resource leveling problems A common approximation framework for early work, late work, and European Journal of Operational Research

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European Journal of Operational Research

journalhomepage:www.elsevier.com/locate/ejor

Discrete Optimization

A common approximation framework for early work, late work, and resource leveling problems ^R

Péter Györgyi, Tamás Kis

^∗

Institute for Computer Science and Control, Kende str 13–17, Budapest 1111, Hungary

a rt i c l e i n f o

Article history:

Received 16 September 2019 Accepted 9 March 2020 Available online 18 March 2020 Keywords:

Scheduling

late work minimization early work maximization resource leveling approximation algorithms

a b s t r a c t

Westudytheapproximabilityoffourschedulingproblemsonidenticalparallelmachines.Inthelatework minimizationproblem,thejobshavearbitraryprocessingtimesandacommonduedate,andtheobjec- tiveistominimizethelatework,definedasthesumoftheportionofthejobsdoneaftertheduedate.

Arelatedproblemisthemaximizationoftheearlywork,definedasthesumoftheportionofthejobs donebeforetheduedate.Wedescribeapolynomialtimeapproximationschemefortheearlyworkmax- imizationproblem,andweextendedittothelateworkminimizationproblemaftershiftingtheobjective functionbyapositivevaluethatdependsontheproblemdata.Wealsoproveaninapproximabilityresult forthelatterproblemiftheobjectivefunctionisshiftedbyaconstantwhichdoesnotdependonthein- put.Theseresultsremainvalidevenifthenumberofthejobsassignedtothesamemachineisbounded.

Thisleadstoanextensionofourapproximationschemetotwovariantsoftheresourcelevelingproblem withunittimejobs,forwhichnoapproximationalgorithmisknown.

© 2020ElsevierB.V.Allrightsreserved.

1. Introduction

Lateworkminimization,introducedbythepioneeringpaperof Bła˙zewicz (1984),isanimportantarea ofmachinescheduling,for anoverviewseeSterna(2011).Thevariantwearegoingtostudyin thispapercan bebrieflystatedasfollows.Wehaveidenticalpar- allel machines anda set of jobswith arbitraryprocessing times, andacommonduedate.Weseekaschedulewhichminimizesthe sumoftheportionofthejobsdoneaftertheduedate.Astrongly relatedproblemisthemaximization oftheearlywork,wherewe havethe samedata andtheobjectiveis tomaximize thesum of the portionofthe jobsdone beforethecommon duedate.How- ever,thelistoftheresultsformaximizingtheearlyworkismuch shorterthanthatforthelateworkminimizationproblem,seee.g., SternaandCzerniachowska(2017),Chen,Liang,Sterna,Wang,and Bła˙zewicz(2020b).

The applications ofthe late work optimizationcriterion range from modeling theloss of informationin computational tasks to the measurement of dissatisfaction of the customers ofa manu-

R This work has been supported by the National Research, Development and Inno- vation Office – NKFIH, grant no. SNN 129178, and ED_18-2-2018-0 0 06. The research of Péter Györgyi was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.

∗ Corresponding author.

E-mail addresses: peter.gyorgyi@sztaki.hu (P. Györgyi), tamas.kis@sztaki.hu (T.

Kis).

facturingcompany. Inparticular,Bła˙zewicz(1984)studiesaparal- lelprocessorschedulingproblemwithpreemptivejobswhereeach jobprocessessomesamplesofdata(ormeasurementpoints),and iftheprocessingcompletesafterthejob’sduedate,thenitcauses alossofinformation.Anaturalobjectiveistominimizetheinfor- mation loss,which isequivalentto the minimization ofthe total late work. A small flexible manufacturing systemis described in Sterna(2007),wherethe applicationofthe latework criterion is motivatedbythe interests ofthecustomersaswell asbythat of theowner of thesystem. The commoninterest ofthe customers istohavetheportions oftheir ordersfinished aftertheduedate minimized. In turn, forthe owner of the system, the amount of latework isa measure ofdissatisfactionofthecustomers.As for early work maximization, we can adapt thesame examples con- sidering gain and satisfaction instead of loss and dissatisfaction, respectively.

Wehavethreemajorsourcesofmotivationforstudyingtheap- proximability ofthe early work maximization, and thelate work minimizationproblems:

(i) Chen,Sterna,Han,andBła˙zewicz (2016)establishthecom- plexityoflateworkminimizationinaparallelmachineenvi- ronment,andthentheauthorsdescribeanonlinealgorithm fortheearlyworkmaximizationproblemofcompetitivera- tio

√2m²−2m+1−1

m−1 ,wheremisthenumberofthemachines.

