Multiprocessor scheduling with rejection

Transcription

1 Multiprocessor scheduling with rejection Bartal, Y.; Leonardi, S.; Marchetti Spaccaela, A.; Sgall, J.; Stougie, L. Published in: SIAM Journal on Discrete Matheatics DOI: /S Published: 01/01/2000 Docuent Version Publisher s PDF, also known as Version of Record includes final page, issue and volue nubers) Please check the docuent version of this publication: A subitted anuscript is the author's version of the article upon subission and before peer-review. There can be iportant differences between the subitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. The final author version and the galley proof are versions of the publication after peer review. The final published version features the final layout of the paper including the volue, issue and page nubers. Link to publication Citation for published version APA): Bartal, Y., Leonardi, S., Marchetti Spaccaela, A., Sgall, J., & Stougie, L. 2000). Multiprocessor scheduling with rejection. SIAM Journal on Discrete Matheatics, 131), DOI: /S General rights Copyright and oral rights for the publications ade accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requireents associated with these rights. Users ay download and print one copy of any publication fro the public portal for the purpose of private study or research. You ay not further distribute the aterial or use it for any profit-aking activity or coercial gain You ay freely distribute the URL identifying the publication in the public portal? Take down policy If you believe that this docuent breaches copyright please contact us providing details, and we will reove access to the work iediately and investigate your clai. Download date: 28. Nov. 2018

2 SIAM J. DISCRETE MATH. Vol. 13, No. 1, pp c 2000 Society for Industrial and Applied Matheatics MULTIPROCESSOR SCHEDULING WITH REJECTION YAIR BARTAL, STEFANO LEONARDI, ALBERTO MARCHETTI-SPACCAMELA, JIŘÍ SGALL, AND LEEN STOUGIE Abstract. We consider a version of ultiprocessor scheduling with the special feature that jobs ay be rejected at a certain penalty. An instance of the proble is given by identical parallel achines and a set of n jobs, with each job characterized by a processing tie and a penalty. In the on-line version the jobs becoe available one by one and we have to schedule or reject a job before we have any inforation about future jobs. The objective is to iniize the akespan of the schedule for accepted jobs plus the su of the penalties of rejected jobs. The ain result is a 1 + φ copetitive algorith for the on-line version of the proble, where φ is the golden ratio. A atching lower bound shows that this is the best possible algorith working for all. For fixed we give iproved bounds; in particular, for = 2 we give a φ copetitive algorith, which is best possible. For the off-line proble we present a fully polynoial approxiation schee for fixed and a polynoial approxiation schee for arbitrary. Moreover, we present an approxiation algorith which runs in tie On log n) for arbitrary and guarantees a 2 1 approxiation ratio. Key words. ultiprocessor scheduling, on-line algoriths, copetitive analysis, approxiation algoriths AMS subject classification. 68Q25 PII. S Introduction. Scheduling jobs on parallel achines is a classical proble that has been widely studied for ore than three decades [6, 12]. In this paper we consider a version of the proble that has the special feature that jobs can be rejected at a certain price. We call this proble ultiprocessor scheduling with rejection and use the abbreviation MSR. Given are identical achines and n jobs. Each job is characterized by its processing tie and its penalty. A job can be either rejected, in which case its penalty is paid, or scheduled on one of the achines, in which case its processing tie contributes to the copletion tie of that achine. The processing tie is the sae Received by the editors March 13, 1996; accepted for publication in revised for) May 7, 1999; published electronically January 13, A preliinary version of this paper appeared in the Proceedings of the Seventh Annual ACM-SIAM Syposiu on Discrete Algoriths, Atlanta, GA, January 28 30, 1996, SIAM, Philadelphia, 1996, pp Departent of Coputer Science, Tel-Aviv University, Tel-Aviv 69978, Israel yairb@ath. tau.ac.il). The research of this author was supported in part by the Ben Gurion Fellowship, Israel Ministry of Science and Arts. Dipartiento di Inforatica Sisteistica, Università di Roa La Sapienza, via Salaria 113, Roa, Italia leonardi@dis.uniroa1.it, archetti@dis.uniroa1.it). The research of this author was partly supported by ESPRIT BRA Alco II under contract 7141, and by Italian Ministry of Scientific Research Project 40% Algoriti, Modelli di Calcolo e Strutture Inforative. Matheatical Institute, AS CR, Žitná 25, Prague 1, Czech Republic, and Departent of Applied Matheatics, Faculty of Matheatics and Physics, Charles University, Prague, Czech Republic sgall@ath.cas.cz). The research of this author was partially supported by grants A and A of GA AV ČR, postdoctoral grant 201/97/P038 of GA ČR, and cooperative research grant INT /ME-103 fro the NSF USA) and the MŠMT Czech Republic). Part of this work was done at the Institute of Coputer Science, Hebrew University, Jerusale, Israel and was supported in part by a Golda Meir Postgraduate Fellowship. Departent of Matheatics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands leen@win.tue.nl). The research of this author was supported by the Huan Capital Mobility Network DONET of the European Counity. 64

