Robust Algorithms for Preemptive Scheduling

Transcription

1 DOI 0.007/s Robust Algorths for Preeptve Schedulng Leah Epsten Asaf Levn Receved: 4 March 0 / Accepted: 9 Noveber 0 Sprnger Scence+Busness Meda New York 0 Abstract Preeptve schedulng probles on parallel achnes are classc probles. Gven the goal of nzng the akespan, they are polynoally solvable even for the ost general odel of unrelated achnes. In these probles, a set of jobs s to be assgned to run on a set of achnes. A job can be splt nto parts arbtrarly and these parts are to be assgned to te slots on the achnes wthout parallels, that s, for every job, at ost one of ts parts can be processed at each te. Motvated by senstvty analyss and onlne algorths, we nvestgate the proble of desgnng robust algorths for constructng preeptve schedules. Robust algorths receve one pece of nput at a te. They ay change a sall porton of the soluton as an addtonal part of the nput s revealed. The capacty of change s based on the sze of the new pece of nput. For schedulng probles, the supreu rato between the total sze of the jobs (or parts of jobs) whch ay be re-scheduled upon the arrval of a new job j, and the sze of j, s called graton factor. We desgn a strongly optal algorth wth the graton factor for dentcal achnes. Strongly optal algorths avod dle te and create solutons where the (non-ncreasngly) sorted vector of copleton tes of the achnes s lexcographcally nal. In the case of dentcal achnes ths results not only n akespan nzaton, but the created soluton s also optal wth respect to any l p nor (for p>). We show that an algorth of a saller graton factor cannot be optal wth respect to akespan or any other l p nor, thus the result s best possble n ths sense as well. We further show that nether unforly related achnes An extended abstract of ths paper appears n Proc. of ESA 0. L. Epsten ( ) Departent of Matheatcs, Unversty of Hafa, 3905 Hafa, Israel e-al: lea@ath.hafa.ac.l A. Levn Faculty of Industral Engneerng and Manageent, The Technon, 3000 Hafa, Israel e-al: levnas@e.technon.ac.l

2 nor dentcal achnes wth restrcted assgnent adt an optal algorth wth a constant graton factor. Ths lower bound holds both for akespan nzaton and for any l p nor. Fnally, we analyze the case of two achnes and show that n ths case t s stll possble to antan an optal schedule wth a sall graton factor n the cases of two unforly related achnes and two dentcal achnes wth restrcted assgnent. Keywords Preeptve schedulng Mgraton factor Robust algorths Introducton We study preeptve schedulng on dentcal achnes wth the goals of akespan nzaton and nzaton of the l p nor (for p>) of the achne copleton tes. A set of n jobs s gven, where p j > 0, the sze (or length) of j, sthe processng te of the j-th job on a unt speed achne. The load or copleton te of a achne s the last te that t processes any job. In akespan nzaton, the goal s to assgn the jobs for processng on the achnes, nzng the axu copleton te of any achne. In the nzaton of the l p nor, the goal s to nze the l p nor of the vector of achne copleton tes. The set of feasble assgnents (also called schedules) s defned as follows. In preeptve schedulng, each achne can execute one job at each te, and every job can be run on at ost one achne at each te. Idle te s allowed. A job s not necessarly assgned to a sngle achne, but ts processng te can be splt aong achnes. The ntervals or te slots n whch t s processed do not necessarly have to be consecutve. That s, a job ay be splt arbtrarly under the constrant that the te ntervals, n whch t runs on dfferent achnes, are dsjont, and ther total length s exactly the processng te of the job. Followng a recent nterest n probles whch possess features of both offlne and onlne scenaros, we study preeptve schedulng wth bounded graton, whch s also called robust preeptve schedulng. The proble s not an offlne proble, n the sense that the nput arrves gradually, but t s not a purely onlne proble ether, snce soe reassgnent of the nput s allowed. In ths varant jobs arrve one by one, and when a new job arrves, ts processng te becoes known. The algorth needs to antan a schedule at all tes, but when a new job j arrves, t s allowed to change the assgnent of prevously assgned jobs n a very restrctve way. More accurately, the total sze of all parts of jobs whch are oved to dfferent te slots (or to the sae te slot on a dfferent achne) should be upper bounded by a constant factor, called the graton factor, tes p j, that s, ther total sze ust be at ost a constant ultplcatve factor away fro the sze of the new job. Algorths whch operate n ths scenaro are called robust. We expect a robust algorth to perfor well not only for the entre nput, but also for every prefx of the nput, coparng the partal output to an optal soluton for ths partal nput. We would lke to stress the fact that n the preeptve varant whch we study, at each step the parts of a gven job ay be scheduled to use dfferent te slots on possbly dfferent achnes. When the schedule s beng odfed, we allow to cut parts of jobs further, and only the total

3 sze of parts of jobs whch are oved ether to a dfferent te slot, or to a dfferent achne (or both), counts towards the graton factor. Next, we revew possble achne envronents. The ost general achne envronent s unrelated achnes [, 5, 6]. The set of achnes s M ={,...,}. In ths odel each job j and achne have a processng te p (j) assocated wth the, whch s the total te that j requres f t s processed on. In the preeptve odel, the job can be splt nto parts, as long as t s not executed n parallel on dfferent achnes. If the total te allocated for the job on achne s t,j, then t,j p (j) = ust be satsfed. In ths n order to coplete the job the equalty = odel the graton factor s not well-defned. We study several portant specal cases n whch the defnton of a graton factor s natural. The ost basc and natural achne envronent s the case of dentcal achnes, for whch p (j) = p j for all values of and j [6,, 8]. We study two addtonal coon odels whch generalze dentcal achnes. In the odel of unforly related achnes [, 4, 0, ], achne has a speed s > 0, and p (j) = p j s for all, j. In the restrcted assgnent odel [3] each job j has a subset of achnes M j assocated wth t, also called a processng set. Only achnes of M j can process job j.wehavep (j) = p j f M j and otherwse p (j) =. In the offlne scenaro of preeptve ultprocessor schedulng [7, 8, 0,, 4, 6 30, 33], jobs are gven as a set, whle n the onlne proble [6], jobs arrve one by one to be assgned n ths order. A job s assgned wthout any knowledge regardng future jobs. The exact te slots allocated to the job ust be reserved durng the process of assgnent, and cannot be odfed later. Idle te ay be created durng the process of assgnent, and as a result, the fnal schedule ay contan dle te as well. Note that n ths varant, unlke non-preeptve schedulng, t s well-known that dle te can be benefcal (for exaple, otherwse the frst job ust be assgned non-preeptvely to one of the achnes). It s known for a whle that the offlne preeptve schedulng proble, unlke the non-preeptve varant, can be solved optally n polynoal te even for ore general achne odels. Frst, we dscuss akespan nzaton on dentcal achnes and on unforly related achnes [, 8, 33]. McNaughton [8] desgned a sple algorth for dentcal achnes (see also [9, 30] for alternatve algorths). The study of ths proble for unforly related achnes started wth the work of Lu and Yang [7], as well as the work of [6] who ntroduced bounds on the cost of optal schedules. These bounds are the average load (that s, the total sze of jobs dvded by the total speed of all achnes), and bounds resultng fro the average load whch ust be acheved by assgnng the k largest jobs to the k fastest achnes (for k ). Horvath et al. [] proved that the optal cost s ndeed the axu of those bounds by constructng an algorth that uses a large (but polynoal) nuber of preeptons. Gonzalez and Sahn [0] devsed an algorth that outputs an optal schedule for whch the nuber of preeptons s at ost ( ). Ths nuber of preeptons was shown to be optal n the sense that there exst nputs for whch every optal schedule nvolves at least that any preeptons. Ths algorth was later generalzed and splfed for jobs of lted splttng constrants by Shachna et al. [33]. Slar results for a wde class of objectve functons, ncludng nzaton of the l p nor, were obtaned by Epsten and

