J Sched (04) 7:87 93 DOI 0007/095-03-036-0 Preemptive olie chedulig with rejectio of uit job o two uiformly related machie Leah Eptei Haa Zebedat-Haider Received: November 0 / Accepted: March 03 / Publihed olie: April 03 Spriger Sciece+Buie Media New York 03 Abtract We coider preemptive olie ad emi-olie chedulig of uit job o two uiformly related machie Job are preeted oe by oe to a algorithm, ad each job ha a rejectio pealty aociated with it A ew job ca either be rejected, i which cae the algorithm pay it rejectio pealty, or it ca be cheduled preemptively o the machie, i which cae it may icreae the maximum completio time of ay machie i the chedule, alo kow a the makepa of the cotructed chedule The objective i to miimize the um of the makepa of the chedule of all accepted job ad the total pealty of all rejected job We tudy two verio of the problem The firt oe i the olie problem where the job arrive uorted, ad the ecod variat i the emi-olie cae, where the job arrive orted by a o-icreaig order of pealtie We alo how that the variat where the job arrive orted by a o-decreaig order of pealtie i equivalet to the uorted oe We deig optimal olie algorithm for both cae Thee algorithm have maller competitive ratio tha the optimal competitive ratio for the more geeral problem with arbitrary proceig time (except for the cae of idetical machie), but larger competitive ratio tha the optimal competitive ratio for preemptive chedulig of uit job without rejectio Keyword Olie chedulig Uiformly related machie Speed ratio Preemptive chedulig L Eptei (B) H Zebedat-Haider Departmet of Mathematic, Uiverity of Haifa, 39050 Haifa, Irael e-mail: lea@mathhaifaacil H Zebedat-Haider e-mail: ha_haider@hotmailcom Itroductio Problem defiitio I thi paper we coider olie preemptive chedulig with rejectio o two uiformly related machie The firt ad ecod machie are deoted by M ad M, repectively, ad machie M i ha a peed i Without lo of geerality we aume that ad, for The value i called the peed ratio of the machie A arrivig tream of job i deoted by J {,,,} (where i ot kow i advace) Each job j ( j,,) ha a uit proceig requiremet (or ize), ad it i characterized by it pealty w j [0, + ] Job are preeted oe by oe to a cheduler A job ca be either rejected, i which cae the cheduler pay it pealty, or it ca be accepted, i which cae it mut be cheduled preemptively o the machie A preemptio i allowed, a cheduled job ca be plit arbitrarily ad aiged to be ru durig time lot, poibly o differet machie, uch that it i ot aiged to ru i parallel o the two machie, ad uch that the total proceig time that ca be completed durig the aiged time lot i equal to it proceig requiremet, that i, to I the olie verio of the problem, job arrive oe by oe, ad the full deciio for each job ha to be made before ay iformatio about the ext job i revealed, that i, ot oly the deciio whether a job i rejected or accepted mut be doe upo the arrival of the job but alo the complete aigmet to machie ad time lot of a job hould be performed before the ext job i preeted The makepa of a chedule i the lat time that ay job i completed, or alteratively, the lat completio time of ay machie The objective i to miimize the um of the makepa of the chedule (which wa created olie for all accepted job) ad the total pealty of all the rejected job 3
88 J Sched (04) 7:87 93 I competitive aalyi, the performace of a olie algorithm i compared to that of the optimal offlie algorithm Thi i a type of wort-cae aalyi For a (olie or offlie) algorithm A we let Z A (I ) deote the cot of A for a iput I Weletopt deote a optimal offlie algorithm For each problem ad iput we coider a fixed optimal olutio opt (i ome cae a pecific optimal olutio i choe for the aalyi), ad the competitive ratio of a olie algorithm Z Alg i up Alg (I ) I A olie (determiitic) algorithm Alg Z opt (I ) i called the bet poible (or a optimal) olie algorithm if there i o determiitic olie algorithm for the dicued problem with competitive ratio maller tha that of AlgWe ay that a algorithm i C-competitive (for ome C ) if