Online scheduling of jobs with favorite machines

Transcription

1 Online cheduling o job with avorite machine Cong Chen 1, Paolo Penna, and Yineng Xu 1,3 arxiv: v1 [c.ds] 4 Dec School o Management, Xi an Jiaotong Univerity, Xi an, China Department o Computer Science, ETH Zurich, Zurich, Switzerland 3 The State Key Lab or Manuacturing Sytem Engineering, Xi an, China December 5, 018 Abtract Thi work introduce a natural variant o the online machine cheduling problem on unrelated machine, which we reer to a the avorite machine model. In thi model, each job ha a minimum proceing time on a certain et o machine, called avorite machine, and ome longer proceing time on other machine. Thi type o cot (proceing time) arie quite naturally in many practical problem. In the online verion, job arrive one by one and mut be allocated irrevocably upon each arrival without knowing the uture job. We conider online algorithm or allocating job in order to minimize the makepan. We obtain tight bound on the competitive ratio o the greedy algorithm and characterize the optimal competitive ratio or the avorite machine model. Our bound generalize the previou reult o the greedy algorithm and the optimal algorithm or the unrelated machine and the identical machine. We alo tudy a urther retriction o the model, called the ymmetric avorite machine model, where the machine are partitioned equally into two group and each job ha one o the group a avorite machine. We obtain a.675-competitive algorithm or thi cae, and the bet poible algorithm or the two machine cae. 1 Introduction Online cheduling on the unrelated machine i a claical and well-tudied problem. In thi problem, there are n job that need to be proceed by one o m dierent machine. The job arrive one by one and mut be aigned to one machine upon their arrival, without knowing the uture job. The time to proce a job change rom machine to machine, and the goal i to allocate all job o a to minimize the makepan, that i, the maximum load over the 1

2 machine. The load o a machine i the um o the proceing time o the job allocated to that machine. Online algorithm are deigned to olve the problem when the input i not known rom the very beginning but releaed piece-by-piece a aorementioned. Since it i generally impoible to guarantee an optimal allocation, online algorithm are evaluated through the competitive ratio. An online algorithm whoe competitive ratio i ρ 1 guarantee an allocation whoe makepan i at mot ρ time the optimal makepan. For cheduling on unrelated machine, online algorithm have a rather bad perormance in term o the competitive ratio. For example, the imple greedy algorithm ha a competitive ratio o m, the number o machine; the bet poible online algorithm achieve a competitive ratio o Θ(log m). However, ome well-known retriction (e.g., the identical machine and related machine) admit a much better perormance, that i, a contant competitive ratio. In act, many practical problem are neither a imple a the identical (or related) machine cae, nor o complicated a the general unrelated machine. In a ene, practical problem are omewhat intermediate a the ollowing example ugget. Example 1 (Two type o job (Vakhania et al., 014)). Thi retriction o unrelated machine arie naturally when there are two type o product. For intance, conider the production o pare part or car. The manuacturer may decide to ue the machine or the production o pare part 1 to produce pare part, and vice vera. The machine or pare part 1 (pare part, repectively) take time p or part 1 (part, repectively), but it can alo manage to produce part (part 1, repectively) in time q > p. Example (CPU-GPU cluter (Chen et al., 014)). A graphic proceing unit (GPU) ha the ability to handle variou tak more eiciently than the central proceing unit (CPU). Thee tak include video proceing, image analyi, and ignal proceing. Neverthele, the CPU i till more uitable or a wide range o other tak. A heterogeneou CPU-GPU ytem conit o a et M 1 o GPU proceor and a et M o CPU proceor. The proceing time o job j i p j1 on a GPU proceor and p j on a CPU proceor. Thereore, ome job are more uitable or GPU and other or CPU. Inpired by thee example, we conider the general unrelated machine cae by oberving that each job in the ytem ha a certain et o avorite machine which correpond to the hortet proceing time or thi particular job. In Example 1, the machine or pare part 1 are the avorite machine or thee part (proceing time p < q), and imilarly or pare part. In Example, we alo have two type o machine (GPU and CPU) and ome job (tak) have GPU a avorite machine and other have CPU a avorite machine. 1.1 Our contribution and connection with prior work We tudy the online cheduling problem on what we call the avorite machine model. Denote the proceing time o job j on machine i by p ji and the minimum proceing time o job j by p j = min i p ji. Thu the et o avorite machine o job j i deined a F j = {i p ji = p j } and the avorite machine model i a ollow:

3 (-avorite machine) Thi model i imply the unrelated machine etting when every job j ha at leat avorite machine ( F j ). The proceing time o job j on any avorite machine i F j i p j, and on any non-avorite machine i / F j i an arbitrary value p ji > p j. Thi model i motivated by everal practical cheduling problem. Beide the two type o job/machine problem mentioned in Example 1 and, the model alo capture the eature o ome real lie problem. For example, worker with dierent level o proiciency or dierent job in manuacturing; tak/data traner cot in cloud computing and o on. It i worth noting that thi model interpolate between the unrelated machine where poibly only one machine ha minimal proceing time or the job ( = 1) and the identical machine cae ( = m). The -avorite machine etting can alo be een a a relaxed verion o retricted aignment problem where each job j can be allocated only to a ubet F j o machine: the retricted aignment problem i eentially the cae where the proceing time o a job on a non-avorite machine i alway. We obtain tight bound on the Greedy algorithm 1 and the well-known Aign-U algorithm (deigned or unrelated machine by Apne et al. (1997)) or the -avorite machine cae, and how the optimality o the Aign-U algorithm by providing a matching lower bound. For the Greedy algorithm, the competitive ratio i m+ 1, which generalize the well-known bound on the competitive ratio o Greedy or unrelated machine ( = 1) and identical machine ( = m), that i, m and 1, repectively. The Aign-U algorithm m ha the optimal competitive ratio Θ(log m ), while it i Θ(log m) or unrelated machine. Eaier intance and the impact o peed ratio. Note that whenever = Θ(m), the competitive ratio i contant. In particular, or = m, Greedy ha a competitive ratio o 3. We conider the ollowing retriction o the model above uch that a iner analyi m i poible. (ymmetric m -avorite machine) All machine are partitioned into two group o equal ize m, and each job ha avorite machine a exactly one o the two group (thereore = m and m i even). Moreover, the proceing time on non-avorite machine i time that on avorite machine, where 1 i the caling actor (the peed ratio between avorite and non-avorite machine). We how that the competitive ratio o Greedy i at mot min{1 + ( ), + ( m +1 ), 3 } (< 3). A modiied greedy algorithm, called GreedyFavorite, which aign m +1 m each job greedily but only among it avorite machine, ha a competitive ratio o (< 3). A one can ee, the Greedy i better than GreedyFavorite or maller, and GreedyFavorite i better or larger. Thu we can combine the two algorithm to obtain a better algorithm (GGF) with a competitive ratio o at mot min{+ +, + 1 } (.675). +1 Indeed, we characterize the optimal competitive ratio or the two machine cae. That i, 1 Thi algorithm aign each job to a machine whoe load ater thi job aignment i minimized. 3