4 Tassa [8]. In that work t was shown that whle for akespan nzaton, the best stuaton that one can hope for s a flat schedule where all achnes have the sae load, ths s not necessarly the case for other functons, such as the l nor. Note that due to the opton of dle te, the proble of axzng the nu load (also known as the Santa Claus proble) does not have a natural defnton of preeptve schedules, and thus we do not dscuss preeptve schedules wth respect to ths goal. The case of unrelated achnes was also shown to adt a polynoal te algorth. Specfcally, Lawler and Labetoulle [4] showed that ths proble can be forulated as a lnear progra that fnds the aount of te that each job should spend on each achne, and afterwards, the proble can be solved usng an algorth for another preeptve schedulng proble, naely, the open-shop preeptve varant. A nuber of artcles consdered the onlne proble for dentcal and unforly related achnes [6, 9, 0,, 6, 7, 35], ncludng any results where algorths of optal (constant) copettve rato were desgned. However, t s possble to desgn an onlne algorth whch outputs an optal soluton wth respect to akespan [6,, 7, 3]. For restrcted assgnent (and unrelated achnes), the lower bound of Ω(log ) gven by Azar, Naor, and Ro [3] s vald for preeptve schedulng. Thus, t s known that n any of the cases, n order to obtan a robust algorth whch produces an optal soluton, a non-zero graton factor ust be used. Another varant of preeptve schedulng whch s on the borderlne between onlne and offlne algorths allows to use a reorderng buffer of fxed sze, where jobs can be stored before they are assgned to te slots on achnes [8]. The non-preeptve verson of the proble was studed as well (see e.g. []). Robust algorths were studed n the past for schedulng and bn packng probles. The odel was ntroduced by Sanders, Svadasan and Skutella [3], who consdered the non-preeptve schedulng proble of nzng the akespan on dentcal achnes. In [3], several sple strateges wth a sall graton factor, but an proved approxaton rato (copared to that of onlne algorths [9]) were presented, as well as a robust polynoal te approxaton schee (PTAS). The bn packng proble was shown to adt an asyptotc robust PTAS [3] (and even the proble of packng d-densonal cubes for any d nto unt cubes of the sae denson adts an asyptotc robust PTAS [5]). The proble of axzng the nu load on dentcal achnes does not adt a robust PTAS, as was shown by Skutella and Verschae [34]. Although the slarty between the akespan objectve and nzng the l p nor for p>, obtanng a robust PTAS for nzng the l p nor of the load vector on dentcal achnes s possble (a constructon slar to the one of [34], usng an ntal batch slar to the exaple of [], gves a lower bound of for l -nor and a lower bound above for every l p - nor [5]). A slar negatve result was shown for bn packng wth cardnalty constrants [5]. In ths verson of bn packng an nteger paraeter t s gven, so that n addton to the restrcton on the total sze of tes n a bn, t ay contan up to t tes. Whle ths proble adts a standard APTAS and an AFPTAS [5, 4], t was shownn[5] that t cannot have a robust APTAS. Many operatng systes receve the jobs to be processed one by one. One crtcs of purely onlne algorths (that cot on the assgnent of a job upon arrval) s

5 that ost systes do not requre a full treatent of each job before the next one s consdered, as t s assued for onlne algorths, but soe flexblty allows to re-arrange the planned soluton. Stll, the syste ust have a concrete plan at all tes, for the case that the nput ternates, and the syste ust start ts operaton wthout delay. Thus, robust algorths allow to sulate real-lfe systes where t s possble to apply nor odfcatons of schedules. An addtonal theoretcal otvaton for the study of robust algorths s senstvty analyss. The queston here s whether t s possble to produce an optal soluton for a gven proble, where a sall odfcaton of the nput would not result n a ajor change n the optal soluton. Coplexty ssues are soetes dsregarded n ths type of study. As shown n [3, 5, 3], for non-preeptve schedulng probles and other parttonng probles, typcally ths s possble, f an exact optu s requred. In ths work we wll show that ths s not the case for preeptve schedulng on dentcal achnes (but not for other knds of achnes). Our an result s a polynoal te algorth of graton factor whch antans a strongly optal soluton on dentcal achnes. Such a soluton s one where the sorted vector of achne copleton tes s lexcographcally nal. Beng strongly optal, the algorth s optal wth respect to akespan nzaton and any l p nor for p>. We show that ths result s tght n the sense that no optal robust algorth for akespan nzaton (or for the nzaton of soe l p nor for p>) can have a saller graton factor. There are several dffcultes n obtanng optal robust algorths for preeptve schedulng. Frst, we note that an algorth whch runs n polynoal te ust have a polynoal nuber of preeptons. A odfcaton to the schedule ust reassgn parts of jobs very carefully, wthout ntroducng a large nuber of preeptons. For exaple, f at each step a new preepton s ntroduced for each part of each job, then the resultng nuber of preeptons would be exponental, and ths stuaton should be avoded. Snce we are nterested n an optal schedule (or even n a strongly optal schedule), the structure of the schedule s strct, and the algorth does not have uch freedo. For exaple, there are jobs that ust be assgned to a specfc achne durng the entre te that any achne s actve. Fnally, snce no parallels s allowed, when a part of a job s oved to soe te slot, an algorth ust ensure that t s not already assgned to any part of ths te slot on any of the other achnes. We also note that all the known optal preeptve algorths for nzng the akespan on dentcal achnes are not robust. In addton to the basc case of dentcal achnes, we study the two other, ore general, achne odels, whch are unforly related achnes and dentcal achnes wth restrcted assgnent. We show that n contrast to the result for dentcal achnes, an optal robust soluton cannot be antaned n the last two achne odels. Specfcally, we show that the graton factor of any optal robust algorth for unforly related achnes s at least, and the graton factor of any robust algorth for dentcal achnes wth restrcted assgnent s Ω(), whch holds for all nors. Thus, the stuaton s very dfferent fro offlne algorths, where the algorths for ore general achne odels are ore coplex, but are based on slar observatons. Note that for dentcal achnes, the set of solutons for (preeptve or non-preeptve) schedulng wth the goal functon of