it competitive ratio i at mot C Previou work Multiproceor chedulig with rejectio (without preemptio, ie, for the variat where each job mut ru cotiuouly o oe machie) wa firt itroduced by Bartal et al (000) They coidered chedulig with rejectio o idetical machie, ad preeted a optimal olie algorithm 5+ whoe competitive ratio i + φ 68, where φ i the golde ratio For a mall umber of machie, they improved thi reult by givig a optimal olie algorithm whoe competitive ratio i φ 68 for m I the ame paper they alo preeted a -competitive olie algorithm for m 3 (i fact, they preeted two ditict algorithm achievig thi reult) The lower boud for m ca be geeralized for ay fixed m, givig a fuctio of m tedig to for large value of m, ad i particular, leadig to a lower boud of 89 for m 3 Thee lat lower boud (for fixed value of m) are valid for uit job The preemptive problem wa tudied by Seide (00), who deiged a algorithm of competitive ratio at mot 38743, ad provided a lower boud of 457 He alo proved that a obliviou algorithm, that i, a algorithm which doe ot take the rejectio pealtie of accepted job ito accout i it chedulig policy, ha a competitive ratio of at leat 3346 For two uiformly related machie, He ad Mi (000) provided a o-preemptive olie algorithm, extedig the algorithm of Bartal et al (000) for two idetical machie, ad howed that the overall competitive ratio (the upremum ratio over all value of )iφ, the ame a for the cae without rejectio (Cho ad Sahi 980; Eptei et al 00) He ad Mi (000) aalyzed the competitive ratio a a fuctio of, ad howed that for ay peed ratio φ, thi pecific geeralizatio of the problem with two machie i equivalet to it pecial cae without rejectio, i the ee that the competitive ratio i equal to the oe for the pecial cae, ie, to + They alo deiged a algorithm for <φ A modificatio of thi algorithm (by a chage of the parameter) wa how to be optimal for 385 <φby Dóa ad He (006), who alo proved lower boud for ubiterval of (, 385) howig that the competitive ratio for the problem with rejectio i trictly higher tha the pecial cae without rejectio Dóa ad He (006) alo tudied preemptive chedulig with rejectio o two uiformly related machie, ad preeted a optimal algorithm of competitive ratio + 4 + for ay Olie preemptive chedulig without rejectio o idetical ad uiformly related machie wa tudied i a equece of paper (Che et al 995; Eptei et al 00; Eptei ad Sgall 000; We ad Du 998; Eptei ad Favrholdt 00; Eptei 00; Seide et al 000; Ebeledr et al 009; Ebeledr ad Sgall 0) For two uiformly related machie a optimal preemptive olie algorithm (for job of arbitrary ize) ha the competitive ratio (+), obtaiig it maximum value 4 3 ++ for The pecial cae of o-icreaig job ize, which are geeralizatio of the cae of uit job, were tudied for idetical machie i Seide et al (000), ad their boud are i fact valid for uit job, a the lower boud i proved for thi cae The tight boud o the competitive ratio for two machie wa how to be 6 5, ad it wa proved that the overall ratio (over all umber of machie) i approximately 366 Ebeledr ad Sgall (0) tudied thi problem for multiple uiformly related machie ad alo howed that their boud for o-icreaig job hold for the cae of uit job They proved a overall upper boud of 8 5 The cae of two uiformly related machie ad oicreaig ize of job wa tudied i Eptei ad Favrholdt (00) Oce agai, the reult are valid for uit job a well The tight boud o the competitive ratio i thi cae are 3+3 (+) 3+ for 3, ad for 3 The overall ratio ++, agai achieved for i 6 5 3 Our reult We give tight boud for the preemptive olie ad emiolie chedulig problem with rejectio o two uiformly related machie with uit proceig time I additio to the olie variat, where job are preeted i a arbitrary order, we tudy a emi-olie verio where we aume that the job arrive orted by their pealtie, i a o-icreaig order The reult for the olie cae are valid for the emi-olie variat where job arrive orted i the oppoite order We preet optimal preemptive olie algorithm for both verio ad prove tight lower boud for every value of The competitive ratio for the olie cae i 9 +0++(+) 4, + while the competitive ratio for the emi-olie cae i The ecod value i trictly maller tha the firt for ay Comparig our boud with the boud of Dóa ad 3