3 MULTIPROCESSOR SCHEDULING WITH REJECTION 65 for all the achines, as they are identical. Preeption is not allowed; i.e., each job is assigned to a single achine and once started is processed without interruption. The objective is to iniize the su of the akespan and the penalties of all rejected jobs. Makespan the length of the schedule) is defined as the axiu copletion tie taken over all achines. In the on-line version of MSR jobs becoe available one by one, and the decision to either reject a job or to schedule it on one of the achines has to be ade before any inforation about following jobs is disclosed. In particular, there ay be no other jobs. On-line algoriths are evaluated by the copetitive ratio; an on-line algorith is c-copetitive if for each input the cost of the solution produced by the algorith is at ost c ties the cost of an optial solution cf. [14]). The ain goal of an on-line MSR algorith is to choose the correct balance between the penalties of the jobs rejected and the increase in the akespan for the accepted jobs. At the beginning, it ight have to reject soe jobs if the penalty for their rejection is sall copared to their processing tie. However, at a certain point it would have been better to schedule soe of the previously rejected jobs since the increase in the akespan due to scheduling those jobs in parallel is less than the total penalty incurred. In this scenario the on-line MSR proble can be seen as a nontrivial generalization of the well-known Rudolph s ski rental proble [11]. In that proble, a skier has to choose whether to rent skis for the cost of 1 per trip or to buy the for the cost of c, without knowing the future nuber of trips. The best possible deterinistic strategy is to rent for the first c trips and buy afterwards. In our proble, rejecting jobs is analogous to renting, while scheduling one job is analogous to buying, as it allows us to schedule 1 ore jobs of no bigger processing tie without extra cost.) Our ain result is a best possible, 1 + φ copetitive algorith for the on-line MSR proble, where φ =1+ 5)/2 is the golden ratio. We prove that no deterinistic algorith that receives as input can achieve a better copetitive ratio independent of. For sall values of we give better upper and lower bounds. In particular, for = 2 we obtain a best possible, φ copetitive algorith. For =3we obtain 2-copetitive algoriths and show a lower bound of Our results should be copared with the current knowledge about on-line algoriths for the classical ultiprocessor scheduling proble. In that proble, each job has to be scheduled; hence it is equivalent to a special case of our proble where each penalty is larger than the corresponding processing tie. Graha s list scheduling algorith schedules each job on the currently least loaded achine and is 2 1 copetitive [7]. It is known that for >3, list scheduling is not optial [5], and in fact there exist 2 ε copetitive algoriths for sall constant ε>0 [2, 10, 1]. The best possible copetitive ratio is known to be between 1.85 and 1.92 see [1]), but its precise value is unknown. In contrast, for the ore general on-line MSR proble we do find the optial copetitive ratio. More surprisingly, our algoriths achieving the optial copetitive ratio schedule the accepted jobs using list scheduling, which is inferior when rejections are not allowed! Next we consider the off-line MSR proble. We present an approxiation algorith with a 2 1 worst-case approxiation ratio running in tie On log n) for arbitrary. We also present a fully polynoial approxiation schee for MSR for any fixed and a polynoial approxiation schee for arbitrary, i.e., where is part of the input.

4 66 BARTAL ET AL. More explicitly, the approxiation schees give algoriths with running tie either polynoial in n and 1/ɛ but exponential in, or polynoial in n and but exponential in 1/ɛ, where ɛ is the axial error allowed. This iplies that for the ore general proble with possible rejection of jobs we have algoriths that are essentially as good as those known for the classical proble without rejection. In fact, our algoriths are based on the techniques used for the proble without rejection, naely, on the fully polynoial approxiation schee for fixed [9] based on a dynaic prograing forulation of the proble) and the polynoial approxiation schee for arbitrary [8]. Obviously, the MSR proble on a single achine is easily solved exactly by scheduling every job whose processing tie does not exceed its penalty, and for 2 it is NP-hard to find the optial solution, siilarly as in the classical case without rejections. The on-line algoriths and lower bounds are presented in sections 3 and 4. Section 5 contains the results of the off-line proble. 2. Notation. An instance of the MSR proble consists of a nuber of achines and a set of jobs J, J = n. We abuse the notation and denote the jth job in the input sequence by j. Each job j J is characterized by a pair p j,w j ), where p j is its processing tie and w j is its penalty. For a set of jobs X J, W X) = j X w j is the total penalty of jobs in X, and MX) = j X p j/ is the su of the loads of the jobs in X, where the load of a job j is defined by p j /. The set B = {j w j p j /} contains jobs with penalty less than or equal to their load. Given a solution produced by an on-line or approxiation algorith, R denotes the set of all rejected jobs, A denotes the set of all accepted job, and T denotes the largest processing tie of all accepted jobs. For their analogues in the optial solution we use R OPT, A OPT, T OPT, respectively. Z OPT denotes the total cost of the optial solution for a given instance of the proble, and Z H is the cost achieved by algorith H. An on-line algorith ON is c-copetitive if Z ON c Z OPT for every input instance. The golden ratio is denoted by φ = 5+1)/ We will often use the property of the golden ratio that φ 1=1/φ. Using list scheduling, the akespan of a schedule is bounded fro above by the processing tie of the job that finishes last plus the su of the loads of all other scheduled jobs [7]. We denote this bound by C LS X) for a set X of scheduled jobs. If l is the job in X that finishes last, then C LS X) =MX {l})+p l MX)+ 1 1 ) 2.1) T. 3. On-line scheduling with rejections. In the first part of this section we present an on-line MSR algorith which works for arbitrary and achieves the best possible copetitive ratio in that case. The corresponding lower bound is given in section 4.2. For fixed 3 this algorith gives the best copetitive ratio we are able to achieve; however, we are not able to prove a atching lower bound. In the second part we present a different algorith which is best possible for the case of two achines. The corresponding lower bound is given in section Arbitrary nuber of achines. Our algorith uses two siple rules. First, all jobs in the set B are rejected, which sees advantageous since their penalty