6 the l p nor of achne copleton tes s the set of strongly optal schedules wth respect to akespan. As stated above, ths s not necessarly the case for other achne odels [8]. In Appendx A, we study a relaxaton of preeptve schedulng whch s fractonal assgnent. Ths varant, where jobs can be splt arbtrarly aong achnes (wthout any restrcton on parallels) s trval for dentcal achnes and unforly related achnes, but not for restrcted assgnent, n whch case the lower bound of Ω() on the graton factor of an optal robust algorth s vald. We desgn an optal algorth for fractonal assgnent of graton factor wth the goal of akespan nzaton. In addton, we study preeptve schedulng for the case of two unforly related achnes and the case of two dentcal achnes wth restrcted assgnent, and desgn robust optal preeptve algorths (wth a graton factor of ) for these cases and all nors. For unforly related achnes, the algorth obtans dfferent schedules for the dfferent nors (snce there does not exst one schedule whch s optal for all nors), and uses the best possble graton factor for antanng an optal schedule for unforly related achnes. For restrcted assgnent, snce no speeds are present, one output can be optal for all nors f t s strongly optal for the akespan, and the algorth produces such an output. The graton factor of s best possble for robust algorths whch antan a strongly optal schedule for two achnes. An Optal Robust Algorth for Identcal Machnes In ths secton we present an algorth whch antans a strongly optal schedule for dentcal achnes. Such a schedule s optal wth respect to akespan and all l p nors for <p<. The algorth has a graton factor of, whch s best possble, as shown n Sect. 3.. Let J denote the sequence of nput jobs, and let n = J. The nstance whch conssts of the frst t jobs s denoted by I t (and so I n = J ). Let P t = t = p and pt ax = ax t p. We say that achne has load L whch s the total processng te of the fractons of jobs whch are assgned to. Note that f a schedule does not have dle tes then the load of a achne s the last te n whch t processes a job. Frst, we sketch the behavor of our algorth. At the arrval te of a job, we use a sple algorth, LOADS whch coputes the sorted vector of achne loads n an optal soluton. Algorth LOADS s based on the ethods of McNaughton [8] and []. The algorth creates a potental schedule of a sple for (whch cannot be used by our algorth, snce our algorth needs to odfy ts exstng schedule rather than creatng one fro scratch). LOADS s a recursve algorth whch assgns the largest reanng job to run non-preeptvely on an epty achne, unless the reanng set of jobs can be scheduled n a balanced way. Thus, the sorted vector of copleton tes has a paraeter κ, where the κ achnes of sallest copleton te have the exact sae copleton te. After fndng the output of LOADS, our algorth fnds whch achnes have an ncreased copleton te, and the new job s assgned. There ay be a te slot n whch the job can be scheduled on one achne, but typcally soe parts of jobs need to be oved to ake roo for the

7 parts of the new job. We carefully ove parts of jobs. In the process t s necessary to ensure that no parts of a job are scheduled to run n parallel (not only parts of the new job), and that parts of jobs are not oved unnecessarly, to avod an ncrease n the graton factor. The soluton of the statc proble for dentcal achnes s relatvely sple. The followng algorth was gven by McNaughton [8]. The optal akespan s gven by OPT = OPTn, where OPTt = ax{ P t,pax t }. The algorth has an actve achne (ntalzed as the frst achne), and keeps assgnng jobs there consecutvely, startng at te zero, untl the assgnent of a job j would exceed the te OPTn. Inths case only a part of the job s assgned, the algorth oves on to the next achne as an actve achne, and the reander of j s assgned there, startng at te zero. The algorth does not use dle te. The algorths whch we desgn n ths secton do not use dle te ether, but they are n fact strongly optal. The algorth descrbed above s not strongly optal, yet strongly optal algorths are known even for unforly related achnes []. We start wth the descrpton of an algorth LOADS whch coputes a strongly optal soluton OPTt for the nput I t (OPTt denotes not only the optal akespan but also a strongly optal soluton for a prefx of t jobs). For an nput I t, we defne an order as follows. For two jobs, j we say that s larger than j f p >p j or f p = p j and <j. When we refer to the largest l jobs of an nput we refer to ths orderng. Recall that J s not necessarly sorted. As stated above, the algorth for an nput I t s recursve. It outputs a schedule as well as a sequence of loads L t Lt Lt. The load Lt s the th load, and t wll be the load of achne. We ntalze a set of ndces J ={,,...,t}, P = P t, k =.. Copute p ax = ax l J p l, and fnd a job j such that p j = p ax (tes are broken n favor of a job of the sallest ndex).. If P k <pax,letl t k = pax, assgn job j to achne k (non-preeptvely, startng at te zero), let J J \{j}, k k, P P p j.ifj, then apply the algorth recursvely, that s, go to step. Otherwse, go to step Let L t = P k for k, assgn the jobs n J usng McNaughton s rule to achnes,,...,k wth a copleton te of P k and halt. The algorth keeps pckng a axu sze job and assgnng t to a achne of axu ndex whch stll dd not receve a job. When the reanng jobs are all relatvely sall and can be scheduled n a balanced way, step 3 s appled. If no nput jobs rean then ths step sply defnes the reanng loads to be zero. If k reaches the value, then step cannot be nvoked. Thus, the fnal value of k s at least. Note that f achne receves a large job j then the recursve call of the algorth acts on achnes,,..., and the jobs I t {j} exactly as t would have acted n the case that ths s the coplete nput. The property that the algorth s strongly optal can be proved by nducton. If a sngle job j s assgned to achne n step, then n every schedule there ust be at least one achne wth a copleton te of at least p j (snce OPTt p j ). The strong optalty of the assgnent to achnes,,...,follows by nducton. If all jobs are assgned n step 3, then all achnes have equal loads, whch s clearly strongly optal. Snce the frst step s executed as ost tes, t s possble to prepare a