J Sched (04) 7:87 93 89 He (006) (which i valid for arbitrary ize) we fid that our boud for the olie variat i lower, except for the cae, for which the boud are both equal to φ Othe other had, the boud for the problem with uit job without rejectio, which i a pecial cae of our emi-olie problem, are trictly lower I particular, our boud for i 44, which i much higher tha (Seide et al 000; Eptei ad Favrholdt 00) I what follow we call the olie problem that we tudy URU, ad the emi-olie problem i called URUD 4 Notatio ad overview For a et of job B, weletw(b) deote the total rejectio pealty of the job i B, that i, w(b) j B w j The algorithm that we preet ever itroduce idle time We let L j ad L j (for j 0) deote the load of the two machie after the jth job wa either cheduled or rejected (jut before the ( j + )th job arrive, or at termiatio if j ) The algorithm ued i thi paper keep the ivariat L j L j L j L j, ad do ot modify the load outide Procedure aig Thi lat procedure aig job a i the algorithm preeted i Bruo ad Gozalez (976), Labetoulle et al (984), that i, M ever ha a maller load tha the load of M, a ew job i aiged uch that a maximal part of it will ru tartig the previou completio time of M ad util the makepa (or util thi job i completed), ad the remaiig part (if the job wa ot aiged completely) i aiged to M The algorithm of Dóa ad He (006) ue thi aigmet procedure a well We ue the propertie that the makepa of opt for oe uit job i (which i achieved by aigig it completely to M, without itroducig ay idle time), ad that for k job it i poible to aig them uch that the makepa i exactly + k [ee for example Horwath et al (977)] I Sect we coider uorted iput, ad i Sect 3 we coider orted iput The algorithm for uorted iput i alog the lie of the algorithm of He ad Mi (000), Dóa ad He (006) (but the parameter i differet) The algorithm for orted iput treat the firt two job differetly The reao for doig thi i that oce the algorithm chedule oe job, it cot for chedulig aother job i fairly mall, while rejectig thi job combied with acceptig the previou job would icur a large cot, ule the ecod job ha a much maller rejectio pealty Uorted iput The algorithm A metioed above, the olie preemptive algorithm that we deig for URU i baically the oe of Dóa ad He (006) (with differet parameter) It coit of two part, a imple rejectio trategy, that decide which job are rejected ad which are accepted, ad a chedulig algorithm, that aig the accepted job to machie a explaied i Bruo ad Gozalez (976) ad Labetoulle et al (984) We ue the ymbol α to deote a parameter i [ +, Let L 0 L0 0 For every ew job j act a follow: If w j α, the reject j, ad let L j L j, L j L j Otherwie, apply procedure aig( j) ] Schedule j a follow Aig a portio of ize j L j L j of j to M (by the required ivariat, proved i Lemma by iductio, we have 0 L j L j, o thi portio i well-defied, but may be empty, if L j L j 0) ad the remaiig part (of ize j 0) to M Thee part of j are aiged tartig the firt time that the machie are free, that i, without itroducig idle time Let L j L j + j created betwee the part of oe job, L j L j No overlap ca be Lemma Ay algorithm that aig all job uig Procedure aig(j) i well-defied, that i, for uch a algorithm, L j L j L j L j hold for all 0 j, where i the legth of the iput I additio, if exactly two job have bee accepted ad cheduled by the time whe j job have arrived ad have bee dealt with, the L j, ad if l job have bee accepted ad cheduled after j job have arrived, the L j + l + (+) Proof The firt property i proved by iductio For j 0we have L 0 L0 j 0 Aume that L L j L j for ome 0 j If job j i rejected, the L j L j, L j L j, ad we are doe Otherwie, L j L j, L j L j + j L j L j, ad o L j L j I additio, L j L j + L j +, ad o L j L j L j Next, we prove the other propertie Aume that out of the firt j job exactly oe job wa accepted, ad aume additioally that job j i accepted ad aiged We have L j 0, ad L j The j L j L j We have L j L j + j + The load do ot chage util aother job i accepted Thi prove the firt part If out of the firt j job exactly l job were accepted ad cheduled, the we fid l L j + L j, ad ice L j L j +, we get l ( + )L j, or alteratively, L j l + + (+) 3