5 MULTIPROCESSOR SCHEDULING WITH REJECTION 67 is saller than their load. The second rule is inspired by the relation of MSR to the ski rental proble and states that a job is rejected unless its penalty added to the total penalty of the hitherto rejected jobs would be higher than soe prescribed fraction of its processing tie. This fraction paraeterizes the algorith; we denote it by α. Algorith RTPα) Reject-Total-Penaltyα)). i) If a job fro B becoes available, reject it. ii) Let W be the total penalty of all jobs fro J B rejected so far. If a job j =p j,w j ) / B becoes available, reject it if W + w j αp j, otherwise accept it and schedule it on a least loaded achine. In Theore 3.1 we will prove that for given, the algorith is c-copetitive if c and α>0 satisfy c ) 1 3.1) α, c 2+α 2. To obtain a best possible algorith for arbitrary, we use α = φ Then c =1+φsatisfies the inequalities above. For a fixed, the best c is obtained if equality is attained in both cases. For = 2 this leads to α = 2/ and c =1+ 2/ , and for = 3 we get α =2/3 and c = 2. For general we obtain α = 1 2 ) , 2 c= ) Theore 3.1. The algorith RTPα) for achines is c-copetitive if c and α satisfy 3.1). Proof. First we notice that since our algorith uses list scheduling for the accepted jobs, its akespan is bounded by C LS A) =1 1 )T+MA) cf. 2.1)). Hence, Z ON 1 1 ) T + MA)+WR). For any set S R, the right-hand side of this inequality can be rewritten as a su of two ters: 3.2) Z ON MA)+WR S)+MS)) ) ) T + W S) MS). Now, we fix an off-line optial solution. We use the above inequality for the set S =R B) A OPT, the set of all jobs rejected by the algorith in step ii) and accepted in the optial solution. First, we bound the first ter in 3.2). Notice that 3.3) MA) =MA A OPT )+MA R OPT ) MA A OPT )+WA R OPT ), since no job of the set B is accepted by the algorith, and thus the load of each job accepted by the algorith is saller than its penalty. Next we notice that S A OPT, iplying that 3.4) MS) =MA OPT S).

6 68 BARTAL ET AL. Since R OPT and A OPT is a partition of the set of all jobs, and B R, we obtain 3.5) R S =[R R OPT ) R A OPT )] [R B) A OPT ] =R R OPT ) B A OPT ). Fro 3.5) and the definition of B we have 3.6) W R S) =WB A OPT )+WR R OPT ) MB A OPT )+WR R OPT ). Inequalities 3.3), 3.4), and 3.6) together iply that MA)+WR S)+MS) MA OPT )+WR OPT ) Z OPT. To finish the proof, it is now sufficient to show 3.7) 1 1 ) T + W S) MS) 1+α 2 ) T OPT ) 1 α WROPT ), and notice that under our conditions 3.1) on c this is at ost c 1)T OPT +c 1)W R OPT ) c 1)Z OPT. All jobs in S are scheduled in the optial solution and hence have processing tie at ost T OPT. The algorith never rejects such a job if this would increase the penalty above αt OPT, and hence 3.8) W S) αt OPT. For any job j that was rejected by step ii) of the algorith we have w j αp j. Suing over all jobs in S we obtain W S) αms), and hence W S) MS) 1 1 ) W S) 1 1 ) αt OPT = α 1 ) 3.9) T OPT. α α Thus, if T T OPT, 3.7) follows. If T>T OPT, let W be the penalty incurred by the jobs rejected in step ii) of the algorith until it schedules the first job with processing tie T, job j say, having penalty w j. By the condition in step ii) of the algorith, αt W + w j. Conversely, W + w j W S) +WR OPT ), as all jobs rejected in step ii) are in S R OPT, and also the job with processing tie T is in R OPT, since T>T OPT. Thus, 3.10) 1 1 ) T 1 1 ) 1 α W S)+WROPT )) 1 1 ) T OPT ) 1 α WROPT ), using 3.8). Adding 3.9) to 3.10) we obtain 3.7), which finishes the proof. Choosing α = φ 1 and c = φ + 1, both inequalities in 3.1) are satisfied for any, which yields our ain result. For arbitrarily large these values are the best possible.