8 sorted lst of the largest jobs n advance. Thus, the algorth can be pleented usng a runnng te of O(t + log ). Ift<then achnes,..., t do not receve jobs, and t s possble to run the algorth on the last t achnes. the runnng te becoes O(+ t log t).. Propertes of Load Vectors n Strongly Optal Solutons Gven the strongly optal soluton OPTt, we say that job j s large n OPTt and t corresponds to achne l, fnoptt job j s assgned to achne k = l n step. Note that n ths case l>. Otherwse, j was assgned n step 3. In ths last case we say that j s sall, and t corresponds to achne κ, whch s the fnal value of k. That s, the last step s a step where L t was defned to be P κ for κ (and p j P κ ). We say that achnes,...,κ are the achnes of sall jobs even f no jobs are assgned to the. The followng holds due to the constructon of the algorth. Proposton L t s a onotoncally non-decreasng functon of. In what follows, we refer to L t as the -th copleton te of the nput I t. Lea L t s a onotoncally non-decreasng functon of t. Proof We need to prove that L t Lt holds for any and t n. Ths s proved by nducton on. Clearly, the cla s true f =. Let j be a (nu ndexed) job of axu sze n I t.ifp j P t,wehave L t = P t for. IfwehaveLt = P t then the requred property clearly holds, snce n ths case L t = P t for, and P t >P t. Otherwse, L t = p j for job j, whch has axu sze n I t.wehavel t L t = p j for, by Proposton, and p j p j by the defnton of j. Snce p j P t = Lt, we are done. Otherwse, we have p j > P t. Assue frst j<t. Then j s assgned to achne and the algorth s executed recursvely on achnes. In ths case, j I t and t s a axu szed job of nu ndex n I t. Snce P t >P t, p j > P t holds too, so n OPTt, j s assgned to achne. Thus we have L t = Lt and L t Lt for follows by the nducton hypothess, as the nput whch results n the loads L t for (that s, I t {j}) s augented wth job for. t to gve the loads L t Assue now that p j > P t satsfes p l P t and Lt = P t and j = t.iflt = P t, then every job l t for. InOPT t,jobt s assgned to achne and then the reanng copleton tes result fro applyng the algorth wth k = and P = P t.wehavep j P t P t for any j t, so n OPTt the other jobs (except for t) are sall and L t = P t P t. In addton, snce p t > P t P t > P t = Lt. = Lt for we get p t >P t + p t or L t = p t >

9 Fnally, f L t = p l for soe job l t, then p l > P t. Snce t s a axu szed job n I t of a nu ndex, we have p t >p l, and thus L t >Lt. For the other achnes the property follows by the nducton hypothess; the set of loads L t for results fro applyng the algorth on achnes for I t {l}, whle the set of loads L t for results fro applyng the algorth on achnes for I t, that s, fro addng one job (job l) tothe nput. In the next lea we consder the dfferences between the solutons OPTt and OPTt obtaned for the nstances I t and I t, respectvely, n the case that the t-th job s scheduled to a dedcated achne k n OPTt. In partcular, we show that n ths case the contents of achnes k +,..., are dentcal n the two solutons (they have the sae large jobs), and the nuber of achnes wth sall jobs n OPTt cannot decrease by ore than copared to OPTt. Lea 3 Assue that t s a large job n OPTt whch s assgned to achne k. Let k t be the axu ndex such that L t = L t for k t (.e., the axu ndex of a achne that receves sall jobs n OPTt ) and let k t be the axu ndex such that L t = Lt for k t. Then the followng propertes hold:. k t <kand k t k.. L t = Lt for k<. 3. k t k t. 4. L t = Lt + for k t + k. 5. L t k t + Lt k t. Proof We clearly have k t <k, snce OPTt assgns a large job to achne k. Wefrst cla L t = Lt for k<, and k t k. We prove the propertes by nducton on.if = k then we are done. Otherwse, let j<tbe the job assgned to achne n OPTt. Snce we have p j > P t, then p j > P t, so achne receves a large job n OPTt.Usngj<t, we fnd that OPTt assgns the sae job j to achne (snce t s a nu ndex job of axu sze n I t ), so L t = p j = L t, and k t. Therefore, we can reove job j and achne fro the nput and the clas for achnes k +,..., follow by nducton (whch can be used snce >k ). We thus neglect the set of jobs assgned to achnes k +,..., and these achnes n both OPTt and OPTt and assue n what follows that k =. We now prove that k t k t. If k t then we are done. If k t =, then every job j t satsfes p j P t P t,solt = P t for, snce job t s assgned to achne as a large job n OPTt.Sowehavek t =. Otherwse, assue by contradcton that k t <k t. That s, achnes k t +,..., are assgned large jobs n both OPTt and OPTt. These large jobs are the largest jobs of each one of the nputs I t and I t, respectvely. Snce job t s assgned to achne, t s the largest job n I t, and the achnes k t +,..., receve (n OPTt )the k t largest jobs of I t (ths set s epty f k t = ).