However,sincethelateworkcanbe0,noapproximationor onlinealgorithmisproposedforthelateworkobjective.

https://doi.org/10.1016/j.ejor.2020.03.032 0377-2217/© 2020 Elsevier B.V. All rights reserved.

(ii) Sterna and Czerniachowska (2017) propose a polynomial timeapproximationschemefortheearlyworkmaximization problemwith2machines,anditisnotobvioushowto get ridofsomeconstantboundonthenumberofthemachines.

Further on,Chenet al.(2020b) describea fullypolynomial time approximation schemeformaximizing theearly work onafixednumberofidenticalparallelmachines.

(iii)We haveobserved that some variants ofthe resource lev- eling problemare equivalentto the early work maximiza- tion and the late work minimization problems. Briefly,the resource leveling problemswe are referring to consist ofa parallel machine environmentand onemore renewablere- sourcerequiredbyasetofunittimejobshavingacommon deadline,andoneaimsattominimize (maximize)thetotal resourceusageabove(below)athreshold.Wearenotaware ofanypublishedapproximationalgorithmsforresourcelev- eling problems ina parallel machine environment, butthe resultsforthe early-andlate work problemscanbe trans- ferredtothisimportantsubclass.

In this paper we propose a polynomial time approximation schemeforthe early work maximization problemin an identical parallelmachine environment,which weextend tothelate work minimizationprobleminthesameprocessingenvironment.Byap- plying a concept of strong equivalence, we obtain analogous re- sultsforthe maximizationaswell astheminimization variantof theresourcelevelingwithunittimejobsproblemonidenticalpar- allelmachines.We emphasizethatthenumberofidenticalparal- lelmachinesispartoftheinputforallproblemsstudied,andthe processingtimesofthejobsarearbitrarypositiveintegernumbers inthe early work maximization, andthe late work minimization problems,whilewehaveunittimejobsandarbitraryresourcere- quirementsintheresourcelevelingproblems.

Theresultsofthispaperaretheoreticalinnature,theproposed algorithmsare not intendedforpractical use.However, they pro- videnewinsightthatcanleadtoefficientalgorithms,andthetech- niquedeveloped,outlinedinthelast section,maybeusedforde- rivingapproximationalgorithmsforotherproblemsaswell.

InSection2wepreciselydefinetheschedulingproblemsstud- ied in this paper, and provide the necessary terminology. In Section 3 we summarize related work from the literature. In Section 4 we prove the equivalence of the late work minimiza- tion problemwith the minimization variantof the resource lev- eling with unit time jobs problem, and an analogous result for theearly workmaximizationproblemandthemaximization vari- antofthe resource leveling problem. An inapproximabilityresult is statedand proved forthe late work minimization problem in Section5.InSection6wedescribeapolynomialtimeapproxima- tion scheme for the early work maximization problem extended withmachinecapacityconstraints,andinSection7weadaptthe resultsof Section 6to the late work minimization problem after shiftingtheobjectivefunctionbyaproblem-datadependentvalue.

BytheresultsofSection4,weobtainpolynomialtimeapproxima- tionschemesforthetwovariantsoftheresourcelevelingproblem aswell.WeconcludethepaperinSection8.

2. Problemformulationandterminology

In the late work minimization problem in a parallel machine environment,thereisasetJ ofnjobsthat havetobe scheduled onmidenticalparallel machines. Ifit isnotnotedotherwise, the numberofthe machinesis partofthe input. Eachjob j∈J has aprocessingtime p_j andthereisa commonduedated.Thelate workobjectiveYisto minimizethe totalamountofwork sched- uled after d, see Chen et al.(2016). Thatis, a schedule S speci- fiesa machine

μ

j(^S)∈

{

¹,...,m

}

^{anda startingtime t}j(S)≥0for

eachjob. Sisfeasibleifforeachpairofdistinctjobsj andk such that

μ

j(^S)=

μ

k(^S), eithert_j(^S)+p_j≤t_k(^S) ^ortk(^S)+p_k≤t_j(^S)^. Throughoutthepaperweassumethattherearenoidletimesbe- tween thejobson anymachine. Thelatework ofa scheduleSis Y(^S)=m

i=1max

{

⁰,

j∈J_i(S)p_j−d

}

,whereJ_i(^S)=

{

^j∈J

| μ

j(^S)= i

}

^{.Laterwe willfrequentlyrefertothesumofthejobprocessing}

timesp_sum:=

j∈Jp_j.