7 MULTIPROCESSOR SCHEDULING WITH REJECTION 69 Theore 3.2. Algorith RTPφ 1) is 1 + φ)-copetitive. For any choice of and α the bounds on c given by the inequalities 3.1) give a tight analysis of Algorith RTPα), as shown by the following two exaples. First, consider the sequence of two jobs 1 1 α,α 1 ) and 1 ε, 1 ) with ɛ>0arbitrarily sall. RTPα) rejects the first job and accepts the second job, while in the optial solution both jobs are rejected. The copetitive ratio attained on this sequence is 1 ε +α 1 ))/α, which for any α>0 and can be ade arbitrarily close to the first inequality of 3.1). Second, consider the sequence fored by one job 1,α), 2 jobs 1, 1 ), and one job 1, 1). RTPα) rejects the first 1 jobs and accepts job 1, 1), while the optial solution accepts all jobs. The copetitive ratio is 2 + α 2, leading to the second inequality of 3.1) Two achines. To obtain a best possible, φ-copetitive algorith for two achines we use another approach. We siply reject all jobs with penalty at ost α ties their processing tie, where α is again a paraeter of the algorith. Again the optial value is α = φ Algorith RPα) Reject-Penaltyα)). If a job j =p j,w j ) becoes available, reject it if w j αp j, otherwise accept it and schedule it on a least loaded achine. Theore 3.3. The algorith RPφ 1) is φ-copetitive for two achines. Proof. If the algorith does not schedule any job, then Z ON = WJ) 2φ 1)MA OPT )+WR OPT ) 2φ 1)Z OPT φz OPT, and the theore is proved. Otherwise denote by l a job that is finished last by the on-line algorith. Since the algorith uses list scheduling, the akespan is bounded by C LS A) =MA {l})+p l, and therefore we have 3.11) Z ON WR)+MA {l})+p l. Notice that 3.12) W R) =WR R OPT )+WR A OPT ) WR R OPT )+2φ 1)MR A OPT ) by direct application of the rejection rule of algorith RPφ 1). For any job that is accepted by the algorith, the rejection rule of RPφ 1) iplies that its load is not greater than its penalty. Therefore, 3.13) MA {l})=ma {l}) A OPT )+MA {l}) R OPT ) MA {l}) A OPT )+WA {l}) R OPT ). Invoking 3.12) and 3.13) in 3.11) yields 3.14) Z ON WR OPT {l})+2φ 1)MA OPT {l})+p l. We distinguish two cases. In the first case the optial solution rejects job l. Since l is scheduled by the algorith, we have p l φw l, and therefore Z ON φw R OPT )+2φ 1)MA OPT ) φz OPT.

8 70 BARTAL ET AL. In the second case l is accepted in the optial solution. Then, we use the identity p l =2φ 1)M{l})+1 φ 1))p l in 3.14) to obtain Z ON 2 φ)p l + W R OPT )+2φ 1)MA OPT ) 2 φ)z OPT +2φ 1)Z OPT = φz OPT, which copletes the proof. The sae approach can be used for larger as well. However, for >3 this is worse than the previous algorith. An interesting situation arises for = 3. Choosing α =1/2we obtain a 2-copetitive algorith, which atches the copetitive ratio of the algorith RTP2/3) for = 3 in the previous subsection. Whereas RP1/2) rejects all jobs with penalty up to 1/2 of their processing tie, RTP2/3) rejects all jobs with penalty up to 1/3 of their processing tie and also jobs with larger penalty as long as the total penalty paid by the jobs with saller or equal processing ties) reains at ost 2/3 ties the processing tie. We can cobine these two approaches and show that for any 1/3 α 1/2, the algorith that rejects each job with penalty at ost α ties its processing tie, and also if the total penalty is up to 1 α ties its processing tie, is 2-copetitive, too. However, no such cobined algorith is better. 4. Lower bounds for on-line algoriths. In the first part of this section we give the lower bound for a sall nuber of achines. In particular it shows that the algorith presented in section 3.2 is best possible for = 2. In the second part we exhibit the lower bound for algoriths working for all Sall nuber of achines. Assue that there exists a c-copetitive on-line algorith for achines. We prove that c satisfies c ρ, where ρ is the solution of the following equation: 4.1) ρ 1 + ρ =ρ. For = 2 we get ρ = φ, and hence prove that the algorith RPφ 1) is best possible. For = 3 we get ρ 1.839, and so on. Notice that for arbitrary this proves only that the copetitive ratio is at least 2. Theore 4.1. For any c-copetitive algorith for MSR on achines, it holds that c ρ, where ρ satisfies 4.1). Proof. Given, let ρ be the solution of 4.1). Consider an adversary providing a sequence of jobs, all with processing tie 1. The first job given has penalty w 1 = 1/ρ. If the on-line algorith accepts this job, the sequence stops and the algorith is ρ-copetitive. Otherwise, a second job is given by the adversary with penalty w 2 =1/ρ 2. Again, accepting this job by the on-line algorith akes the sequence stop and the copetitive ratio is ρ. Rejection akes the sequence continue with a third job. This process is repeated for at ost 1 jobs with penalties w j =1/ρ j for 1 j = 1. If the on-line algorith accepts any job in this sequence, job k say, the adversary stops the sequence at that job, yielding a copetitive ratio of the on-line algorith on this sequence of k jobs of Z ON Z k 1 j=1 1 ρ j 1+ = OPT k = ρ, j=1 1 ρ j since for any such k 1 in the optial solution all jobs are rejected.