10 These jobs are assgned to achnes k t +,..., n OPTt, so the reanng nput for achnes,,...,k t + noptt s the sae as the reanng nput for achnes,,...,k t n OPTt, and the largest job of ths reanng nput s assgned to achne k t + as a large job n OPTt. Thus, the reanng nput for achnes,...,k t n the two nputs dffers by one job y, whch s the largest job of the reanng nput n OPTt. Snce we assued that k t k t, achne k t receves a large job n OPTt,soy s assgned to achne k t. Consder the reanng jobs (whch are not assgned to achnes k t +,...,n OPTt, that s, whch are not assgned to achnes k t,..., n OPTt ). Let Y be the total sze of these jobs and X the axu sze of any job. OPTt assgns a large job to achne k t, but OPTt does not assgn a large job to achne k t,sowehave Y k t X, whch s a contradcton, so k t k t. Y k t <Xand We next argue that L t = Lt + for k t + k. In OPTt, achnes k t +,..., receve large jobs, and n OPTt, achnes k t +,..., receve large jobs. Slarly to the proof above, the job whch OPTt assgns to achne + sthe job whch OPTt assgns to achne (k t + k ), snce job t s assgned to achne k by OPTt. It reans to prove that L t k t + Lt k t.ifk t = k t +, then n OPTt, achnes,,...,k t receve the sae jobs as achnes,,...,k t = k t + noptt, all assgned as sall jobs, so L t k t L t k t. Otherwse, n OPTt, achne k t + >k t receves one large job, j. Ths job s ncluded n the set of jobs that OPTt assgns to achnes,,...,k t,sol t k t p j = L t k t +. The next lea shows that f the t-th job s sall then the nuber of achnes contanng sall jobs can only ncrease n OPTt copared to OPTt. Lea 4 Assue that t s a sall job n OPTt. Let k be the achne ndex to whch t corresponds. Let k t denote the axu ndex such that L t = L t. Then k k t. Proof Assue by contradcton k<k t. Then the sae set of k t large jobs are assgned to achnes k t +,...,for both nputs. Let Y denote the total sze of jobs n I t whch are not assgned to achnes k t +,...,.LetZ denote the Y axu sze of any job n ths set. By defnton of k t, k t Z.Ifp t >Z, snce t s a sall job, t s not assgned as a large job to achne k t n OPTt.Ifp t Z, usng Y +p t k t Z we fnd that the job of sze Z s not assgned as a large job ether, contradctng the assupton k<k t.. The Procedure ASSIGN We descrbe a procedure ASSIGN whch wll be used by our algorth. For each new job, ASSIGN s nvoked at ost once. We wll show that the graton factor of ths procedure s at ost, whch ples the claed graton factor. One of the paraeters n the runnng te of the algorth s the te to construct the output. Ths s the te to produce the schedule and specfy the te slots for each

11 job. Naturally, the nuber of such te slots for each job s the nuber of tes that t s preepted plus. We are therefore nterested n the total nuber of tes that any job s preepted, that s, the total nuber of preeptons. In what follows, we wll copute an upper bound on the nuber of preepton tes rather than on the exact nuber of preeptons. Those are tes at whch soe achne possbly stops runnng one job and starts runnng another one, or a achne copletes all parts of jobs assgned to t. The total nuber of preeptons s obvously at ost tes the nuber of preepton tes. We wll show that each te that ASSIGN s nvoked durng the assgnent of job t, atost t new preepton tes are created. Our algorth wll use the procedure at ost once for each newly arrvng job, and the nuber of addtonal new preepton tes for each job wll be constant. Therefore the nuber of preeptons wll be O( 3 n ) for n jobs. The nput to ASSIGN conssts of the followng paraeters: a job or a part x of sze X of job t, an ndex of a achne l, a te L X, a preeptve assgnent of a set of jobs to achnes,,...,, a set of tes L L l for achnes,,...,l, wth L l L, such that each achne l s free durng the te nterval [L, L], and ll = l = L + X. Machne l ay already have a part of job t whch was assgned the startng te L, and no other achne has any dle te. Moreover, ths te after L on achne l s the only te durng whch parts of job t are possbly assgned, and all other achnes of ndces l +,..., are copletely occuped durng the te [0, L] runnng other jobs (not parts of job t). An addtonal requreent s that there are no preepton tes durng (L,L + ) for l. However, n all cases that ASSIGN s nvoked, the tes L and L for l are already defned to be preepton tes. After the applcaton of ASSIGN, no achne wll have any dle te, all achnes, where <lwll be busy durng exactly [0, L], and achne l wll be busy durng ths te perod and possbly later as well. We start wth assgnng a part of x, ofszel L, to achne durng the te nterval [L, L]. Note that X = l = (L L ), so the reander of x wth sze X satsfes X = X (L L ) = l = (L L ). So far no graton was used and no addtonal dle te was ntroduced. Snce L L for l, L L L L for l,sox X X l X.Ifl = then we are done. Assue therefore l, n whch case the reander has a postve sze (that s, X>0). Cla 5 L l = (L + L )( ) + (L L l )(l ) = X. Proof Usng ll = l = L + X, wehave, l = (L + L )( ) + (L L l ) (l ) = l = L + L(l ) = L + X Ll + L(l ) = L + X L L, as X L. Snce X X = L L, we are done. We let = X = l = (L + L )( ) + (L L l )(l ), δ = L + L for l and δ l = L L l. That s, = l = δ ( ). Snce X X X l,we have = X ( l )X ( )X. In what follows we use the ters cells and strpes. Both ters correspond to te ntervals. A cell s an epty te slot on a specfc achne, whle a strpe s a te slot whch s occuped on all the frst l achnes.

12 Frst, we create the followng cells. For l, f L + >L, there are cells of the te nterval [L,L + ], on achnes,...,. In addton, there are l cells of the nterval [L l, L] on achnes,...,l,fl >L l. The total length of all these cells on all achnes s l = (L + L )( ) + (L L l )(l ). Before we apply reassgnent of parts of jobs, all these cells are epty. We wll explan how to use the for the assgnent of parts of jobs. We would lke to assgn the reanng part of the job (of sze X) durng the te [0, = X]. The job ay be assgned preeptvely, and t wll run durng ths entre te perod. Recall that currently all achnes are busy durng ths te. We splt ths te nterval further nto strpes, and create strpes fro te 0 untl te L, where for =,...,l there are strpes of length δ. The strpes are non-overlappng, the frst strpe s created at the botto, and each addtonal strpe (for whch we have >) s created just above the prevous one so that all strpes are consecutve. Next, we defne a one-to-one correspondence of the strpes defned here to the cells defned before, such that each strpe has one cell of the sae length assocated wth t,.e., a strpe of length δ has a cell of ths length assocated wth t. Recall that there are strpes of length δ, and there are cells of ths length. The correspondence s defned such that the y-th strpe (fro the botto) of length δ corresponds to the cell of length δ on achne y +. For each strpe, parts of jobs wll grate fro the strpe to the cell assocated wth t. The strpes wll be consdered n a botto up anner, and the total sze of parts of jobs that wll grate n one strpe s exactly ts heght, so all cells wll becoe copletely occuped, and the total sze of gratng jobs s exactly ( )X. We defne the breakpont tes between strpes to be new preepton tes. Ths adds at ost ( )/ new preepton tes. We deal wth strpes fro botto to top. Consder one strpe of heght δ,ofthe te nterval [α, β] (β α = δ ) and show how to assgn a part of x of length δ durng the te [α, β] (possbly preeptvely, that s, ths part ay be splt further), whle soe parts of jobs,...,t assgned durng ths te nterval are oved to the cell assocated wth ths strpe. We denote the te slot of ths cell by [α,β ] (β α = δ ) and the achne of ths cell by q. The cell of the sae te slot for achnes, 3,...,q was already dealt wth, as ts strpes are located earler (.e., lower) n the schedule. Our algorth antans for each job j the sze of the part of j whch was rescheduled nto the te slot [α,β ] whle further parttonng of these (parts of) jobs nto parts and ther schedule are not deterned yet, and wll be deterned after all cells of ths te slot have been consdered. Just before consderng the cell of achne q we create a vrtual schedule of the jobs, whch were reassgned to [α,β ] so far (onto achnes, 3,...,q ) by applyng McNaughton s algorth on ths set of (parts of) jobs. Ths vrtual schedule defnes a set of (vrtual) preepton tes. Durng the process of the algorth where t decdes whch parts to ove to the cells wth the sae te slot of further achnes, ths set of vrtual preepton tes s consdered as preepton tes even though the jobs are not actually scheduled. Later, when the algorth fnshes consderng all cells of ths te slot, the schedule of the parts of jobs n ths te slot s deterned (see below). Partton the te nterval [α, β] nto axu length te ntervals [α + t,α+ t + ], defned for 0 r, where t <t +, such that t 0 = 0 and β = α + t r+,