We add a further constraint to this problem. We introduce a boundNonthenumberofthejobsthatcanbescheduledonany ofthemachines.Thisiscalledmachinecapacity,seee.g.Woeginger (2005).Throughoutthepaperweassumethatm·N≥n,otherwise there is no feasible solution forthe problem. Notethat machine capacityisnot a commonconstraintforthelate work minimiza- tionproblem,butitwillbeusefullater.However,bysettingN=n, thecapacityconstraintsbecomevoid,andwegetbackthefamiliar lateworkminimizationproblem.

Since thelate workobjective can be 0,and decidingwhether a feasible schedule of zero late work exists or not is a strongly NP-hard decision problem (Chen et al., 2016), no approximation algorithm exists for this objective.However, by applying a stan- dard trick,we can ensure that theobjective function value isal- ways positive, andapproximating it becomespossible. We intro- ducea probleminstance-dependentpositivenumberT,andwhen approximatingtheoptimumlatework,wewillconsidertheobjec- tivefunctionT+Y.

Thereisanotherwaytomodifytheobjectivefunctionsothatit allowsustoachieve approximationresults.Theearlywork objec- tiveX,introducedbyBła˙zewicz,Pesch,Sterna,andWerner(2005), whichmeasures the total amountofwork scheduled onthe ma- chinesbefored,iscloselyrelatedtoYbytheequation

X

(

^S

)

=psum−Y

(

^S

)

^{foranyfeasiblescheduleS}. (1) Intheresourcelevelingproblem,wehavenjobswithunitprocess- ingtimestobescheduledonmidenticalparallelmachinesinthe timeinterval[0,C],whereCisacommondeadlineofallthejobs.

Additionally,thereisarenewableresourcealongwithasoftlimitL fortheresourceusage.Eachjobjhassomerequirementa_j≥0from theresource.All problemdataisintegral. AscheduleSspecifies a machine

μ

j(^S)∈

{

¹,...,m

}

^{andstarting timet}j(^S)∈

{

⁰,...,C−1

}

for each job j. Without loss of generality, m·C≥n, otherwise no feasiblescheduleexists.Throughoutthe paperwe assume thatin anyschedule, ifk<m jobsstart at some time point t, then they occupythe first kmachines. The goal isto finda feasiblesched- ule S, where each job starts in [0,C−1], and the total resource usage above L is minimized, i.e., we have to minimize Y˜(^S)^:= _C−1

t=0max

{

⁰,

j∈J_t(S)a_j−L

}

,whereJt(^S)=

{

^j∈J

|

^tj(^S)=t

}

,and

j∈J_t(^S)a_jisthetotalresourceusageofthosejobsstartingattime point t. Acloselyrelated problemisthe maximization ofthe to- tal resource usage below L over the scheduling horizon [0, C], i.e.,maximizeX˜(^S)^:=C−1

t=0min

{

^L,

j∈J_t(^S)a_j

}

^.Letasum:=

j∈Ja_j. Thetwoobjectivefunctionsarerelatedbytheequation

X˜

(

^S

)

^=asum−Y˜

(

^S

)

^{foranyfeasiblescheduleS. (2)}

Noticethesimilarityof(1)and(2).Aswewillsee,thisisnotaco- incidence.Furthermore, since checkingwhethera feasible sched- ule S with Y˜(^S)=0 exists is a strongly NP-hard decision prob- lem(Neumann&Zimmermann, 2000), forapproximatingtheop- timal solutionwe will usethe objectivefunction T˜+Y˜, whereT˜ is an instance-dependent positive number. If m≥n, then we get the project scheduling version of the resource leveling problem, i.e., there are no machines andarbitrary number of jobscan be startedatthesametime.

Thispaperusesthe

α

^|

β

^|

γ

^{notationofGraham,Lawler,Lenstra,}

andRinnooy Kan (1979), where

α

^{denotes the machine environ-}

ment,

β

^{the additional} constraints,and

γ

^{theobjective function.}

Inthe

α

^{fieldwe usePforarbitrarynumberofparallelmachines}

and P2 in case oftwo machines. In the

β

^{field, d}j=d indicates that thejobshavea commonduedate,whilen_i≤N indicatesthe capacityconstraintsofthemachines. ThesymbolsXandYinthe

γ

^{fieldrefertotheearly work,andto thelateworkcriterion,re-}

spectively, and we use the symbols X˜ andY˜ to denote the total resourceusagebelowandabovethelimitL,respectively,incaseof theresourcelevelingproblem.