9 MULTIPROCESSOR SCHEDULING WITH REJECTION 71 Otherwise, if none of the first 1 jobs are accepted by the on-line algorith, another job is presented with penalty w = 1. In the optial solution all jobs are accepted and scheduled in parallel, giving cost 1. The on-line cost is equal to the su of the penalties of the first 1 jobs plus 1, independent of whether the last job is accepted or rejected. Thus, Z ON Z OPT 1 =1+ j=1 1 ρ j. By 4.1), this is exactly ρ, and the theore follows. Corollary 4.2. For two achines, no on-line algorith has copetitive ratio less than φ Arbitrary nuber of achines. Now we prove the lower bound on algoriths working for arbitrary. The sequence of jobs starts as in the previous section, but additional ideas are necessary. Theore 4.3. There exists no on-line algorith that is β-copetitive for soe constant β<1+φ and all. Proof. All jobs in the proof have processing tie 1. All logariths are base 2. For contradiction, we assue that the on-line algorith is β-copetitive for a constant β < 1+φ, and is a sufficiently large power of two. Let a i = log ) i+1, and let k be the largest integer such that log + k i=1 a i <. Calculation gives k = log / log log 1. Consider again an adversary that intends to provide the following sequence of at ost jobs all with processing tie 1): 1 job with penalty 1/1 + φ), 1 job with penalty 1/1 + φ) 2,. 1 job with penalty 1/1 + φ) log, a 1 jobs with penalty 1/a 1,. a k jobs with penalty 1/a k. As in the proof of Theore 4.1 we argue that if the on-line algorith accepts one of the first log jobs, the adversary stops the sequence and the copetitive ratio is 1+φ. Therefore, any β-copetitive algorith has to reject the first log jobs. Now, let b i be the nuber of jobs with penalty 1/a i that are rejected by the β-copetitive algorith. The penalty the algorith pays on those jobs is b i /a i. Since there are less than jobs, the optial cost is at ost 1. Thus the total penalty incurred by the on-line algorith has to be at ost β, and in particular there has to exist l k such that b l /a l β/k < 3/k. Fix such l. Now consider the following odified sequence of at ost 2 jobs again all with

10 72 BARTAL ET AL. processing tie 1): 1 job with penalty 1/1 + φ), 1 job with penalty 1/1 + φ) 2,. 1 job with penalty 1/1 + φ) log, a 1 jobs with penalty 1/a 1,. a l jobs with penalty 1/a l, M jobs with penalty 6, where M = +1 l i=1 a i b i ). The sequence is identical up to the jobs with penalty 1/a l, and hence the on-line algorith behaves identically on this initial subsequence. In particular, it also rejects all first log jobs paying a penalty of at least log j=1 1+φ) j =1 1+φ) log )/φ φ 1 1/ for the. Then it also rejects b i jobs with penalty 1/a i, for i l, paying penalty l i=1 b i/a i for the. The on-line algorith has to accept all jobs with penalty 6, since the adversary will present at ost 2 jobs, and hence scheduling the all would lead to a cost of at ost 2. By suing the nubers, it follows that the on-line algorith schedules exactly + 1 jobs. Thus, its akespan is at least 2, and its total cost is at least 1+φ 1/. To finish the proof, it is sufficient to present a solution with cost 1+o1). Consider the solution that rejects 1 + log jobs with penalty 1/a 1, b 1 jobs with penalty 1/a 2, b 2 jobs with penalty 1/a 3,. b l 2 jobs with penalty 1/a l 1, b l 1 + b l jobs with penalty 1/a l and schedules all reaining jobs optially. First we verify that this description is legal, i.e., there are always sufficiently any jobs with given penalty. By definition, b i a i a i+1. For sufficiently large, we have 1 + log <a 1, and due to our choice of l, we also have b l 1 + b l a l 1 +3a l /k a l. In the presented schedule one ore job is rejected than in the solution produced by the on-line algorith, and hence there are only jobs to be scheduled. Thus, the akespan is 1. The penalty paid is 1 + log l 1 b i + + b l = 1 + log a 1 a i=1 i+1 a l log ) l 1 b i + b l. log a i=1 i a l The su in the second ter is less than the penalty paid by the on-line algorith, and hence this ter is bounded by O1/ log ). The last ter is bounded due to our choice of l; naely, it is O1/k) =Olog log / log ). Thus, the total penalty paid is Olog log / log ) =o1), and the total cost is 1 + o1).