13 Fg. The ntal schedule for the exaple for ASSIGN and there are no preeptons durng (α + t,α + t + ) and no vrtual preepton tes durng (α + t,α + t + ),.e., every achne executes one job durng all ntervals of ths for. The nuber of such ntervals s at ost the nuber of preepton tes durng (α, β] (α,β ], whch nclude prevously exstng preepton tes durng (α, β], and durng (α,β ] there are only vrtual preepton tes, snce the ntal schedule was such that there were no preepton tes nsde cells. Now, for 0 r, fnd a achne whch executes a job durng [α + t,α + t + ] that s not executed on any achne durng [α + t,α + t + ]. There are two types of achnes: those of ndex hgher than q whch run a job contnuously durng [α,β ], and n partcular n the shorter te nterval, and achnes partcpatng n the vrtual schedule on the achnes,...,q. As for achne q, t has an epty cell ntally, so the te slot [α + t,α + t + ] on t s stll epty, and there are at ost jobs runnng durng [α + t,α + t + ]. Thus, there s at least one achne q runnng a dfferent job durng [α + t,α+ t + ], and ths part of job s oved fro [α + t,α+ t + ] on q to [α + t,α + t + ] on q to jon the current vrtual schedule (that s, the job s not assgned yet, but s added to the lst of parts of jobs to be oved). A part of x s assgned durng the te [α + t,α+ t + ] on q. If the cell whch was dealt wth s the last cell for [α,β ], then the jobs whch are planned to be oved to ths te slot are assgned usng McNaughton s algorth nto all cells of [α,β ] before ovng to cells of other te slots. Note that the nuber of preepton tes for [α, β] ncreased by at ost ( )(t ), and t addtonal preepton tes ay be created durng [α,β ] by the last applcaton of McNaughton s algorth (the one n whch jobs are actually assgned). Snce there are at ost te slots havng non-epty sets of cells, we conclude that the nuber of preepton tes ncreased by at ost ( )(t ). The runnng te of ASSIGN s polynoal n the nput sze. In Fgs.,, 3, 4, 5, and 6 we llustrate an exaple run of ASSIGN. Letl = 5, t = 9, L = L = L 3 = 3, L 4 = 5, L 5 = L = 8, and X = 8. Intally job s assgned to achne 5 durng the te slot [0, 8], job s assgned to achne 4 durng the te slot [0, 5], job 3 s assgned to achne durng the te slot [0, 5], job 4 s assgned to achne durng the te slot [5, ], job 5 s assgned to achne 3 durng the te slot [0, ] and to achne durng the te slot [, 3], job 6 s assgned to achne 3 durng the te slot [, 9], job 7 s assgned to achne

14 Fg. The defnton of cells and strpes for the exaple for ASSIGN Fg. 3 The schedule after dealng wth the frst two strpes n the exaple for ASSIGN 3 durng the te slot [9, 3], and fnally job 8 s assgned to achne durng the te slot [0, 3]. See Fg. for an llustraton of ths ntal schedule. We have δ = δ = δ 5 = 0, δ 3 =, and δ 4 = 3, we have two strpes of length δ 3 and three strpes of length δ 4, such that each strpe has a cell of the sae length assocated wth t as llustrated n Fg.. Note that job 9 s scheduled onto achne durng the te slot [3, 8] before defnng the cells. Fgures 3, 4, and 5 show nteredate steps durng the assgnent of x. Fgure 3 shows the partal assgnent of x nto the frst two strpes, and the vrtual schedule for the frst two cells after the assgnent of x nto the frst two strpes s decded. Note that the frst strpe was splt nto two ntervals, and as a result, the second one was splt as well (based on a preepton n the vrtual schedule for the frst cell). Fgure 4 shows the partal assgnent of x nto the frst four strpes, and the thrd and fourth cells are n the status that the vrtual schedule for both of the together s about to be coputed. Note that the thrd strpe was splt nto two ntervals, and as a result, the fourth strpe was splt nto three ntervals, due to one preepton n the schedule of ths strpe, and one preepton n the vrtual schedule of the thrd cell. Fgure 5 shows the coplete assgnent of x nto strpes, the thrd and fourth cells contan a vrtual schedule, and the last three cells are n the status that the vrtual schedule for all three of the together s about to be coputed. Once agan the partton of the cell s based on the vrtual schedule of the precedng

15 Fg. 4 The schedule durng the process of dealng wth the fourth strpe n the exaple for ASSIGN. The schedule for the frst two cells s fnal, the next cell has a vrtual schedule, and the algorth already decded whch parts of jobs are oved to the fourth cell, but the vrtual schedule for these two cells together was not coputed yet Fg. 5 The schedule durng the process of dealng wth the ffth strpe n the exaple for ASSIGN. There s a vrtual schedule for the thrd and fourth cells together, and the algorth already decded whch parts of jobs are oved to the ffth cell, but the schedule for these three cells together was not coputed yet two cells, and on the schedule of the ffth strpe. In Fg. 6, McNaughton s algorth was appled to the parts of jobs assgned nto the last three cells, creatng the fnal schedule. Corollary 6 If L X, L l L and ll = l = L + X, procedure ASSIGN returns a feasble schedule of the jobs whch satsfes the requred propertes..3 A Robust Algorth We wll descrbe a greedy algorth for odfyng the schedule upon arrval of a job. Ths algorth keeps the nvarants that after the assgnent of each job there s no dle te, and there are no preeptons after the copleton te of the least loaded achne. That s, t receves as an nput an arbtrary strongly optal schedule for I t, that has no preeptons after te L t and converts t nto a strongly optal schedule for I t wth no preeptons after te L t, usng a graton factor of. In order to defne the assgnent of job t and possble graton of a total sze of at ost ( )p t of other jobs we consder two cases for OPTt, whch needs to be coputed. (Recall that the jobs are not assgned accordng to OPTt.)