4 or ymmetric 1-avorite machine, we how that the GGF algorithm i min{1 + +1, }- competitive and there i a matching lower bound. For thi problem, our reult how the impact o the peed ratio on the competitive ratio. Thi i intereting becaue thi problem generalize the cae o two related machine (Eptein et al., 001), or which i the peed ratio between the two machine. The two machine cae ha alo been tudied earlier rom a game theoretic point o view and compared to the two related machine (Chen et al., 017; Eptein, 010). Relation with prior work. A mentioned above, the -avorite machine i a pecial cae o the unrelated machine and a general cae o the identical machine. The ymmetric 1-avorite machine i a generalization o the related machine. Beide, there are everal other intermediate problem that have connection with our model and reult in the literature (ee Figure 1 or an overview). Speciically, in the two type o machine cae (Imreh, 003; Chen et al., 014), they have two et o machine, M 1 and M, and the machine in each et are identical. Each job j ha proceing time p j1 or any machine in M 1, and p j or any machine in M. The o-called balanced cae i the retriction in which the number o machine in the two et are equal. Note that our m -avorite machine i a generalization o the balanced cae, becaue each job ha either M 1 or M a avorite machine, and thu = M 1 = M = m. The ymmetric m -avorite machine cae i the retriction o the balance cae in which the proceing time on non-avorite machine i time that o avorite one, i.e., either p j1 = p j or p j = p j1. 1. Related work The Greedy algorithm (alo known a Lit algorithm) i a natural and imple algorithm, oten with a provable good perormance. Becaue o it implicity, it i widely ued in many cheduling problem. In ome cae, however, better algorithm exit and Greedy i not optimal. Identical machine. For m identical machine, the Greedy algorithm ha a competitive ratio exactly 1/m (Graham, 1966), and thi i optimal or m =, 3 (Faigle et al., 1989). For arbitrary larger m, better online algorithm exit and the bound i till improving (Karger et al., 1996; Bartal et al., 1995; Alber, 1999). Till now the bet-known upper bound i (Fleicher and Wahl, 000) and the lower bound i 1.88 (Rudin III, 001). Related machine (uniorm proceor). The competitive ratio o Greedy i (1 + 5)/ or two machine (m = ), and it i at mot 1 + m / or m 3. The bound are tight or m 6 (Cho and Sahni, 1980). When the number o machine m become larger, the Greedy algorithm i ar rom optimal, a it competitive ratio i Θ(log m) or arbitrary m (Apne et al., 1997). Apne et al. (1997) alo devie the Aign-R algorithm with the irt contant competitive ratio o 8 or related machine. The contant i improved to 5.88 by Berman et al. (000). Some reearch ocue on the dependence o the competitive ratio on peed when m i rather mall. For the related machine cae, Greedy ha a competitive ratio o 1 + 4

5 unrelated machine m, Θ(log m) ( ) -avorite machine 1 + m 1, Θ(log m ) ( ) m-avorite machine 3 m, two type o machine 1 + m 1 (or m ), 3.85 balanced two type 3 m, ymmetric m -avorite min{1 + ( ), + ( ), 3 },.675 m +1 m +1 m ymmetric 1-avorite 1 + min{, 1}, 1 + min{, 1} ( ) two related machine 1 + min{, 1 }, 1 + min{, 1} ( ) Figure 1: Comparion between prior problem and reult (in red) and our problem and reult (in blue). The two bound below each problem are the competitive ratio or Greedy and bet-known algorithm, repectively, and the ( ) mark repreent the optimality o the bet-known algorithm. Arrow go rom a problem to a more general one, and thereore the upper bound or the general problem apply to the pecial one a well. 5

6 min{, 1 } and there i a matching lower bound (Eptein et al., 001). +1 Unrelated machine. A to the unrelated machine, Apne et al. (1997) how that the competitive ratio o Greedy i rather large, namely m. In the ame work, they preent the algorithm Aign-U with a competitive ratio o O(log m). A matching lower bound i given by Azar et al. (199) in the problem o online retricted aignment. Retricted aignment. The online retricted aignment problem i alo known a online cheduling ubject to arbitrary machine eligibility contraint. Azar et al. (199) how that Greedy ha a competitive ratio le than log m +1, and there i a matching lower bound log (m + 1). For other reult o online cheduling with machine eligibility we reer the reader to the urvey by Lee et al. (013) and reerence therein. Two type o machine (CPU-GPU cluter, hybrid ytem). Imreh (003) prove that Greedy algorithm i ( + m 1 1 m )-competitive or thi cae, where m 1 and m ( m 1 ) are the number o two et o machine, repectively. In our terminology, = m and m 1 = m, meaning that Greedy i (1 + m 1 )-competitive or m. The ame work alo improve the bound to 4 m 1 with a modiied Greedy algorithm. Chen et al. (014) give a competitive algorithm or the problem, and a imple 3-competitive algorithm and a more involved competitive algorithm or the balanced cae, that i, the cae m 1 = m. Further model and reult. Some work conider retriction on the proceing time in the oline verion o cheduling problem. Speciically, Vakhania et al. (014) conider the cae o two proceing time, where each proceing time p ji {q, p} or all i and j. Kedad- Sidhoum et al. (018) conider the two type o machine (CPU-GPU cluter) problem, and Gehrke et al. (018) conider a generalization, the ew type o machine problem. Some work alo tudy imilar model in a game-theoretic etting (Lavi and Swamy, 009; Auletta et al., 015). Regarding online algorithm, everal work conider retricted aignment with additional aumption on the problem tructure like hierarchical erver topologie (Bar- Noy et al., 001) (ee alo Crecenzi et al. (007)). For other reult o proceing time retriction, we reer the reader to the urvey by Leung and Li (008). Finally, the -avorite machine model in our paper ha been recently analyzed in a ollow-up paper in the oline game-theoretic etting (Chen and Xu, 018). Model and preliminary deinition The avorite machine etting. There are m machine, M := {1,,..., m}, to proce n job, J := {1,,..., n}. Denote the proceing time o job j on machine i by p ji, and the minimum proceing time o job j by p j = min i M p ji. We deine the avorite machine o a job a the machine with the minimum proceing time or the job. Let F j M be the et o avorite machine o job j. Aume that each job ha at leat avorite machine, i.e., F j or j J. Thu, the proceing time o job j on it avorite machine equal to the minimum proceing time p j (i.e., p ji = p j or i F j ), while the proceing time on it non-avorite machine can be any value that greater than p j (i.e., p ji > p j or i / F j ). 6

