after the completion of job d i. The length of an optimal schedle inclding all jobs from the list p to (and inclding) job k i is at most p k j + i 1 j=1 p d j. If the instance ends with the ith d-job, the length of an optimal schedle is at most p ki + ij=1 p d j. We prove the lower bond of.5 for the competitive ratio by complete enmeration over the possible delays z ki and z di, i = 1,...,10, introdced by Algorithm A. Note that no delay can be too large becase the competitive ratio.5 mst be satisfied at any time dring the rn of the Algorithm (i.e., for every sb-instance). Using a compter program, it trns ot that there is no way for A to create delays in sch a manner that its competitive ratio is.5 for all sb-instances. The following algorithm is the first algorithm with a constant competitive ratio for schedling parallel jobs one by one. Algorithm Partition the time axis into the intervals I i := [ i, i+1 ], i = 0, 1,.... Schedle the arriving job j of length p j and width m j in the earliest interval I i that is more than twice as long as p j and in which job j can feasibly be schedled, as follows:

16 448 J Sched (006) 9: If m j > m/, schedle job j as late as possible within I i... If m j m/, schedle job j as early as possible within I i. Theorem 5.5. The competitive ratio of Algorithm 5.4 for the online version schedling jobs one by one of the schedling problem P m j C max is smaller than 1. Proof: Let S be the schedle constrcted by Algorithm 5.4. We denote by Cmax S the length of S and by Cmax the optimal offline makespan. Let I l = [ l, l+1 ] be the last and therefore the longest nonempty interval of S. Hence, Cmax S l+1 1. We distingish two cases depending on whether I l contains a job j with p j l /4 or not, in which case I l contains only jobs that are shorter, bt did not fit into the preceding interval. Case 1. There is j I l with p j l. An optimal schedle cannot be shorter than the longest job of the instance. Ths, the optimal makespan is at least C max l. Conseqently C S max /C max (l+1 1)/ l < 8. Case. For all jobs j I l we have p j < l. In this case, every job j I l did not fit anymore into the interval I l 1. We consider the interval I l 1. Its length is l 1. We partition the set of time-slots of the interval I l 1 into the disjoint sets K, H, L, and D. Let K be the set of time-slots that are filled to more than m/ with small jobs, i.e., jobs with m j m/. These time-slots had been filled dring Step.. of Algorithm 5.4 and are located at the beginning of the interval. Let H be the set of time-slots that contain only small jobs, bt in which at most m/ machines are bsy. These time-slots are located right after the time-slots in K. All jobs in H start no later than in the first time-slot of H. Let L be the set of empty time-slots. They are located between the time-slots of H and the time-slots belonging to D. Let D be the set of time-slots that contain a wide job, i.e., a job with m j > m/. These time-slots were filled dring Step.1. of Algorithm 5.4 and are the last time-slots of the interval I l 1. The sets K, H, L, and D are disjoint and we have K + H + L + D = l 1. The timeslots in K and D are filled to more than half. Note that there cannot be more time-slots in H than in the rest of the interval since all jobs in H start no later than in the first time-slot of H and all jobs are no longer than half of the length of the interval they belong to. We consider a job j that cold have been processed within the interval I l 1 (since I l 1 is more than twice as long as job j), bt was forced into the next interval I l becase there was insfficient space. We distingish two frther cases depending on whether job j is wide or small. Case.1. m j > m/. Obviosly, p j > L since otherwise j wold have fitted into the empty time-slots in I l 1. The total load of job j and of jobs schedled in the interval I l 1 is more than ( K + D ) m + L m = m ( K + D + L ) = m (l 1 H ). It follows that C max (l 1 H )/. We also have C max H. These facts combined lead to C max l 1 /3. It follows that C S max /C max 3( l+1 1) l 1 < 1. Case.. m j m/. This time p j > H + L since otherwise j cold have been assigned to time-slots in H and L. Ths, C max p j > H + L. Moreover, the time-slots in K and D together contain more load than ( K + D ) m = (l 1 ( H + L )) m, leading s to C max > (l 1 ( H + L ))/. We combine both lower bonds for the optimm to obtain C max l 1 /3. This reslts in C S max /C max 3(l+1 1) l 1 < 1.