16 Fg. 6 The schedule resultng fro the assgnent of x n the exaple for ASSIGN Case. t s a large job n OPTt Let k be the achne ndex to whch t corresponds. Let k t, k t beasnlea3. We start wth the assgnent to achne k t +. Let b = L t k t, and recall that p t = L t k, snce t s assgned as a large job to achne k n OPTt. By Lea 3, L kt L t k t +. Defne L t k and Lt k t to be new preepton tes. The other copleton tes are copleton tes n OPTt as well. Assgn a part of the job t to achne k t + durng the te [L t k t,l t k = p t]. After applcaton of the procedure ASSIGN of achnes,...,k t + (see below), the te nterval [L t k t +,Lt k t ] wll be occuped as well (so no dle te would be ntroduced) and we wll change the ndexng of achnes so that achne k t + becoes achne k, and each achne k t + k becoes achne (fk t + = k then no change of ndexng s perfored). The new loads of these achnes are ( L t k t +,...,Lt k,p t,l t ) k+,lt k+,...,lt = ( L t k t +,...,Lt k,lt k,lt k+,lt k+ ),...,Lt. Snce k t k t +, these are exactly the szes of the k t largest jobs n {,,...,t}. We are left wth a part of sze b of job t, whch needs to be assgned to achnes,,...,k t +. We nvoke the procedure ASSIGN wth l = k t +, X = b, L = b, and L = L t for k t +. By ths assgnent, no preeptons are ntroduced after te L t k t (and such preeptons dd not exst prevously). By these defntons, X = L and L L l. The property ll = l = L + X holds snce by the resultng loads of achnes wth large jobs, achne k t + needs to be assgned a total sze of L t k t L kt + to fll the dle te, and achne where k t needs to be assgned a total sze of L t k t L. Ths total sze ust be equal to the total sze of the prevous jobs assgned to achnes,...,k t + plus the reander of job t of sze p t (L t k Lt k t ), thus (k t +)b = k t + = L +p t (L t k Lt k t ) = k t + = L +b. Case. t s a sall job n OPTt Denote by k the achne ndex to whch t corresponds. We nvoke the procedure ASSIGN wth l = k, X = p t, L = L t for

17 l and L = Y +p t l, where Y denotes the total sze of jobs not assgned to achnes k +,..., n OPTt (.e., the jobs assgned to achnes,...,k n OPTt ). Defne L to be a new preepton te. All other copleton tes are szes of large jobs that rean large, that s, they were copleton tes n OPTt as well. We need to show X L, L L l, and ll = l = L +X. Snce job t s assocated wth achne l = k (as a sall job), we have Y +p t k p t so L X. Snce L = L t k, we get L L k. Fnally, ll = Y + p t = Y + X. Snce Y s exactly the total sze of jobs assgned to achnes,,...,k pror to the assgnent of job t, wehave Y = l = L. We have proved the followng theore. Theore 7 There exsts a polynoal te algorth of graton factor whch antans a strongly optal schedule on dentcal achnes wth a polynoal nuber of preeptons. 3 Lower Bounds on the Mgraton Factor of Optal Algorths 3. Identcal Machnes We show that an optal algorth for akespan nzaton (whch s not necessarly strongly optal) ust have a graton factor of at least. Recall that ths s exactly the graton factor of the algorth gven n Sect.. An optal algorth whch nzes the l p nor for soe <p< on dentcal achnes s sply a strongly optal soluton wth respect to akespan, and therefore t ust nze the akespan as well. Thus, we also fnd that the algorth of Sect. has the optal graton factor for nzaton of any l p nor. Proposton 8 Let A be an optal robust algorth for preeptve schedulng on dentcal achnes wth the goal of akespan nzaton or nzaton of an l p nor. The graton factor of A s no saller than. Proof We use an nput slar to the one of Graha []. Consder an nput consstng of jobs of sze. An optal soluton for ths nput has akespan. Next, a job of sze arrves. The optal akespan becoes. In an optal schedule for the augented nput, the new job ust run at every te durng the te nterval [0,]. Before the arrval of the longer job, the te nterval [0, ] s copletely occuped on all achnes. Thus, at each te durng the te nterval [0, ], there ust be a part of a job whch s reassgned, for a total sze of of reassgned parts of jobs. Ths ples the requred graton factor. 3. Unforly Related Machnes In ths secton we show a lower bound of on the graton factor of any optal soluton for the nu akespan proble on unforly related achnes, and for the nzaton proble of the l p nor of the loads.