In this paper we describe approximation algorithms for the above mentioned, and some other combinatorial optimization problems.OurterminologycloselyfollowsthatofGareyandJohn- son(1979).Aminimization(resp.maximization)problem^isgiven by asetofinstancesI,andeach instanceI∈Ihasa setofsolu- tionsS^I,andan objectivefunctionc^I:S^I→Q.Givenanyinstance I,the goalis to finda feasible solutions^∗∈S^I such that c^I(^s∗)= min

{

^cI(^s)

|

^s∈S^I

}

^(cI(^s∗)=max

{

^cI(^s)

|

^s∈S^I

}

^{). LetOPT(I) denote}

theoptimumobjectivefunctionvalueofprobleminstanceI.Afac- tor

ρ

approximation algorithmfor a minimization (maximization) problem^{isapolynomial timealgorithmAsuch thattheobjec-} tive function value,denoted by A(I), ofthesolution found bythe algorithmAonanyprobleminstanceI∈I satisfiesA(I)≤

ρ

·OPT(I) (A(I)≥

ρ

·OPT(I)). Naturally,

ρ

≥1 for minimization problems, and 0<

ρ

≤1 for maximization problems. Furthermore, a polynomial time approximation scheme(PTAS) for ^{isa familyof algorithms} {A_ε}_ε>0 suchthatA_ε isafactor1+

ε

approximationalgorithmfor ^ifitisaminimizationproblem,orafactor1−

ε

approximation algorithm(0<

ε

<1),for^ifitisamaximization problem.Inad- dition,afullypolynomialtimeapproximationscheme(FPTAS)islike aPTAS,butthetimecomplexityofeachA_εmustbepolynomialin 1/

ε

^aswell.

Let 1 and 2 be two optimization problems. We say that they are stronglyequivalentifthereexistbijective functionsfand g, where f establishes a one-to-one correspondence betweenthe instances of 1 and that of 2, whereas g establishes a one- to-one correspondence between the set of solutions of each in- stance Iof 1 and that of f(I) of 2 such that for each S∈S^I, c^I(^S)=c^f(I)(^g(^S))^.

3. Previouswork

In this section first we overview existing complexityand ap- proximability results for scheduling problems withthe total late work minimization,andthetotalearly workmaximization objec- tive functions, but we abandon exact and heuristic methods as theyarenotdirectlyrelatedtoourwork.Thenwebrieflyoverview whatisknownaboutresourcelevelinginaparallelmachineenvi- ronment.

The total late work objective function (late work for short) is proposed by Bła˙zewicz (1984), where the complexity of min- imizing the total late work in a parallel machine environment is investigated. For non-preemptive jobs it is mentioned that minimizing the late work is NP-hard, while for preemptive jobs, a polynomial-time algorithm, based on network flows, is de- scribed. This approach is extended to uniform machinesas well.

Subsequently, several papers have appeared discussing the late work minimization problem in various processing environments.

For the single machine environment, Potts and Van Wassenhove (1992b) describe an O(nlogn) time algorithm for the problem with preemptive jobs, where each job has its own due date.

Furthermore,thenon-preemptivevariantisshowntobe NP-hard, and amongother results, a pseudo-polynomial time algorithm is proposedforfindingoptimalsolutions.PottsandVanWassenhove (1992a)devise afullypolynomial time approximationscheme for thesinglemachine non-preemptivelate workminimization prob- lem,whichisextendedtothetotalweightedlateworkproblemby Kovalyov,Potts, andVan Wassenhove(1994),wherethelatework of each job is weighted by a job-specific positive number. For a

two-machineflowshop,Bła˙zewiczetal.(2005)provethatthelate workminimizationproblemisNP-hard evenifallthejobshavea commonduedate,andtheyalsodescribeadynamicprogramming based exact algorithm. A more complicated dynamic program is proposed for the two-machine job shop problem with the late work criterion by Bła˙zewicz, Pesch, Sterna, and Werner (2007). Late work minimization in an open shop environment, with preemptiveorwithnon-preemptive jobs, isstudied inBła˙zewicz, Pesch,Sterna,andWerner(2004),whereanumberofcomplexity results are proved. For the parallel machine environment, Chen etal. (2016) provethat decidingwhether a schedulewith 0late work exists is a strongly NP-hard decision problem, while if the numberof machinesis only 2, then it is binaryNP-hard even if thejobshaveacommonduedate.Furthermore,they describean onlinealgorithmformaximizingthe earlyworkofjobsthat have to be scheduled in a given order. For several other complexity resultsnotmentionedhere,werefertoSterna(2000,2006,2011). Arelatedproblemistheminimizationofthetotaltardinesson identical parallel machines, when the jobs have a common due dated. Kovalyov andWerner (2002)observe that without modi- fyingtheobjectivefunction, thereisno hope foranyapproxima- tionalgorithm,likeinthecaseofminimizing thetotallate work.