7 The ymmetric avorite machine etting. Thi etting i the ollowing retriction o the avorite machine. There are m = machine partitioned into two ubet o machine each, that i, M = M 1 M, where M 1 = {1,,..., } and M = { + 1, +,..., }. Each job j ha either M 1 or M a it avorite machine, i.e., F j {M 1, M }. The proceing time o job j i p j on it avorite machine and p j on non-avorite machine, where 1. Further notation. We ay that job j i a good job i it i allocated to one o it avorite machine, and it i a bad job otherwie. Let l i (j) denote the load on machine i ater job 1 through j are allocated by online algorithm: { l i (j 1) + p ji, i job j i aigned to machine i, l i (j) = l i (j 1), otherwie. In the analyi, we hall ometime conidered the machine in non-increaing order o their load. For a equence job, we denote by l i (j) the i th highet load over all machine ater the irt j job are allocated, i.e., l 1 (j) l (j) l m (j), or any j J. The job arrive one by one (over a lit) and mut be allocated irrevocably upon each arrival without knowing the uture job. the goal i to minimize the makepan, the maximum load over all machine. The competitive ratio o an algorithm A i deined a ρ A := C up A (I), where I i taken over all poible equence o job, C I C opt(i) A(I) i the cot (makepan) o algorithm A on equence I, and C opt (I) i the optimal cot on the ame equence. We write C A and C opt or implicity whenever the job equence i clear rom the context. 3 The avorite machine model In thi ection, we irt analyze the perormance o the Greedy algorithm and how it competitive ratio i preciely m+ 1. We then how that no online algorithm can be better than Ω(log m ) and that algorithm Aign-U (Apne et al., 1997) ha an optimal competitive ratio o O(log m ) or our problem. 3.1 Greedy Algorithm Algorithm Greedy: Every job i aigned to a machine that minimize the completion time o thi job (the completion time o job j i allocated to machine i i the load l i (j) o machine i ater the job i allocated). Theorem 3. The competitive ratio o Greedy i at mot m+ 1. The key part in the proo o Theorem 3 i the ollowing lemma which ay that Greedy maintain the ollowing invariant: the um o the larget machine load never exceed the um o all job minimal proceing time. 7

8 Lemma 4. For -avorite machine, and or every equence o n job, the allocation o Greedy atiie the ollowing condition: l 1 (n) + l (n) + + l (n) p 1 + p + + p n. (1) Proo. The proo i by induction on the number n o job releaed o ar. The bae cae i n = 1. Since Greedy allocate the irt job to one o it avorite machine, and the other machine are empty, we have i=1 l i(1) = l i (1) = p 1, and thu (1) hold or n = 1. A or the inductive tep, we aume that (1) hold or n 1, i.e., l 1 (n 1) + l (n 1) + + l (n 1) p 1 + p + + p n 1, () and how that the ame condition hold or n, i.e., ater job n i allocated. I job n i allocated a a good job, then the let-hand ide o () will increae by at mot p n, while the right-hand ide will increae by exactly p n. Thu, equation (1) ollow rom the inductive hypothei (). I job n i allocated a a bad job on ome machine b, then the ollowing obervation allow to prove the tatement: beore job n i allocated, the load o each avorite machine or job n mut be higher than l b (n 1) (otherwie Greedy would allocate n a a good job). Since there are at leat avorite machine or job n, there mut be a avorite machine a with l b (n 1) < l a (n 1) l (n 1). (3) Thu, l b (n 1) i not one o the larget load beore job n i allocated. Ater allocating job n, the load o machine b increae to l b (n) = l b (n 1) + p nb. We then have two cae depending on whether l b (n) i one o the larget load ater job n i allocated: Cae 1 (l b (n) l (n 1)). In thi cae, the larget load remain the ame ater job n i allocated, meaning that the let-hand ide o () doe not change, while the right-hand ide increae (when adding job n). In other word, (1) ollow rom the inductive hypothei (). Cae (l b (n) > l (n 1)). In thi cae, l (n 1) will be no longer included in the irt larget load, ater job n i allocated, while l b (n) will enter the et o larget load: i=1 l i (n) = i=1 l i (n 1) l (n 1) + l b (n). (4) Since Greedy allocate job n to machine b, it mut be the cae l b (n) = l b (n 1) + p nb l a (n 1) + p n l (n 1) + p n, (5) where a i the avorite machine atiying (3). Subtituting (5) into (4) and by inductive hypothei (), we obtain i=1 l i(n) i=1 l i(n 1) + p n n 1 i=1 p i + p n, and thu (1) hold. Thi complete the proo o the lemma. 8

9 Proo o Theorem 3. Without lo o generality we, aume that the makepan o the allocation o Greedy i determined by the lat job n (otherwie, we can conider the lat job n which determine the makepan, and ignore all job ater n ince they do not increae C Greedy nor decreae C opt ). Ater allocating the lat job n, the cot o Greedy become C Greedy min { min i F n l i (n 1) + p n, min i M\F n l i (n 1) + p ni min i F n l i (n 1) + p n l (n 1) + p n, (6) where the lat inequality hold becaue job n ha at leat avorite machine, and thu the leat loaded among them mut have load at mot l (n 1). Since the optimum mut allocate all job on m machine, and job n itel require p n on a ingle machine, we have { n j=1 C opt max p } { } j m, p l (n 1) + p n n max, p n. (7) m where the econd inequality i due to Lemma 4 applied to the irt n 1 job only (peciically, we have n 1 j=1 p j i=1 l i(n 1) l (n 1)). By combining (6) and (7), we obtain C Greedy C opt l (n 1) + p n { } l (n 1)+p max n, p m n ( ) l (n 1) m p n + 1 = min l (n 1) p n + 1, l (n 1) + 1 p n m + 1, where the lat inequality hold becaue the irt term decreae in l (n 1)/p n and the econd term increae in l (n 1)/p n. Theorem 5. The competitive ratio o Greedy i at leat m+ 1. Proo. We will provide a equence o job whoe chedule by Greedy ha a makepan m+ 1, and the optimal makepan i 1. For implicity, we aume that in cae o a tie, the Greedy algorithm will allocate the job a a bad job to a machine with the mallet index. Without lo o generality, uppoe m i diviible by. We partition the m machine into m := m group, M 1, M,..., M m, each o them containing machine, i.e., M i = {(i 1) + 1, (i 1) +,..., (i 1) + } or i = 1,..., m. The job are releaed in two phae (decribed in detail below): In the irt phae, we orce Greedy to allocate each machine in M i a load o bad job equal to i 1, and a load o 9 }