90 J Sched (04) 7:87 93 I thi ectio we chooe α a follow: let α 9 +0+ (+) (+), for ay Thi i the uique poitive olutio of the followig equatio with the variable x: x + + x () The algorithm uorted(α) with thi pecific choice of α will called uorted i what follow Uig imple algebra, it ca be how that + <α< for ay (uig (3 + ) < 9 + 0 + <(3 + 3) ) I additio, we fid (uig α> + + ), > α, ad therefore, uig the defiitio of α, wehave( + )α + + α, which give ( + )α α + < α, provig α< (+) 9 +0++(+) Let R α α+ 4 + the pecial cae, the boud R φ Note that i 5+ 68 i the ame a the boud of o-preemptive geeral cae, ad the algorithm i a pecial cae of the oe of Bartal et al (000) (our algorithm doe ot ue preemptio for ) The value of R i mootoically decreaig a a fuctio of Aalyi Lemma The competitive ratio of ay olie algorithm for URU i at leat R for ay Thi hold alo for iput orted by a o-decreaig order of rejectio pealtie Proof Let H be a olie algorithm, w α, ad w Deote the iput coitig of oly the firt job by I, ad let the iput coitig of two job be I If the firt job i accepted by H, the Z opt (I ) α, Z H (I ), ad Z H (I ) Z opt (I ) α R If it rejected, the Z opt (I ) +, while Z H (I ) α + Itfollowthat Z H(I ) Z opt (I ) α+ R, + by the defiitio of α ad R Coider a fixed optimal olutio opt Aume by cotradictio that the competitive ratio of uorted for ome itace i trictly above R, that i, there exit a itace j that i a couterexample Aume that j i a couterexample itace with a miimum umber of job We partitio the iput job ito four et j(c, N), j(c, A), j(e, A), j(e, N) The firt et coit of the job rejected both by the algorithm ad by opt The ecod et coit of the job rejected by the algorithm, but ot by opt The third et coit of the job accepted both by the algorithm ad by opt The fourth et coit of the job accepted by the algorithm, but ot by opt The otatio C correpod to cheap job while E correpod to expeive oe (a the algorithm reject exactly the et of job with mall rejectio pealtie, baed o the threhold α), ad A ad N correpod to job that are accepted ad ot accepted (repectively) by opt A i previou work (Bartal et al 000; He ad Mi 000; Dóa ad He 006, we ca prove the followig claim Claim 3 j(c, N) Proof If there exit j j(c, N), the lettig j j { j} reult i Z opt (j ) w j, Z uorted (j ) Z uorted (j) w j, ad thu Z uorted (j ) Z opt (j ) Z uorted (j) R, oj i couterexample with a maller umber of job, cotradictig the choice of j Let > 0 deote the umber of job i j Lemma 4 + Moreover, if j(c, A) j(e, A), the α k + Proof The makepa of opt for k 0 job i at leat (it i equal to + k if k, ad larger if k ) Thu, we have j(c,a) j(e,a) + + w(j(e, N)) Every j j(e, N) ha w j >α> +, which implie the firt boud, ice w(j(e, N)) α j(e, N) > j(e,n) + Next, we prove the ecod boud If optdoe ot accept ay job, the all job belog to j(e, N) The rejectio pealty of each uch job i at leat α, ad the claim follow If optaccept oe job, ie, j(c, A) j(e, A), the it ru it o oe of the machie, ad i thi cae, the makepa of opti, while the other job all belog to j(e, N), which give + w(j(e, N)) + α( ) >α, a α< Lemma 5 Z uorted(α) (j) α( ) + Proof If the algorithm rejected all job, the the oly oempty et i j(c, A), ad the rejectio pealty of each job doe ot exceed α Thu, Z uorted (α)(j) α <α( ) +,aα< Otherwie, we additioally boud the makepa uig j(e, A) j(e, N) Sice L L, by Lemma, ad ice j(e, A) j(e, N) job were cheduled by the algorithm, by the ame lemma, the makepa i L j(e,a) j(e,n) + + (+) We fid that Z uorted (α)(j) j(e,a) j(e,n) + + (+) +α j(c, A) j(e,a) j(e,n) + + + α j(c, A) α( ) +,aα> + Theorem 6 The competitive ratio of uorted i at mot R, ad thi i the bet poible competitive ratio for ay 3