Hence, they augment the objective function value by a positive constant b, andprove that the problem does not admit a factor (¹+

ε

)approximationalgorithmforany0<

ε

<1/bunlessP=NP.

Itfollowsthatinordertohavean(F)PTAS,bmustdependpolyno- miallyondorthejobprocessingtimes.Theyalsodescribeafully polynomialtime approximation schemeifb=d, andthenumber ofthemachinesisfixed.

Asfortheearly work,besidesthe paperofChenetal.(2016), we mention Sterna andCzerniachowska (2017),where a PTAS is proposedformaximizingtheearlyworkinaparallelmachineen- vironment with 2 machines, where all the jobs have a common duedate.Chenetal.(2020b)describeafullypolynomialtimeap- proximationschemeifthenumberofidenticalparallelmachineis fixed.TheyalsoprovidecomputationresultsforthepreviousPTAS aswellasforthe FPTASon probleminstanceswith2and3 ma- chinesandupto65and13jobs,respectively.

Resource leveling isa well studied area of projectscheduling, whereanumberofexactandheuristicmethodsare proposedfor solving it for various objective functions and under various as- sumptions,seee.g.,Kis(2005),NeumannandZimmermann(2000), Rieck,Zimmermann, andGather(2012),Verbeeck, VanPeteghem, Vanhoucke,Vansteenwegen, andAghezzaf (2017). Drótosand Kis (2011) consider a dedicated parallel machine environment, and proposeandexact methodforsolving resourceleveling problems optimally with hundreds of jobs. In the same paper, some new complexityresultsareobtained.

Chen,Kovalev,Sterna,andBła˙zewicz(2020a)introducetheno- tion of mirror scheduling problems, which is a kind of strong equivalence. Two scheduling problems, 1 and 2, constitute a pairof mirror scheduling problems if there is a bijective mapping betweentheir instances,andanysolutionS₁ ofanyinstanceI₁ of 1 canbemappedtoasolutionS₂ ofthecorresponding instance I₂ of2suchthat theobjectivefunctionvaluesofthetwosched- ulesareequal,andthereisamirrortimepointTandifamachine processesjob jattimetinS₁,thenthesamemachineprocessesj attimeT−tinS₂.

4. Equivalenceofthelateworkminimizationproblemandthe resourcelevelingproblem

Inthissectionweprovetheequivalenceofthelateworkmin- imizationproblemandtheresourcelevelingprobleminthesense definedattheendofSection2.

Fig. 1. Corresponding schedules for late work minimization problem and resource leveling problem.

Theorem1. ThelateworkminimizationproblemP

|

^dj=d,n_i≤N

|

^Y, andtheresourcelevelingproblemP

|

^pj=1

|

^Y˜areequivalent.

Proof.The proofconsistsoftwo parts.First,we defineabijective functionbetweenthesetofinstances ofthelate workminimiza- tionproblemandthesetofinstancesoftheresourcelevelingprob- lem with unit time jobs. Then, we consider an arbitrary pair of instancesofthetwoproblems(thepairisdeterminedbythepre- viousfunction)andwe define anotherbijectivefunction between theschedulesofthetwoinstances.

Consider an arbitrary instance I of the late work minimiza- tion problem (m machines, n jobs withprocessing times p_j (j∈

{

¹,...,n

}

^{) andcommon due date d, and upper bound N on the}

numberofjobson each machine). Thecorresponding instance of the resource leveling problem has N machines, n jobs with pro- cessingtimes1,resource requirementsa_j:=p_j(j=1,. . .,n),com- mondeadlineC:=m,andresourcelimitL:=d.Nowweverifythat thegivenmappingbetweentheinstances ofthetwoproblems is abijection.Indeed,thefunctionisinjective(differentinstances of thelate work minimization problemare mapped to different in- stancesoftheresourcelevelingproblem),andsurjective(forevery instanceI oftheresource levelingproblemthereisan instanceI ofthe late work minimization problemsuch that Iis mappedto I),thusitisbijective.