10 1 {}}{ M m m 1 M m 1 M 3 M M m m 1 1 }{{} machine } {{ } ( 1) Figure : Proo o Theorem 5 (bad job in white). good job equal to 1 i or a uitable > 1 (except or M m ); In the econd phae, everal job with avorite machine in M m are releaed and contribute an additional 1 to the load o one machine in M m, and thu the makepan i m+ 1. We ue the notation r (p, F ) to repreent a equence o r identical job whoe avorite machine are F, the proceing time on avorite machine i p, and on non-avorite machine i p with 1. Phae 1. For each i rom 1 to m 1, releae a equence o job (1 i, M i) ollowed by a equence ( i, M i). In thi phae, we take > m 1 + (m 1)(m ), o that, or each i, the job (1 i, M i) will be aigned a good job to the machine in M i, one per machine, and the job ( i, M i) will be aigned a bad job to the machine in M i+1, one per machine (a hown in Figure ). At the end o Phae 1, each machine in M i (i [1, m 1]) will have load i i = i(1 1 ), and each machine in M m will have load m 1. The optimal chedule aign every job to ome avorite machine evenly o that all machine have load 1, except or the machine in the lat group M m which are let empty. Phae. In thi phae, all job releaed have M m a their avorite machine, peciically ( 1) ( 1, M m ) ollowed by a ingle job (1, M m ). By taking > m, no job will be allocated a bad job by Greedy. Thu, thi phae can be een a cheduling on identical machine. Conequently, all job will be allocated a in Figure increaing the maximum load o machine in M m by 1, while the optimal chedule can have a makepan 1. At the end o Phae, the maximum load o machine in M m i m = m+ 1, while the optimal makepan i A general lower bound In thi ection, we provide a general lower bound or the -avorite machine problem. The bound borrow the baic idea o the lower bound or the retricted aignment (Azar et al., 199). The trick o our proo i that we alway partition the elected machine into arbitrary group o machine each, and apply the idea to each o the group repectively. 10

11 Theorem 6. Every determinitic online algorithm mut have a competitive ratio at leat 1 log m + 1. Proo. Suppoe irt that m i a power o, i.e., m = u 1. We provide a equence o m job having optimal makepan equal to 1, while any online algorithm will have a makepan at leat u+1 = 1 log m + 1. We denote by r (1, F ) a equence o r identical job whoe avorite machine are F, the proceing time on avorite machine i 1, and the proceing time on non-avorite machine i greater than u+1 = 1 log m + 1, o that all job mut be allocated to avorite machine. We conider ubet o machine M 1, M,..., M u, where the irt ubet i M 1 = M and the other atiy M i M i 1 and M i = 1 M i 1 = m or i =,... u. We then i 1 releae u et o job iteratively. At each iteration i (i = 1,,..., u), a et o job J i with avorite machine M u are releaed or allocation, and ater the allocation, hal o the highly loaded machine in M i are choen or the next et M i+1. More in detail, or each iteration i (i = 1,,..., u) we proceed a ollow: Partition the current et M i o machine into arbitrary group o machine each, that i, into group M i,1, M i,,..., M i,m i where m i = M i = m = u i. Then, releae a et i 1 o job J i : { J i,1 J i,... J i,m J i := i, i 1 i u 1 ; { (1, M u )}, i i = u ; where J i,l := { (1, M i,l) }. The next et M i+1 o machine conit o the / mot loaded machine in each ubgroup ater job J i have been allocated. Speciically, we let M i+1 := M i,1 M i, M i,m i, or M i,l M i,l being the ubet o / mot loaded machine in group M i,l ater job J i have been allocated. We then how that the ollowing invariant hold at every iteration. Note that the ollowing lemma how a eparation o our method and the one o Azar et al. (199). Speciically, the maximum load increae by 1/ at each iteration in our lower bound, but increae by 1 in Azar et al. (199). Lemma 7. The average load o machine in M i i at leat i 1 beore the job in J i are aigned or 1 i u. Proo. The proo i done by induction on i. For the bae cae i = 1, the lemma i trivial. A or the inductive tep, uppoe the lemma hold or i. Denote the average load o M i,l beore the allocation o job J i by Avg(i, l), and the average load o M i,j ater the allocation o job J i by Avg (i, l). We claim that Avg (i, l) Avg(i, l) + 1/ ater the allocation o job J i. Note that M i and M i+1 are the union o all group M i,l and M i,l, repectively. Thu, we have that the average load o M i+1 i at leat i = i, ince the average load o M i i i 1 (the inductive hypothei). Thi conclude 11

12 the proo o the inductive tep, and thu the lemma ollow. Next, we prove that the claim Avg (i, l) Avg(i, l) + 1/ hold: Cae 1. There i a machine in M i,l \M i,l that ha load at leat Avg(i, l) + 1/. Since M i,l conit o the / mot loaded machine in M i,l, each machine in M i,l will alo have load at leat Avg(i, l) + 1/. Thu, we obtain that Avg (i, l) Avg(i, l) + 1/. Cae. Each machine in M i,l \M i,l ha load no greater than Avg(i, l) + 1/. Oberve that the total load o M i,l i k Ave(i, l) + k/ ater the allocation o J i. Since M i,l \M i,l ha load at mot k/ (Avg(i, l) + 1/), the average load o the / machine in M i,l mut be at leat Avg (i, l) k Avg(i, l) + k k (Avg(i, l) + 1 ) k/ = Avg(i, l) + 1. By applying Lemma 7 with i = u, we have that the average load o M u i at leat u 1 beore the allocation o job J u. Thu, ater job in J u are allocated, the average load o M u i at leat u = u+1 u+1, i.e., the online cot i at leat. An optimal cot o 1 can be achieved by uing the machine in M i,l \M i,l or J i (or 1 i u 1 and 1 l m i) and uing machine in M u or J u. Thereore, the competitive ratio i at leat u+1 = 1 log m + 1. Finally, i m i not a power o, we imply apply the previou contruction to a ubet o m machine, or m = u 1 m and u = 1 + log. Thi give the deired lower bound u +1 = 1 log m General upper bound (Algorithm Aign-U i optimal) In thi ection, we prove a matching upper bound or -avorite machine. Speciically, we how that the optimal online algorithm or unrelated machine (algorithm Aign-U by Apne et al. (1997) decribed below) yield an optimal upper bound: Theorem 8. Algorithm Aign-U can be ued to achieve O(log m ) competitive ratio. Following Apne et al. (1997), we how that the algorithm ha an optimal competitive ratio ρ (Lemma 9) or the cae the optimal cot C opt (J ) i known, and then apply a tandard doubling approach to get an algorithm with competitive ratio 4ρ or the cae the optimum i not known in advance (Theorem 8). In the ollowing, we ue a tilde notation to denote a normalization by the optimal cot C opt (J ), i.e., x = x/c opt (J ). Algorithm Aign-U (Apne et al., 1997): Each job j i aigned to machine i, where i i the index minimizing i = a l i (j 1)+ p ji a l i (j 1), where a > 1 i a uitable contant. Lemma 9. I the optimal cot i known, Aign-U ha a competitive ratio o O(log m ). Proo. Without lo o generality, we aume that the makepan o the allocation o Aign- U i determined by the lat job n (thi i the ame a the proo o Theorem 3 above). Let 1