18 Theore 9 Any optal algorth for preeptve schedulng on unforly related achnes so as to nze the akespan has a graton factor of at least. Proof Consder an nstance wth achnes such that achne has speed s = (+ε) where ε>0sanarbtrarly sall constant. The nput starts wth a sequence of jobs where job j has processng te p j = ( + ε) j. For ths nstance the unque optal soluton (wth akespan of ) assgns job j copletely to achne j. We show by nducton that no other soluton can be optal. Specfcally, we show by nducton that for 0, n order to coplete all jobs at te, every job j {,...,} ust run on achne j durng the te nterval [0, ]. The base case s for = 0. Assue that ths holds for an nteger (where 0 ). We prove the cla for +. For that we need to consder only the assgnent of job +. Snce achnes,..., are copletely occuped durng the te nterval [0, ], ths job ust run on achnes +,...,.Lett be the total te durng whch job + runs on achne + and t s the te durng whch t runs on other (slower) t achnes. We get that f t < then p + + t (+ε) + = p (+ε) + + (t + t +ε )< p +, snce ( + ε)t + t < + ε. The nput now contnues wth an addtonal job of ndex of sze. It can be proved, slarly to the above, that the unque optal soluton of the augented nput has a akespan of + ε; t assgns each job j to achne j + for j, and the new job to achne. Thus every job needs to be oved copletely to a dfferent achne, and the graton factor s at least and ths bound tends to asε tends to zero. = (+ε) (+ε), Theore 0 Any optal algorth for preeptve schedulng on unforly related achnes so as to nze the l p nor of the achne loads has a graton factor of at least. Proof Let p>. It was shown n [8] that any optal soluton for the l p nor does not use dle te, and the achne copleton tes of two achnes of speed rato s have rato s p, that s, the total szes of jobs assgned to these achnes have a rato of s p p. The last property s true only f such an assgnent exsts, otherwse these ratos ay be nexact. Consder an nstance wth achnes where achne has speed s = ( + ε) where ε>0 s an arbtrarly sall constant. The nput starts wth a sequence of jobs where job j has processng te p j = ( + ε) jp p. For ths nstance the unque optal soluton assgns job j copletely to achne j. Ths results n a jp copleton te of (+ε) p (+ε) j = ( + ε) j p for achne j. For a par of achnes < j, the speed rato s s s j = (+ε) = ( + ε) j, and the rato between the copleton (+ε) j tes s (( + ε) p )/(( + ε) j p ) = ( + ε) j p. Slarly to the prevous proof, t s possble to show that no other schedule can be optal for l p. The nput s now augented wth a job of sze. As a result, n the unque optal schedule, job j s assgned to achne j +for j and the new job s assgned to achne.

19 The graton factor s at least p = ( + ε) p+, and (for a gven (+ε) ( )p/(p+) value of p) ths bound tends to asε tends to zero. 3.3 Restrcted Assgnent In ths secton we show a lower bound of Ω() on the graton factor of any optal soluton for akespan nzaton and l p nor nzaton on dentcal achnes wth restrcted assgnent. Theore Any optal algorth to the proble of preeptve schedulng on parallel achnes wth restrcted assgnent so as to nze the akespan, has a graton factor of at least. The sae holds for the proble of nzng the l p nor. Proof Consder an nput consstng of unt sze jobs of types where a job of type can be processed on ether achne or achne +, and such that there are jobs of each type. For ths nput, we denote by x the total processng te of achne on jobs of type, and by y the total processng te of jobs of type on achne +. Clearly n any optal soluton (for any nor) of ths proble we have x = and y = and thus the copleton te of every achne s (snce the copleton te of achne for sx + y = + = and ths s also the copleton te of achnes and ). Now, the nput contnues wth one addtonal job of sze whch ust be processed on achne. The new nput has a unque optal soluton (for any nor), where all achnes are actve untl te. Ths soluton assgns all jobs of type to achne (for =,,..., ). Thus the graton factor of any optal soluton s at least = y = = = ( ) =, and the cla follows. Recall that an optal algorth for nzaton of the l p nor for soe < p< cannot use dle te and ust be n fact strongly optal wth respect to akespan. We show the followng lower bound for two achnes. Theore Any strongly optal algorth to the proble of preeptve schedulng on two achnes wth restrcted assgnent has a graton factor of at least. Proof Consder an nput whch conssts of three jobs: p =, and M ={}, p = and M ={}, and fnally, p 3 = 3 and M 3 ={, }. The akespan of any schedule s at least 3, and a strongly optal schedule has one achne of copleton te 3 and one achne of copleton te. Such a schedule ndeed exsts (see dscusson on optal schedules n Sect. B.), and one possble such schedule s where durng the te slot [0, ] the frst achne processes the frst job and the second achne processes the thrd job, durng [, 3] the frst achne processes the thrd job, and durng [, ] the second achne processes the second job. Any strongly optal schedule ust be of the for that one achne runs the thrd job durng the te slot [, 3] whle the other achne s dle (and copleted the parts of jobs assgned to t by te ). Assue wthout loss of generalty that the frst achne s dle durng ths

20 te. Let p 4 = and M 4 ={}. For the nputs of four jobs the akespan reans 3, and both achnes have loads of 3 (t s possble to odfy the schedule above by runnng the last job on the second achne durng the te slot [, 3]). However, snce we assue that pror to the arrval of the fourth job, the second achne s busy durng the te [0, 3], and the new job ust run on ths achne, the total sze of jobs ovng fro t to the frst achne ust be at least, whch gves a graton factor of at least. Note that the last lower bound does not exclude the opton of an algorth whch uses dle te and nzes the akespan (but t s not strongly optal), whch has a graton factor of (the lower bounds shown n Proposton 8 and Theore hold for ths case). 3.4 Non-Preeptve Schedulng Our work s concerned wth preeptve schedulng. The case of non-preeptve schedulng (where every job ust run contnuously on the achne on whch t s scheduled) was nvestgated [3], and t was shown that a graton factor of Ω() s requred n order to antan an optal schedule (even f the coputatonal coplexty of the algorth s not polynoal). We further show that no fnte graton factor s possble for non-preeptve schedulng, even for two dentcal achnes. Consder two dentcal achnes. Let N be a large nteger. The frst four jobs are of szes N +, N +, N, and N. The unque optal soluton assgns the two jobs of szes N + and N to one achne and the other two jobs to the other achne. The optal akespan s N. Next, a job of sze 6 arrves. The unque optal soluton assgns the three sallest jobs to one achne and the two largest jobs to the other achne, and ts akespan s N + 3. To acheve ths soluton, two of the frst four jobs ust grate to another achne. We get a graton factor of at least N 6 whch can be arbtrarly large. To generalze ths exaple for an arbtrary nuber of achnes, the ntal nput s augented by jobs of sze N. Optal solutons never cobne one of these jobs wth any of the other jobs on the sae achne. Snce the schedules have sple structures, the proof holds both for akespan and for any l p nor. Appendx A: Fractonal Restrcted Assgnent In ths secton we dscuss a relaxaton where a job can be splt arbtrarly aong achnes, as long as the total te dedcated to t s suffcent. That s, the constrant that a job cannot be processed on dfferent achnes n parallel s reoved. In the cases of dentcal achnes and unforly related achnes ths akes the proble trval even n the onlne envronent; n order to obtan an optal soluton, each new job s sply splt nto parts of proportonal szes, accordng to the requred rato between achne loads. Ths algorth does not grate any jobs or parts of jobs (so ts graton factor s zero). As for restrcted assgnent, sply splttng a job equally aong ts allowed achnes leads to solutons whose akespan can be Ω() tes the optal akespan