Now,wedescribeamappingfromthesetoffeasibleschedules ofanyinstance ofthelate workminimization problemtothat of the corresponding instance of the resource leveling problem. Let instanceIofthelate workminimizationproblembe fixedandlet I bethecorresponding instanceofresourcelevelingproblem. Let SbeanyfeasibleschedulefortheinstanceI,ourfunction defines ascheduleSforI basedonSasfollows.Ifa jobj isthethjob scheduledonmachinei inSthen schedulethecorrespondingjob ofI on machine at time t_j(^S)^:=i−1, for an illustration, see Fig.1.

Thefollowingseriesofclaimswillprovethetheorem:

Claim1. SisfeasibleforI.

Proof.Since thereareatmostN jobsscheduled onamachine in S,thusweassigneachjobtooneoftheNmachinesofI.Further- more,eachjobinIhasaunitprocessingtime,hencethejobsdo notoverlap.

Claim2. ThemappingbetweentheschedulesforIandthatforIisa bijection.

Proof.Itiseasytoseethatthegivenmappingofschedulesisin- jective.Moreover, let S be any scheduleforI.We define S forI such that S is mapped to S asfollows. Suppose job j starts on M at time point i−1 for some i∈

{

¹,...,C

}

^{in S,then j is the}

^th job on

μ

j(^S)=i.Since inS,there isno idlemachine among M₁,...,Mbydefinition,Sisfeasible,andthevalueoft_j(S)iswell defined.

Claim3. IfthelateworkofsomescheduleSforinstanceIisY,then theobjectivefunctionvalueofthecorrespondingscheduleSforIis alsoY.

Proof. Considerthei^th machineM_i(i∈

{

¹,...,m

}

^{) inS,let} Jide- note the set of jobs scheduled on M_i in S. The late work on M_iismax

{

⁰,

j∈J_ip_j−d

}

,thusY=m

i=1max

{

⁰,

j∈J_ip_j−d

}

^.On

the other hand,observethat the jobsof Ji are mappedto those jobs of the resource leveling problem that start at time point i−1 in S. The total resource requirementof these jobs exceeds L by max

{

⁰,

j∈Jia_j−L

}

, thus the objective function value of S isC

i=1max

{

⁰,

j∈Jia_j−L

}

=m

i=1max

{

⁰,

j∈Jip_j−d

}

=Y,since L=d,C=m,andp_j=a_jbythemappingdefinedabove.

Theaboveclaimsprovethetheorem.

By(1)and(2),wehavethefollowing:

Corollary 1. The early work maximization problem P

|

^dj=d,n_i≤ N

|

^X,andtheresourcelevelingproblemP

|

^pj=1

|

^{X˜ are}equivalent.

5. InapproximabilityofP2

|

^dj=d

|

^c+Y

Inthissection we provethat ifwesimply adda value c toY inthe objective functionof the late work minimization problem, wherecisafixedpositivenumber,thenitisimpossibletogetan approximationalgorithmoffactorsmallerthan ^c_c⁺¹ unlessP=NP. WewillusethefollowingresultofChenetal.(2016):

Theorem2(Theorem2inChenetal.,2016). TheproblemP2

|

^dj= d

|

^{Y is NP-hard. In} particular, it is NP-hard to decide if a feasible scheduleoftotallatework0exists.

The following statementand its proof isanalogous to that of Theorem2ofKovalyovandWerner(2002)fortheinapproximabil- ityofPm

|

^dj=d

|

^b+T_j.¹

Proposition1. Letcbeapositiveconstant.Thenforany0<

ε

<1/c, thereisno(¹+

ε

)-approximationalgorithmforP2

|

^dj=d

|

^c+Y un- lessP=NP.

Proof. Supposewehaveafactor1+

ε

approximationalgorithmfor P2

|

^dj=d

|

^c+Y forsome 0<

ε

<1/c.Weshow howtoapply this approximation algorithm to decide iffor anyinstance of P2

|

^dj= d

|

^{Y a feasible schedule of total late work 0 exists. However, the}

latter decisionproblem is NP-hard by Theorem 2,which implies ourclaim.

Consider anyinstance ofP2

|

^dj=d

|

^c+Y.Iftheapproximation algorithm returns a solution of value c, then clearly, there is a scheduleof0latework.Nowsupposetheapproximationalgorithm returnsasolutionofvalueatleastc+1(novaluebetweencand c+1ispossible, becauseall problemdata isintegral).Indirectly, assumethatthereisascheduleoftotallatework0,andhence,the optimum solution value is c. But then c+1≤(¹+

ε

)^c<c+1 musthold,wherethefirstinequalityfollowsfromtheapproxima- tion factor and the second form

ε

<1/c. This is a contradiction, thusallfeasibleschedulesmusthavetotallateworkatleast1.