13 li (j) be the load o machine i in the optimal chedule ater the irt j job are aigned. Deine the potential unction: m ( Φ(j) = a l i (j) γ l ) i (j), (8) i=1 where a, γ > 1 are contant. Similarly to the proo in Apne et al. (1997), we have that the potential unction (8) i non-increaing or a = 1 + 1/γ (ee Appendix A.1 or detail). Since Φ(0) Φ(n 1) and l i (n 1) 1, we have γ m m i=1 Thu, it ollow that ( a l i (n 1) γ l ) i (n 1) Claim 10. C Aign-U l (n 1) + p n. m a l i (n 1) (γ 1) a l (n 1) (γ 1). i=1 ( γ l (n 1) log a γ 1 m ) C opt. (9) Proo. Suppoe Aign-U aign the lat job n to machine i, it hold that C Aign-U = l i (n 1) + p ni, ince we uppoe job n i the lat completed job. Cae 1. Job n i allocated a a good job, i.e., p ni = p n. According to the rule o Aign-U, machine i mut be one o the machine in F n that ha the minimum load. Since there are at leat avorite machine, it hold that l i (n 1) l (n 1). Thu, C Aign-U l (n 1) + p n. Cae. Job n i allocated a a bad job, i.e., p ni > p n. Let i be the machine who ha the minimum load in F n. Note that l i (n 1) l (n 1). According to the rule o Aign-U, it hold that l i (n 1) l i (n 1); otherwie, we have i i, which i contradictory to that Aign-U aign job n to machine i. Since i i and l i (n 1) l i (n 1) l (n 1), we have a l i (n 1)+p ni Copt a l i (n 1) Copt a l i (n 1)+p ni Copt a l i (n 1) Copt, o that l i (n 1)+p ni C opt l i (n 1)+p ni C opt l (n 1)+p ni C opt, implying C Aign-U = l i (n 1) + p ni l (n 1) + p ni = l (n 1) + p n. According to Claim 10 and (9), we have C Aign-U C opt l (n 1) + p ( n γ log C a opt γ 1 m ) Since the latter quantity i O(log m ), thi prove Lemma 9. + p ( n γ log C a opt γ 1 m ) + 1. By uing the tandard doubling approach, Lemma 9 implie Theorem 8 (ee Appendix A.). 13

14 4 The ymmetric avorite machine model Thi ection ocue on the ymmetric avorite machine problem, where M = {M 1, M } ( M 1 = M = ), F j {M 1, M } or each job j, and the proceing time o job j on a nonavorite machine i ( 1) time that on a avorite machine. We analyze the competitive ratio o Greedy a a unction o the parameter. Though Greedy ha a contant competitive ratio or thi problem, another natural algorithm called GreedyFavorite perorm better or larger. At lat, a combination o the two algorithm, GreedyOrGreedyFavorite, obtain a better competitive ratio, and the algorithm i optimal or the two machine cae. 4.1 Greedy Algorithm The next two theorem regard the competitive ratio o the Greedy algorithm or the ymmetric avorite machine cae. Theorem 11. For the ymmetric m -avorite machine cae, the Greedy algorithm ha a competitive ratio at mot { ( ˆρ Greedy := min ) ( + 1, + 1 ) + 1, 3 1 }, where = m and m i the number o machine. Since thi upper bound increae in ( 1), we have the ollowing inequalitie: 1 m ˆρ Greedy 3 m, and both bound can be achieved by the actual competitive ratio ρ Greedy o algorithm Greedy. Note that or = 1, the lower bound i indeed tight a it correpond to the analyi o Greedy on m identical machine (Graham, 1966). More generally, we have the ollowing reult (whoe proo i deerred to Appendix A.3 or readability ake): Theorem 1. The upper bound in Theorem 11 i tight (ρ Greedy = ˆρ Greedy ) in each o the ollowing cae: 1. For = 1, ρ Greedy = min{1 + +1, };. For = and , ρ Greedy = (+1) ; 3. For 3 1 (implying 1 1.5), ρ Greedy = 1 + ( 1 ) +1 ; 4. For > 1 and , ρ Greedy = + ( 1 ) +1 ; 5. For <, ρ Greedy =

15 4.1.1 Proo o Theorem 11 Becaue thi i a pecial cae o the general model, we have ˆρ Greedy 3 1 (by recalling that m = ). Thu, we jut need to prove C Greedy C opt { min 1 + ( 1 ) + 1, + ( 1 ) }. + 1 rom Theorem 3 Without lo o generality, we aume that the makepan o the allocation o Greedy i determined by the lat job n. Suppoe that the irt n 1 job have been already allocated by Greedy, and denote l α = min i M1 l (n 1) i, L α = i M 1 l (n 1) i, l = min i M l (n 1) i, L = i M l (n 1) i. Alo let L good α and L good be the total load o good job o L α and L, repectively. Oberve that l α Lα and l L. Without lo o generality, we aume that machine in M 1 are the avorite machine o the lat job n. We irt give a lower bound o the optimal cot C opt : { } lα+ l Claim 13. C opt max + p n, lα+ l + p n, p + + n Proo. Denote by P α and P the total minimum proceing time o the job which have M 1 and M a their avorite machine, repectively, i.e., P α = p j = L good α j: F j =M 1 P = p j = L good j: F j =M + 1 (L L good ) + p n, (10) + 1 (L α L good α ). (11) A lower bound on the optimal cot can be obtained by conidering the ollowing ractional aignment. Firt, allocate all job a good job, i.e., aign P α to M 1 and P to M. Then, reaign a raction o them to make all machine to have the ame load. We next ditinguih two cae: Cae 1 (P α P ). In thi cae, the reaignment i to move 1 (P +1 α P ) o P α to machine in M o that P α (P α P ) = P (P α P ). Thereore, along with (10) and (11), the optimal cot i at leat: C opt P α + P ( + 1) = L α + L + p n + ( 1)L good α +. (1) Furthermore, ubtituting (10) and (11) into P α P we have L good α L good (L α L ) + 1 p n. 15