J Sched (04) 7:87 93 9 Proof The lower boud follow from Lemma Next,we prove the upper boud If, the uig the boud of Lemma 5 ad the firt boud of Lemma 4 we have Z uorted (j) )(α + / α ) ( + )(α + / α α( )+/ /(+) (+ ) α+/ /(+) R, where the lat iequality hold ice the boud i a mootoically decreaig fuctio of Otherwie,, ad uig the boud of Lemma 5 ad the ecod boud of Lemma 4 we have α, ad Z uorted (j) Thu, Z uorted (j) 3 Sorted iput 3 The algorithm α R We deig a algorithm for URUD which alo coit of rejectio trategy ad a algorithm for chedulig the accepted job The chedulig algorithm i the ame a for URU The rejectio trategy alo deped o threhold, but the threhold i fairly large for the firt job, maller for the ecod job, ad it i a itermediate value for all additioal job We ue the ymbol β to deote a parameter i The parameter β atifie 0 <β <β Let L 0 L0 0 Iitial mode For a ew job of idex j ( +, If j, the act a follow If w >β, the chedule the job uig aig() Otherwie reject it, let L L 0, ad move to the fial mode If j, the act a follow If w >β, the chedule the job uig aig() Otherwie reject it, let L L, L L, ad move to the fial mode 3 If j 3, the act a follow If w j > +, the chedule the job uig aig( j) Otherwie reject it, let L j L j, L j L j, ad move to the fial mode Fial mode Reject every a ew job of idex j, ad let L j L j, L j L j We defie the parameter a follow β ) (+), which i the uique poitive olutio of the followig equatio with repect to x: x x + It i eay to ee that + < β < for ay We let β β Wehave0<β < + ice β > 4 i equivalet to (+) >, ad ice β < + +, that hold ice β i the geometric average of ad +, ad thu it i maller tha the algebraic average Fially, we let R β β( + ) + Note that β > α ad R < R (o we will prove that orted ha a trictly maller competitive ratio tha uorted) The algorithm orted(β, β ) with thee choice of β ad β will be called orted i what follow 3 Aalyi Lemma 7 The competitive ratio of ay olie algorithm for URUD i at leat R for ay Proof Let H be a olie algorithm, ad let w w β The iput coitig of oe job i called I, ad the iput coitig of two job i called I If the firt job i accepted by H, the Z opt (I ) β, Z H (I ), ad Z H (I ) Z opt (I ) β R If it rejected, the Z opt (I ) +, while Z H (I ) β +mi{β, }β(ice β< ) It follow that Z H(I ) β + R, by the defiitio of β ad R Z opt (I ) Theorem 8 The competitive ratio of ortedi at mot R, ad thi i the bet poible competitive ratio for ay Proof The lower boud follow from Lemma 7 Next,we prove the upper boud Oce agai, we will deal with a couterexample j We aume that j i miimal i the ee that it ha the mallet umber of job amog example with at leat two job If j ha a igle job, the we add aother job of rejectio pealty zero to the itace Thu, the umber of job i j,, atifie A the rejectio pealtie are o-icreaig, it ca be aumed (by imple exchage argumet) that opt accept a (poibly empty) prefix of the job, ad reject the remaiig job (if ay exit) Moreover, we coider a optimal olutio opt that reject a maximum umber of job (out of the optimal olutio that accept a prefix of the job) Lemma 9 If 3, the w > +, ad optaccept all job If the lat job i accepted Proof Aume firt that w + by opt, the all job are accepted ad + The alterative olutio where oly the firt job are accepted ha the cot + + w, thu it i optimal too, it alo accept a prefix of the job, but it reject a larger umber of job, cotradictig the choice of opt Thu, the lat job mut be rejected by opt We how that by the defiitio of the algorithm it reject job a well Aume that i ot 3