6. APTASforP

|

^dj=d,n_i≤N

|

^X

Inthissection we describe a PTASforP

|

^dj=d,n_i≤N

|

^X.Note

that the machine capacity N is a positive integer such that m·N≥n,where n isthe numberof thejobs, and m isthe num- berofidenticalparallelmachines.

Wewilldevise twoalgorithms (bothparameterized by

ε

^),and

we will run both of them on the same input, and finally, we will choosethe betterof thetwo schedules obtainedasthe out- put ofthe algorithm. The first familyof algorithms, describedin Section6.1,hasanapproximationfactorof(¹−4

ε

)^iftheoptimum value isatleast

ε

·m·d.In contrast,theapproximation algorithm

1We thank a referee for calling our attention to this paper.

presented inSection 6.2is offactor1−2

ε

^{if theoptimum value}

issmallerthan

ε

·m·d.Runningboth methodsonthesameinput guaranteesanapproximationfactorof(¹−4

ε

)^.

After somepreliminary observations,wewilldescribe thetwo algorithms along with the proofs of their soundness,and in the endwecombinethemtoobtainthePTAS.

Throughoutthissection,S^∗denotesanoptimalscheduleforan instanceofP

|

^dj=d,n_i≤N

|

^X.

6.1. FamilyofalgorithmsforthecaseX(S^∗)≥

ε

·m·d

Inthissectionwedescribeafamilyofalgorithms

{

Aε

| ε

>0

}

, such thatAε isafactor(¹−4

ε

)approximationalgorithm forthe problemP

|

^dj=d,n_i≤N

|

^{XundertheconditionX(S∗)}≥

ε

·m·d.

We start by observingthat ifa job starts after dthen we do nothavetodealwithitsexactstartingtime andwithitsmachine assignment, because the total processing time of this job is late work.We canschedulethesejobsfromanytime pointafterdon anymachine wherewe donot violatethe machine capacitycon- straints.

Let

ε

>0befixed.Wedividethesetofjobsintothreesubsets, huge,bigandsmall.The setofhugejobsisH:=

{

^j∈J

|

^pj≥d

}

, thesetofbigjobsisB:=

{

^j∈J

| ε

^{2 d}≤p_j<d

}

,andtheremain- ingjobsaresmall.

Proposition2. Ifthereareatleast mhugejobs,thenschedulingm, arbitrarilychosen hugejobson m distinctmachines, andtherest of thejobsarbitrarily,yieldsanoptimalschedulebothforthemaximum earlyworkandtheminimumlateworkobjectives.

Proof. LetSbethescheduleconstructedasdescribedinthestate- ment of the proposition. Then X(^S)=m·d, which is the maxi- mum possible early work. By Eq.(1),S hasminimum late work aswell,thusitisoptimalforbothobjectivefunctions.

Proposition3. If

|

H

|

≤m−1,thenthereexistsanoptimalschedule forthemaximumearly work aswell asfortheminimumlatework objectivessuchthatthehugejobsarescheduledon

|

H

|

^{distinct ma-}

chines.

Proof. LetS^∗ be an optimalschedule forthe earlywork (as well asforthelatework)objectivewiththemaximumnumberofma- chines onwhich a hugejob isscheduled.Indirectly, suppose less than

|

H

|

^{machinesprocessatleastonehugejob,hence,thereex-}

ists a machine M₁ processingat least two huge jobs, say j₁ and j₂, in this order. Since there are at mostm−1 huge jobs, there exists a machine M^∗ (in fact thereare atleast two), which does not process any huge jobs. If lessthan N jobs are scheduled on M^∗,then movejob j₂ fromM₁ toM^∗,otherwiseswapjobj₂ with anyofthejobsscheduledonM^∗,andletSbetheresultingsched- ule. Clearly,the machine capacities arerespected by S,andboth ofthemachinesM^∗ andM₁ workintheperiod[0,d]inS,while theworkassignedtoanyothermachineisthesameinbothsched- ules.Hence,X(S)≥X(S^∗).Therefore,Sisoptimalfortheearlywork objective,andbyEq.(1),forthelateworkobjectiveaswell.How- ever,inSmoremachinesprocessatleastonehugejobthaninS^∗, acontradiction.

From now on, we assume that there are at mostm−1 huge jobs,andwefixanoptimalscheduleS^∗inwhichthehugejobsare scheduledondistinctmachines.

Ouralgorithmhasthree mainphases:first,we scheduleallof thehugejobs,andsomeofthebigjobssuchthattheygetastart- ingtime smallerthand,thenweschedulesome ofthesmalljobs such that they getastartingtime smaller thand,andfinally,we scheduletheremainingbigandsmalljobs,ifany,arbitrarilywhile respectingthemachinecapacityconstraints.