16 Thereore, { 1 L good α max + 1 (L α L ) } + 1 p n, 0. (13) Subtituting (13) into (1), we have { Lα + L + p n C opt max, L } α + L + p n. + + Along with C opt p n, L α l α and L l, we obtain the inequation o thi claim. Cae (P α < P ). Similarly to the previou cae, we have C opt P α + P ( + 1) In thi cae, (10) and (11) imply L good max = L α + L + p n + ( 1)L good + (14) { (L α L ) + } + 1 p n, 0, (15) Subtituting (15) into(14), we have { Lα + L + p n C opt max, L } α + L + p n. + + Along with C opt p n, L α l α and L l, we obtain the inequation o thi claim and complete the proo. We next conider the cot o the Greedy algorithm. Recall that job n ha M 1 a avorite machine. Ater job n i allocated, we have C Greedy min{l α + p n, l + p n }. (16) Two cae arie depending on the larget between the two quantitie in (16): Cae 1 (l α + p n l + p n ). Thi cae implie Thereore, we have l l α ( 1) p n, (17) C Greedy l α + p n. (18) C Greedy C opt l α + p { n } lα+ l max + p n, lα+ l + p n, p + + n l α + p n { } max (+1)lα+(+ )pn, (+1)lα+(+ )p n, p + + n { } ( + )(x + 1) = min ( + 1)x + +, ( + )(x + 1) ( + 1)x + +, x + 1 { min 1 + ( 1 ) + 1, + ( 1 } ),

17 where the irt inequality i by (18) and Claim 13; the econd inequality i by (17); the third equation i obtained by deining x := l α /p n ; the irt term o the lat inequality i obtained by the econd and third term o the third equation (one decreae in x and one increae in x); imilarly, the econd term o the lat inequality i obtained by the irt and third term o the third equation. Cae (l α + p n l + p n ). Thi cae implie l α l + ( 1) p n and C Greedy l + p n. Similarly to the previou cae, we can obtain C Greedy C opt min { 1 + ( 1 ) + 1, + ( 1 } ) + 1 The above two cae conclude the proo o the theorem. 4. GreedyFavorite Algorithm We next conider another algorithm called GreedyFavorite which imply aign each job j to one o it avorite machine in F j. Algorithm GreedyFavorite: Aign each job to one o it avorite machine, choen in a greedy ahion (minimum load). It turn out that thi natural variant o Greedy perorm better or large. Theorem 14. For ymmetric m -avorite machine, the GreedyFavorite algorithm ha a competitive ratio o 1 + 1, where = m and m i the number o machine. Proo. Note that in GreedyFavorite all job are aigned a good job. Suppoe the overall maximum load occurred on machine 1 in M 1. Moreover, i there i any job executed on machine in M, we can remove all o it, which will not decreae C GreedyFavorite C opt. Thereore, all job have the ame avorite machine M 1, and GreedyFavorite aign all o them to M 1. We alo ue l α to repreent the minimum load over M 1 beore job n i allocated. Denote by P α the total minimum proceing time o the job who have M 1 a their avorite machine, which i alo the total minimum proceing time o all the job here. Obviouly, P α l α + p n (19) C GreedyFavorite l α + p n. (0) The optimal chedule can allocate ome o the job to M to balance the load over all machine. Thu, the optimal cot will be at leat { C opt max (P α x) 1, x } + 1 P α 1 ( lα + p n ), (1)

18 where x i the load o job that are aigned to M to balance the load over all machine. According to (0), (1) and C opt p n, we have { } C GreedyFavorite l α + p n l α + p n min C opt (l +1 α + pn ), 1 p n + 1. () Thu the upper bound on the competitive ratio i proved. To ee that thi bound i tight or any and, conider the ollowing equence o job: ( 1) ( 1, M 1), ( 1, M 1), (1, M 1 ). According to algorithm GreedyFavorite, all thee job are aigned to M 1 in a greedy ahion, and thu C GreedyFavorite = 1 ( 1) = The optimal olution will intead aign the job ( 1, M 1) to M, thu implying C opt = A better algorithm A one can ee, the Greedy i better than GreedyFavorite or maller, and Greedy- Favorite i better or larger. Thu we can combine the two algorithm to obtain a better algorithm. Algorithm GreedyOrGreedyFavorite (GGF): I, run Greedy; otherwie run GreedyFavorite. Corollary 15. For ymmetric m -avorite machine, i then ρ GGF min{ , + 1 }.675. Proo. By Theorem 11, we have ρ Greedy + By Theorem 14, we have ( 1 ) = ρ GreedyFavorite Note that i 1.481, , otherwie + + > + 1, thu { ρ GGF min + +, + 1 }

19 4.4 Tight bound or two machine (ymmetric 1-avorite machine) In thi ection, we how that the GGF algorithm i optimal or the ymmetric cae with two machine, i.e., the ymmetric 1-avorite machine. Theorem 16. For ymmetric { 1-avorite machine, } any determinitic online algorithm ha competitive ratio ρ min 1 +, Proo. Conider a generic algorithm Alg. Note that we have two machine, M 1 contain machine 1 and M contain machine only. Without lo o generality, aume the irt job i aigned to machine 1 and thi machine then ha load 1, that i, job 1 i either (1, M 1 ) or (1/, M ). Job i (, M 1 ). I Alg aign job to machine 1, then C Alg = 1 + while C opt =, thu implying ρ Alg Otherwie, i job i aigned to machine, then a third job ( + 1, M ) arrive. No matter where Alg aign job 3, the cot or Alg will be C Alg = A the optimum i C opt = + 1, we have ρ Alg 1 + in the latter +1 cae. By combining Theorem 11, Theorem 14, and Theorem 16, we obtain the ollowing: Corollary 17. For ymmetric 1-avorite machine, i then ρ GGF = min{1 +, } Thereore, the GGF algorithm i optimal An extenion o our model In thi ection, we dicu a imple extenion, which explain why the intance, where i mall, till have a good competitive ratio. The main idea i to conider avorite machine a the machine which have approximately the minimal proceing time or the job. For example, a job with proceing time (0.99, 1, 1, 1, ) might be conidered to have approximate proceing time (1, 1, 1, 1, ). In the latter cae, the job ha 4 avorite machine, intead o 1. More ormally, we conider the ollowing modiied algorithm  o a generic online algorithm A. For a et o job J, ix a parameter c 1 and denote ˆF j := {i : p ji c p j }. Run algorithm A auming proceing time are ˆp ji := { p j i i ˆF j, p ji otherwie. Note that in the recaled proceing time above the number ˆ o avorite machine per job atiie ˆ and p ji c ˆp ji. Ideally, we would like ˆ a big a poible and c a mall a poible, a the ollowing obervation indicate. Obervation 18. I algorithm A i ρ()-competitive or a certain cla o intance, where denote the minimum number o avorite machine per job in the input intance, then the modiied algorithm  i at mot c ρ( ˆ)-competitive on the ame cla o intance. 19