9 J Sched (04) 7:87 93 rejected by the algorithm I thi cae, ice 3, job i accepted i tep 3 of the iitial mode, which implie w > +, cotradictig the aumptio w + Removig job from j we get a itace with at leat two job, ad the cot of both orted ad opt are reduced by the ame amout Similar to Claim 3, we fid that a maller couterexample with at leat two job exit Thu, we fid w > + Now aume that opt reject job Letl<deote the umber of job accepted by opt Wehave l + + jl+ w j > +, ice w l+ w > + Replacig optwith a olutio that accept all job reduce it cot to cotradictig optimality + Lemma 0 If, the optchedule both job of j if ad oly if mi{w, }+w > + If the ecod job i rejected, the the firt job i accepted if ad oly if w > Proof There are three optio, which are acceptig both job, acceptig oly the firt job, ad rejectig both job Their cot are +, + w, ad w + w, repectively, ad i a cae of a tie, optprefer to reject a job Before we proceed, we ummarize the poible actio of the algorithm, whe it i applied o j It ca reject all job, if w β It ca accept oly the firt job, if w >βad w β, or it ca accept the firt two job, i which cae it accept all job (ice if 3, the w > + ad therefore all rejectio pealtie are above +, by Lemma 9) Cae : The firt job i rejected by the algorithm I thi cae the algorithm reject all the job Sice the job arrive orted by a o-icreaig order of pealtie, every job j atifie w j w, ad w β hold ice the algorithm reject the firt job Thu Z orted (j) β Ifoptaccept all job, the β ( + ) + R +, ad we get Z orted (j) β + If 3, the opt accept all job, thu we are left with the cae, uch that optreject the ecod job (ad poibly alo the firt oe) By Lemma 0, the firt job mut be rejected too by opt, aw β< Therefore, orted produce a optimal olutio i thi cae Cae : The firt job i accepted by the algorithm, ad the ecod job i rejected Thu, w w β < +, ad by Lemma 9, I thi cae we have Z orted + w If opt alo accept the firt job ad reject the ecod oe, the it cot i + w, ad the ortedproduce a optimal olutio i thi cae If optreject both job, the w + w, ad Z orted (j) /+w w +w < /+w β+w < / β R, where the ecod iequality i by w >β(ice the algorithm accept the firt job), ad the third iequality hold ice 0 <β< ad w 0 If optaccept both job, the Z opt +, ad Z orted (j) /+w /(+) /+(β /) /(+) β( + ) R, where the ecod iequality i by w β β (ice the algorithm reject the ecod job) Cae 3: The firt two job are accepted by the algorithm If 3, the by Lemma 9, w > + j > + for all 3 j, ad both the algorithm ad optaccept all job Z By Lemma, (j) /(+)+/((+)) /(+) + + 3 < R, ice 3 ad 3+ 3 < + i equivalet to (3 + ) < 9( + ), that hold for If, Z ad optaccept both job, the by Lemma, (j) / / /(+) < R, ice ( )(+) + i ( )(+) < equivalet to ( + ) < 4 3 ( + ), which hold for We are left with the cae that, ad optreject the ecod job By Lemma 0 mi{,w }+w > β +(β ), ice w >β, >β, ad w >β β, due to the actio of orted Oce agai we ue Lemma ad get, Z orted (j) We fid that Z orted (j) 3β < β R, where the lat iequality i equivalet to < 3 β Referece that hold ice β> + Bartal, Y, Leoardi, S, Marchetti-Spaccamela, A, Sgall, J, & Stougie, L (000) Multiproceor chedulig with rejectio SIAM Joural o Dicrete Mathematic, 3(), 64 78 Bruo, J, & Gozalez, T (976) Schedulig idepedet tak with releae date ad due date o parallel machie Techical report 3 Uiverity Park: Computer Sciece Departmet, The Peylvaia State Uiverity Che, B, va Vliet, A, & Woegiger, G J (995) A optimal algorithm for preemptive o-lie chedulig Operatio Reearch Letter, 8(3), 7 3 Cho, Y, & Sahi, S (980) Boud for lit chedule o uiform proceor SIAM Joural o Computig, 9(), 9 03 Dóa, G, & He, Y (006) Preemptive ad o-preemptive o-lie algorithm for chedulig with rejectio o two uiform machie Computig, 76( ), 49 64 Ebeledr, T, Jawor, W, & Sgall, J (009) Preemptive olie chedulig: Optimal algorithm for all peed Algorithmica, 53(4), 504 5 Ebeledr, T, & Sgall, J (0) Semi-olie preemptive chedulig: Oe algorithm for all variat Theory of Computig Sytem, 48(3), 577 63 Eptei, L (00) Optimal preemptive o-lie chedulig o uiform proceor with o-decreaig peed ratio Operatio Reearch Letter, 9(), 93 98 Eptei, L, & Favrholdt, L M (00) Optimal preemptive emi-olie chedulig to miimize makepa o two related machie Operatio Reearch Letter, 30(4), 69 75 Eptei, L, Noga, J, Seide, S S, Sgall, J, & Woegiger, G J (00) Radomized olie chedulig o two uiform machie Joural of Schedulig, 4(), 7 9 Eptei, L, & Sgall, J (000) A lower boud for o-lie chedulig o uiformly related machie Operatio Reearch Letter, 6(), 7 3
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