11 MULTIPROCESSOR SCHEDULING WITH REJECTION Off-line scheduling with rejection An approxiation algorith for arbitrary nuber of achines. In this section we give a 2 1 )-approxiation algorith for MSR on achines. Our lower bounds iply that such a ratio cannot be achieved by an on-line algorith. The algorith rejects all jobs in the set B = {j w j p j /}. Fro all other jobs it accepts soe nuber of jobs with the sallest processing tie and chooses the best aong such solutions. Algorith APPROX. i) Sort all jobs in J B according to their processing ties in nondecreasing order. ii) Let S i,0 i J B, be the solution that schedules the first i jobs fro J B using list scheduling and rejects all other jobs. Choose the solution S i with the sallest cost. Note that step ii) of the algorith takes tie On log ) or On) in case n), as we can build the schedules increentally, and the bookkeeping of penalties for rejected jobs is siple. Thus, the whole algorith runs in tie On log n), independent of. A perforance analysis leads to the following worst-case ratio. Theore 5.1. Algorith APPROX achieves Z H 2 1 )ZOPT, where Z H is the cost of the solution found by the algorith. Proof. We assue that the jobs fro J B are ordered 1, 2,..., J B, according to the ordering given by step i) of the algorith. If the optial solution rejects all jobs fro J B, by the definition of B it is optial to reject all jobs fro B as well. Thus the solution S 0 that rejects all jobs is optial and Z H = Z OPT. Otherwise let l be the last job fro J B accepted in the optial solution. Consider the solution S l, which schedules all jobs up to l. Let A = {1,...,l} be the set of all jobs scheduled in S l. Job l has the largest running tie of all scheduled jobs, and since we use list scheduling, the akespan of S l is at ost C LS A) =MA)+ 1 1 ) p l MA)+ 1 1 ) Z OPT. Since the cost of the algorith is at ost the cost of S l,wehave Z H WJ A)+MA)+ 1 1 ) Z OPT = WA OPT J A)) + W R OPT J A)) + MR OPT A)+MA OPT A)+ 1 1 ) Z OPT. By the choice of l, A OPT J A) B, and thus W A OPT J A)) MA OPT J A)). Moreover, since A does not contain any job of B, MR OPT A) WR OPT A). These observations inserted in the above inequality yield Z H W R OPT )+MA OPT )+ 1 1 ) Z OPT 2 1 ) Z OPT. That the ratio is tight is shown by the following instance with jobs and achines): p 1 = = p =1,w 1 =1 ɛ, and w 2 = = w = 1 1 ɛ). The heuristic will reject all jobs resulting in Z H =1+ 1 )1 ɛ). In the optial solution all jobs are accepted; hence Z OPT = 1. Therefore, Z H /Z OPT can be ade arbitrarily close to 2 1.

12 74 BARTAL ET AL. This exaple also shows that any heuristic that rejects all jobs in the set B has a worst-case ratio no better than 2 1, since there is no scheduling at all involved in it. Thus, the only way in which an iproveent ight be obtained is by also possibly accepting jobs in the set B A fully polynoial approxiation schee for fixed. For the offline MSR proble there exists a fully polynoial approxiation schee for fixed. The proof uses a rounding technique based on dynaic prograing, as was developed in [9] for the classical akespan proble. Lea 5.2. The MSR proble with integer processing ties and penalties can be solved in tie polynoial in n and Z OPT ). Proof. We use dynaic prograing. Let M i represent the current load of achine i, i =1,...,. We copute for each M 1,...,M Z OPT the inial value of total penalty to be paid that can be achieved with these loads. We denote this value after the first j jobs are rejected or scheduled by W j M 1,...,M ) and define it to be whenever M i < 0 for soe i. At the sae tie we copute the inial cost of a schedule that can be achieved with given loads M 1,...,M, denoted ZM 1,...,M ). For M 1,...,M 0 these values can be coputed recursively as follows: W 0 M 1,...,M )=0, W j M 1,...,M ) = in{ w j + W j 1 M 1,...,M ), in W j 1 M 1,...,M i 1,M i p j,m i+1,...,m )}, i ZM 1,...,M )=W n M 1,...,M ) + ax M i. i We copute the values in the order of increasing ax i M i. Assoonasax i M i reaches the cost of the current optial solution, which is the sallest value of Z coputed so far, we stop, as we know it is a global optiu. Theore 5.3. For any ε 0, there exists an ε-approxiation algorith for the MSR proble that runs in tie polynoial in the size of the input instance, n and 1/ε. Proof. Given an instance I of the MSR proble with n jobs and achines, we first use the approxiation algorith fro section 5.1 to obtain the cost Z H. Now we define an instance I by rounding the processing ties and the penalties of the jobs in I. Naely, the processing tie p j and the penalty w j of job j in I are p j = p j/k and w j = w j/k, where k = εz H /2n. We obtain the optial solution of I by the dynaic prograing algorith presented in the proof of Lea 5.2 and derive an approxiate solution for I by scheduling the respective jobs on the sae achines as in the optial solution for I. The cost Z Ak) of the approxiate solution deviates fro the optial solution for I by at ost nk = εz H /2. Therefore, by applying the lower bound Z OPT Z H /2 we obtain Z Ak) Z OPT Z OPT 2nk Z H = ε. By Lea 5.2 it follows that the running tie of the approxiation algorith is polynoial in n and Z OPT I )). The theore follows since Z OPT I ) Z OPT I)/k 2Z H /k, and hence Z OPT I ) 4n/ε.