20 6 Concluion and open quetion Thi work tudie online cheduling or the avorite machine model. Our reult are upplement to everal claical problem and reveal the relation among them (a indicated in Figure 1). For the general -avorite machine cae, we provide tight bound on both Greedy and Aign-U algorithm and how that the latter i the bet-poible online algorithm. To ome extent, the key actor in our model capture ome o the main eature that make the model perorm well or badly: low or high competitive ratio. In particular, when = 1, the model i exactly the unrelated machine; when = m, the model i exactly the identical machine. Finally, the analyi o ymmetric avorite machine allow a direct comparion with the two related machine. Acknowledgment Thi work wa upported by the National Natural Science Foundation o China [ ]; and the China Potdoctoral Science Foundation [016M59811]. Part o thi work ha been done while the irt author wa viiting ETH Zurich. Reerence Suanne Alber. Better bound or online cheduling. SIAM Journal on Computing, 9(): , Jame Apne, Yoi Azar, Amo Fiat, Serge Plotkin, and Orli Waart. On-line routing o virtual circuit with application to load balancing and machine cheduling. Journal o the ACM, 44(3): , Vincenzo Auletta, George Chritodoulou, and Paolo Penna. Mechanim or cheduling with ingle-bit private value. Theory Comput. Syt., 57(3):53 548, 015. Yoi Azar, Joeph Sei Naor, and Raphael Rom. The competitivene o on-line aignment. In Proceeding o the third annual ACM-SIAM ympoium on Dicrete algorithm, page Society or Indutrial and Applied Mathematic, 199. Amotz Bar-Noy, Ari Freund, and Joeph Naor. On-line load balancing in a hierarchical erver topology. SIAM Journal on Computing, 31():57 549, 001. Yair Bartal, Amo Fiat, Howard Karlo, and Rakeh Vohra. New algorithm or an ancient cheduling problem. Journal o Computer and Sytem Science, 51(3): , Piotr Berman, Moe Charikar, and Marek Karpinki. On-line load balancing or related machine. Journal o Algorithm, 35(1):108 11,

21 Cong Chen and Yineng Xu. Selih load balancing or job with avorite machine. Operation Reearch Letter, 018. ISSN doi: Cong Chen, Paolo Penna, and Yineng Xu. Selih job with avorite machine: Price o anarchy v. trong price o anarchy. In Proc. o the 11th Annual International Conerence on Combinatorial Optimization and Application (COCOA), page 6 40, 017. Lin Chen, Dehi Ye, and Guochuan Zhang. Online cheduling o mixed cpu-gpu job. International Journal o Foundation o Computer Science, 5(06): , 014. Yookun Cho and Sartaj Sahni. Bound or lit chedule on uniorm proceor. SIAM Journal on Computing, 9(1):91 103, Pilu Crecenzi, Giorgio Gamboi, Gaia Nicoia, Paolo Penna, and Walter Unger. On-line load balancing made imple: Greedy trike back. J. Dicrete Algorithm, 5(1):16 175, 007. Leah Eptein. Equilibria or two parallel link: the trong price o anarchy veru the price o anarchy. Acta Inormatica, 47(7): , 010. ISSN Leah Eptein, John Noga, Steve Seiden, Jiří Sgall, and Gerhard Woeginger. Randomized on-line cheduling on two uniorm machine. Journal o Scheduling, 4():71 9, 001. Ulrich Faigle, Walter Kern, and György Turán. On the perormance o on-line algorithm or partition problem. Acta cybernetica, 9(): , Rudol Fleicher and Michaela Wahl. On-line cheduling reviited. Journal o Scheduling, 3 (6): , 000. Jan Clemen Gehrke, Klau Janen, Stean EJ Krat, and Jakob Schikowki. A pta or cheduling unrelated machine o ew dierent type. International Journal o Foundation o Computer Science, 9(04):591 61, 018. Ronald L Graham. Bound or certain multiproceing anomalie. Bell Sytem Technical Journal, 45(9): , Canád Imreh. Scheduling problem on two et o identical machine. Computing, 70(4): 77 94, Aug 003. ISSN David R Karger, Steven J Phillip, and Eric Torng. A better algorithm or an ancient cheduling problem. Journal o Algorithm, 0(): , Saia Kedad-Sidhoum, Florence Monna, Grégory Mounié, and Deni Trytram. A amily o cheduling algorithm or hybrid parallel platorm. International Journal o Foundation o Computer Science, 9(01):63 90,

22 Ron Lavi and Chaitanya Swamy. Truthul mechanim deign or multidimenional cheduling via cycle monotonicity. Game and Economic Behavior, 67(1):99 14, 009. ISSN Kangbok Lee, Joeph Y.-T. Leung, and Michael L. Pinedo. Makepan minimization in online cheduling with machine eligibility. Annal o Operation Reearch, 04(1):189, Apr 013. Joeph Y.-T. Leung and Chung-Lun Li. Scheduling with proceing et retriction: A urvey. International Journal o Production Economic, 116():51 6, 008. ISSN John F Rudin III. Improved bound or the online cheduling problem. PhD thei, The Univerity o Texa at Dalla, 001. Nodari Vakhania, Joe Alberto Hernandez, and Frank Werner. Scheduling unrelated machine with two type o job. International Journal o Production Reearch, 5(13): , 014. A Potponed proo A.1 The potential unction i non-increaing (or proo o Lemma 9) Recall that the potential unction i deined a Φ(j) = m i=1 a l i (j) (γ l i (j)). Aume that job j i aigned to machine i by algorithm Aign-U and to machine i by the optimal chedule, i.e., l i (j) = l i (j 1) + p ji and l i (j) = l i (j 1) + p ji. Then we have Φ(j) Φ(j 1) = (γ l i (j 1))(a l i (j) a l i (j 1) ) a l i (j 1) p ji γ(a l i (j 1)+ p ji a l i (j 1) ) a l i (j 1) p ji γ(a l i (j 1)+ p ji a l i (j 1) ) a l i (j 1) p ji (by i i ) = a l i (j 1) (γ(a p ji 1) p ji ). By taking a = 1 + 1/γ, we get γ(a p ji 1) p ji 0 ince 0 p ji 1, o that the potential unction i non-increaing. A. Doubling approach (proo o Theorem 8) Let ρ be the competitive ratio o Aign-U when the optimal cot C opt i known. By uing doubling approach one can eaily get a 4ρ-competitive algorithm or the cae optimal cot i not known. Thi approach ha been ued in Apne et al. (1997). We report the detail below or completene. We run Aign-U in phae, and let Λ i be the etimation o C opt at the beginning o phae i. Initially (beginning o phae 1) when the irt job arrive, let Λ 1 be the minimum proceing

23 time o the irt job. Whenever the makepan exceed ρ time the current etimation, ρλ i, the current phae i end and the next phae i + 1 begin with doubled etimation Λ i+1 = Λ i a the new etimation o the C opt to run Aign-U. During a ingle phae, job are aigned independently o the job aigned in the previou phae. It i eay to ee that thi approach increae the competitive ratio ρ by at mot a multiplicative actor 4 (a actor o due to the load in all but the lat phae, and another actor o due to imprecie etimation o C opt ). More in detail, each phae i can increae the load o every machine by at mot ρλ i. I u denote the number o phae, then the inal makepan will be no more than ρ u i=1 Λ i. Note that u i=1 Λ i = ( )Λ u 1 u = ( 1 )Λ u 1 u, ince Λ i+1 = Λ i. We alo have Λ u = Λ u 1 < C opt, becaue Λ u 1 < C opt (otherwie in phae u 1 the makepan will not exceed ρλ u 1 according to Lemma 9). Thu we have ρ u i=1 Λ i = ( 1 )ρλ u 1 u < 4ρC opt. A.3 Proo o Theorem 1 In ome o the proo we hall make ue o the ollowing initial et o mall job: (ɛ, M 1 ), ( ɛ, M 1), ( ɛ, M ), ( ɛ, M 1), ( ɛ, M )... }{{} t/ɛ block (3) where the total number o job i t/ɛ, and ɛ i choen o that t/ɛ i integer. According to the algorithm Greedy, only the irt irt job o length ɛ are aigned a good job, while all other job are aigned a bad job. Moreover, all machine in each cla will have the ame load t and t ɛ. Thee job can be reditributed to the machine in order to built arbitrary load (up to ome arbitrarily mall additive ɛ). Taking ɛ 0, we can obtain the ollowing reult. Lemma 19. At the beginning o a chedule by algorithm Greedy, i <, each machine can have a load o t o that all job executed during [0, t] are bad job, and each bad job i extremely mall o that they can be reditributed to create an arbitrary load on any machine. Proo o Theorem 1. We give ive intance each o them reulting in a lower bound or the correponding cae. Cae 1 ( = 1). I , the job equence i ( 1, M +1 ), (, M +1 ) and (1, M 1 ). The Greedy algorithm aign the irt job to machine, and the lat two job to machine 1, which lead to C Greedy = 1 +. In optimal chedule all job are allocated a good job, +1 i.e., C opt = 1. I > 1+ 5, the job equence i ( 1, M ), ( 1, M ) and (1, M 1 ). The Greedy aign the irt job to machine, and the lat two job to machine in 1, which lead to C Greedy =. Again, in optimal chedule all job are allocated a good job, i.e., C opt = 1. Thereore, ρ Greedy = min{1 +, } i tight or any Cae ( = and ). Let l α = 3, S (+1) 6 = 6 i=1 ( 1)i = 1(1 ( 1)6 ) and l good =. The equence o job correpond to the ollowing three tep: (+1) 3

24 Step 1: We ue Lemma 19 to let each machine have and initial load l α S 6 l good + ( 1) 6 o bad job, where l α S 6 l good + ( 1) 6 0 due to Step : Four job arrive: (l good ( 1) 6, M ) and ( lgood, M ). According to Greedy, the irt two job will be aigned to machine 3 and 4 repectively a good job. But the lat two job will be aigned to machine 1 and a bad job. At thi point, all the our machine have the ame load l α S 6. ( 1) 6 Step 3: Thi equence o job arrive: (( 1) 6, M ), (( 1) 5, M ), (( 1) 4, M 1 ), (( 1) 3, M ), (( 1), M 1 ), ( 1, M ). According to Greedy, the irt two job are allocated a good job, while the other are allocated a bad job. At thi point, the load o machine 1 and i l α, while machine 3 and 4 have load l α ( 1). Step 4: Job (1, M 1 ) arrive, which will be aigned to machine 3 a a bad job. Thereore, C Greedy = l α + 1 = 1 +. The optimal chedule i to aign all job a good job. By 3 (+1) calculation we have C opt = 1. Thu, ρ Greedy = (+1) Cae 3 (3 and l good = +, where u i even number. Suppoe 3 (+1) Note that when u, we have a u 0, i.e. 3 i tight or ). Let l α = ( 1/) +1, a i = ( 1) i, S u = u i=1 a i = 1. 1 (1 a u) 1+(+1)a u and Step 1: We ue Lemma 19 to let each machine have an initial load l α S u l good + a u o bad job, where l α S u l good + a u 0 due to 3 1+(+1)a u and Step : Thee job arrive: (l good a u, M ) and ( lgood a u, M ). According to Greedy, the irt job will be aigned to M a good job with one machine each, while the lat job will be aigned to M 1 a bad job with one machine each. At thi point, all the machine ha the ame load o l α S u. Step 3: Thi equence o job arrive: (a u, M ), (a u 1, M ), (a u, M 1 ), (a u 3, M ), (a u 4, M 1 ),..., (a, M 1 ), (a 1, M ). According to Greedy, the irt job will be allocated a good job while the other a bad job. At thi point, each machine in M 1 ha load l α, and each machine in M ha load l α ( 1). Step 4: Job (1, M 1 ) arrive, which i aigned to one machine in M a a bad job. Thereore, C Greedy = l α + 1 = 1 + ( 1/). +1 The optimal chedule will allocate all job a good job. We next give uch an optimal chedule to how that C opt = 1 i achievable: (Step 1 job.) All the job in Step 1 will be allocated a good job in optimal chedule, meaning lα Su lgood +a u or each o M 1 and M ; (Step job.) All the job in Step will be aigned to M ; (Step 3 job.) All the job in Step 3 will be allocate a good job, meaning (a + a a u ) will be aigned to M 1, while the ret o them to M ; (Step 4 job.) The lat job in Step 4 will be aigned to M 1. For the job allocated to M 1, notice that every job o (a +a 4 + +a u ) hould be aigned to one machine, i.e., the load o ome machine are all (a + a a u ), 4