13 MULTIPROCESSOR SCHEDULING WITH REJECTION A polynoial approxiation schee for arbitrary. For arbitrary we will design a polynoial approxiation schee PAS) based on the PAS for the akespan proble in [8]. Given an instance with n jobs, achines, and ɛ>0, we are to find an ɛ- approxiate solution. As an upper bound U on the solution value we use the outcoe Z H of the heuristic presented in section 5.1. Notice that all jobs with p j >Uwill be rejected. Thus, all jobs that are possibly scheduled have processing ties in the interval [0,U]. Fro Theore 5.1 we have a lower bound on the optial solution that we denote by L = Z H /2=U/2. We define the set S = {j p j [0, ɛl/3]}, a set of jobs with relatively sall processing ties. Let D = {j j/ S}. The reaining interval ɛl/3,u] is partitioned into s 18 1/ɛ 2 subintervals l 1,l 2 ], l 2,l 3 ],..., l s,l s+1 ] of length ɛ 2 L/9 each, with l 1 = ɛl/3 and l s+1 U. Let D i be the set of jobs with processing tie in the interval l i,l i+1 ], and let the jobs in each such set be ordered so that the penalties are nonincreasing. As before, define the set B = {j w j p j /}. First we will describe how, for any subset of D, we generate an approxiate solution with value Z Hɛ) ). For any such set we deterine a schedule for all the jobs in with an ɛ/3-approxiate akespan using the PAS in [8]. All other jobs in D, i.e., all jobs in D, are rejected. Jobs in the set S that have w j 1 p j, i.e., jobs in the set S B, are scheduled in any order according to the list scheduling rule starting fro the ɛ/3-approxiate schedule deterined before. The reaining jobs, j S B, are considered in any order. Each next job is rejected if its assignent to a least loaded achine would cause an increase of the akespan; otherwise it is assigned to a least loaded achine as indicated by list scheduling. This procedure is applied to every set Dy 1,...,y s ) D, where Dy 1,...,y s ) denotes the set that is coposed of the first y i eleents in the ordered set D i, i = 1,...,s. In this way an approxiate solution Z Hɛ) Dy 1,...,y s )) is found for each set Dy 1,...,y s ). The iniu value over all these sets, Z Hɛ) = in y ZHɛ) Dy 1,...,y s )), 1,...,y s) is taken as the output of our procedure. Theore 5.4. For any ɛ>0the algorith Hɛ) described above runs in tie polynoial in n and and yields Z Hɛ) Z OPT 1+ɛ. Proof. The proof consists of two steps. First, consider the set A OPT D of jobs in D that are accepted in the optial solution. Applying the heuristic procedure described above to this set of jobs yields the approxiate solution Z Hɛ) A OPT D). We will prove that Z Hɛ) A OPT 5.1) D) Z OPT 1+ ɛ 3. In the second step we analyze how uch the set A OPT D ay differ fro Dy 1,...,y s ). Assue that for i =1,...,s, A OPT D consists of yi OPT jobs fro the set D i. These yi OPT jobs are not necessarily the first yi OPT jobs in the ordered set D i, but we will show that 5.2) Z Hɛ) Dy1 OPT,...,ys OPT )) Z Hɛ) A OPT D)+ 2 3 ɛl.

14 76 BARTAL ET AL. Inequalities 5.1) and 5.2) iply that Z Hɛ) Dy1 OPT,...,ys OPT )) Z OPT 1+ɛ. Since, obviously, Z Hɛ) Z Hɛ) Dy1 OPT,...,ys OPT )), the theore follows. In order to prove inequality 5.1) two cases are distinguished. 1) The copletion ties of the various achines in the heuristic solution corresponding to Z Hɛ) A OPT D)) differ by no ore than ɛl/3. The resulting akespan is the sae as the akespan after scheduling the jobs in A OPT D and S B, and due to our assuption it is at ost MA OPT D)+MS B)+ɛL/3. The weight of all rejected jobs is at ost W S B)+WD A OPT ). Thus Z Hɛ) A OPT D) MA OPT D)+MS B)+ ɛl 3 + W S B)+WD AOPT ). Using the definition of the set B, we have for the optial solution Z OPT MA OPT D)+MS B)+WS B)+WD A OPT ). Fro these two inequalities 5.1) follows iediately. 2) The copletion ties of the achines differ by ore than ɛl/3. Since the processing tie of each job in S is less than ɛl/3, we know that no job in the set S B is rejected, and scheduling all jobs in S has not increased the akespan coputed for the set A OPT D. Let C Hɛ) A OPT D) and C OPT A OPT D) denote, respectively, the ɛ/3-approxiate and the optial akespan for the jobs in A OPT D. In this case and Z Hɛ) A OPT D) =C Hɛ) A OPT D)+WD A OPT ) Z OPT C OPT A OPT D)+WD A OPT ). Moreover, since we have used an ɛ/3-approxiate algorith for scheduling the jobs in A OPT D, wehave C Hɛ) A OPT D) 1+ ɛ ) C OPT A OPT D). 3 Inequality 5.1) results fro the above three inequalities. In order to prove 5.2) we need to bound the extra error that ight occur due to the fact that A OPT D Dy1 OPT,...,ys OPT ). Notice that, for any D i, i =1,...,s, the difference in processing tie between any two jobs in D i is at ost ɛ 2 L/9, and that Dy1 OPT,...,ys OPT ) contains the jobs with larger penalties in D i. The latter iplies that the extra error can be due only to the fact that the first yi OPT jobs in D i have longer processing ties than those in A OPT D i. Since the processing tie of a job in D is at least ɛl/3 and U 2L, no ore than 6/ɛ jobs fro D are scheduled on any achine. Therefore the overall extra contribution to the akespan due to the fact that A OPT D Dy1 OPT,...,ys OPT ) can be no ore than 6/ɛ)ɛ 2 L/9)=2ɛL/3, which iplies inequality 5.2). This copletes the proof of correctness of the approxiation. The running tie of the algorith is doinated by the tie required to copute the heuristic Z Hɛ) Dy 1,...,y s )) for each possible set of values y 1,...,y s, such that

15 MULTIPROCESSOR SCHEDULING WITH REJECTION 77 0 y i D i,i=1,...,s. Since y i, i =1,...,s, satisfies 1 y i n, there are at ost n s = On 18 1/ɛ2 ) possible sets of values y 1,...,y s. For each of these sets an ɛ-approxiate schedule is coputed using the algorith in [8], taking On/ɛ) 9/ɛ2 ); attaching the jobs in the set S just adds On 2 ) tie to each of these coputations. Hence, the overall running tie of the algorith is On 3 /ɛ) 9/ɛ2 ). This establishes that the algorith is a polynoial approxiation schee for the proble with arbitrary. 6. Open probles and recent developents. Soe open probles reain. For the on-line proble tight algoriths for the case of fixed other than =2 are still to be established. For the off-line proble perhaps better heuristics ay be found by iproving the rejection strategy proposed in the algorith in section 5.1. Seiden [13] has proved new results related to our proble. For the variant of deterinistic preeptive scheduling with rejection he gives a )/ copetitive algorith for any nuber of achines, thus showing that allowing preeption can provably be exploited. Interestingly, this yields yet another 2-copetitive algorith for three achines. Also, Seiden notes that our Theore 4.1 yields a lower bound for preeptive scheduling as well and hence yields a lower bound of 2 for general nuber of achines. For two achines, this shows that our algorith RPφ 1) is best possible even aong all preeptive algoriths. For three achines, an interesting open proble is to establish whether preeption allows a better copetitive ratio. The best upper bound of 2 and the best lower bound of for preeptive algoriths still coincide with those shown in this paper for nonpreeptive algoriths. Seiden [13] also studies randoized scheduling with rejection, both preeptive and nonpreeptive. He gives algoriths which are better than deterinistic for a sall nuber of achines, and in particular are 1.5-copetitive for two achines, both preeptive and nonpreeptive; this is best possible for two achines. In both cases the question of whether randoized algoriths for any nuber of achines can be better than their deterinistic counterparts reains open. Epstein and Sgall [4] presented polynoial tie approxiation schees for related achines for various objectives, including MSR, thus generalizing the polynoial tie approxiation schee given in this paper. Engels et al. [3] study scheduling with rejection where, in the objective, the akespan is replaced by the su of the copletions ties. Acknowledgents. We thank Giorgio Gallo for having drawn our attention to this scheduling proble. We thank the anonyous referees for nuerous helpful coents. REFERENCES [1] S. Albers, Better bounds for online scheduling, in Proceedings of the 29th Annual ACM Syposiu on Theory of Coputing, ACM, New York, 1997, pp [2] Y. Bartal, A. Fiat, H. Karloff, and R. Vohra, New algoriths for an ancient scheduling proble, J. Coput. Systes Sci., ), pp [3] D. W. Engels, D. R. Karger, S. G. Kolliopoulos, S. Sengupta, R. N. Ua, and J. Wein, Techniques for scheduling with rejection, in Proceedings of the 6th Annual European Syposiu on Algoriths, Lecture Notes in Coput. Sci. 1461, Springer-Verlag, New York, 1998, pp [4] L. Epstein and J. Sgall, Approxiation Schees for Scheduling on Uniforly Related and Identical Parallel Machines, Technical Report KAM-DIMATIA Series , Charles University, Prague, Czech Republic, 1998.

16 78 BARTAL ET AL. [5] G. Galabos and G. J. Woeginger, An on-line scheduling heuristic with better worst case ratio than Graha s list scheduling, SIAM J. Coput., ), pp [6] M. R. Garey and D. S. Johnson, Coputers and Intractability: A Guide to the Theory of NP copleteness, W. H. Freean, San Francisco, CA, [7] R. L. Graha, Bounds for certain ultiprocessor anoalies, Bell Syste Tech., ), pp [8] D. S. Hochbau and D. B. Shoys, Using dual approxiation algoriths for scheduling probles: Theoretical and practical results, J. Assoc. Coput. Mach., ), pp [9] E. Horowitz and S. Sahni, Exact and approxiate algoriths for scheduling non-identical processors, J. Assoc. Coput. Mach., ), pp [10] D. R. Karger, S. J. Phillips, and E. Torng, A better algorith for an ancient scheduling proble, J. Algoriths, ), pp [11] R. M. Karp, On-line algoriths versus off-line algoriths: How uch is it worth to know the future?, in Proceedings of the IFIP 12th World Coputer Congress. Vol. 1: Algoriths, Software, Architecture, J. van Leeuwen, ed., Elsevier Science Publishers, Asterda, 1992, pp [12] E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shoys, Sequencing and scheduling: Algoriths and coplexity, in Handbooks in Operations Research and Manageent Science, Vol. 4: Logistics of Production and Inventory, S. C. Graves, A. H. G. Rinnooy Kan, and P. Zipkin, eds., North Holland, Asterda, 1993, pp [13] S. S. Seiden, More Multiprocessor Scheduling with Rejection, Technical Report Woe-16, Departent of Matheatics, TU Graz, Graz, Austria, [14] D. D. Sleator and R. E. Tarjan, Aortized efficiency of list update and paging rules, Co. Assoc. Coput. Mach., ), pp