For each big job j we round down its processing time p_j to the greatest integer p_j:=

ε

^{2 d}(¹+

ε

)^k

^{by selecting k}∈Z such that p_j≤p_j. Since we have

ε

^2d≤p_j<d for each big job j, the number of the different p_j values is bounded by the constant k1:=

^log1+ε(¹/

ε

²)

+1 that depends on the fixed

ε

only. Let B1,B2,...,Bk₁ denote the sets of the big jobs with thesame roundedprocessingtimes,i.e., Bh:=

{

^j∈J :p_j= (¹+

ε

)^h−1·

ε

^2d

}

⁽Bh=∅ispossible).

For each machine without a huge job, we guess the number of the big jobs from each set Bh that start before d. This guess canbedescribedbyanassignmentA,whichconsistsofk₁numbers (

γ

1,

γ

2,...,

γ

k₁),where

γ

h describesthenumberofthejobsfrom Bh.Abigjob assignment(

γ

1,

γ

2,...,

γ

k₁)^{isfeasible,ifitdoesnot} violate the constraint on the number of the jobs on a machine, i.e.,k₁

h=1

γ

h≤N,andalltheselectedjobscanbestartedbefored. Toverify thelatter condition, it suffices toschedule the selected jobsinanyordersuchthatthelongestjobisscheduledlast,which ensuresthat thelast job startsasearly aspossible.Letk₂ be the numberofpossiblebigjobassignments.Sincethetotalnumberof bigjobsthatmaystartbeforedonamachineisatmost

^1/

ε

²

^,we

have k₂≤k₁^1/ε2. Let A₁,A₂,...,A_k

2 denote the different feasible bigjobassignments.

Alayoutisak₂tuple(^t1,t₂,...,t_k

2)^{thatspecifiesforeachfea-} sibleassignmentthenumberofthemachinesthatusesit.Let

γ

ih

denote the number of big jobs from Bh assigned by A_i. A lay- outisfeasibleifandonlyifk₂

i=1t_i

γ

ih≤

|

Bh

|

^foreachh=1,...,k₁. The number of feasible tuples is bounded by the number of non-negative,integersolutionsoftheinequalityk₂

i=1t_i≤m−

|

H

|

, whichis bounded by

_m−|H|+k2

k₂

, a polynomial inthe size of the input,sincek₂isaconstant(thatdependson

ε

^{only).InAlgorithm}

A,weexamineeachbigjoblayoutandgetacompleteschedulefor eachofthem.

AlgorithmA

1.Determinethesetoffeasiblelayouts.

2.Foreachlayoutt,performSteps3–6.

3.AssignthehugejobsofHtomachinesM₁,...,M_|H|arbitrarily, andbigjobstotheremainingm−

|

H

|

^{machinesaccordingtot}

(t_imachinesuseassignmentA_i)

4.Oneachmachine,scheduletheassignedjobsfromtimepoint0 oninarbitraryorder.

5.IfN≥n,theninvokeAlgorithmB,otherwiseinvokeAlgorithmC toschedulesmalljobs.

6.Scheduletheremainingjobs(smallandbig,ifany)onthema- chines arbitrarily such that no machine receives morethan N jobsintotal(includingthepre-assignedhugeandbigjobs).

7.OutputS_A,whichisthebestschedulefoundinSteps2–6.

Nowwe turntoAlgorithmsBandCforschedulingsmalljobs.

AlgorithmBisasimplegreedymethodwhichworksonlyifthere arenomachinecapacityconstraints,i.e.,N≥n.

AlgorithmB

Input:partialscheduleofbigjobs

1.Fori=1,...,mdo:

2.Schedule amaximal subset ofsmalljobson machineM_i after the bigjobswithout idletimesuch that nosmalljob finishes afterd.

Observethat theabove methodmayassignalotofsmalljobs toamachine,thusitmaynotyieldafeasiblescheduleifN<n.

AlgorithmC ismuch more complicated. LetJ^small denote the setof small jobs, P_i^small≥0 the idletime on machine i before d,

andn^small_i thenumber ofthe jobsthat canbe scheduled onma- chineiafterthepartial scheduleofbigjobs,i.e., n^small_i isthedif- ferencebetweenNandthenumberofthebigjobsassignedtoma- chineM_i.NotethatP_i^small=0ifahugejobisassignedtomachine M_i.

Ourgoalistomaximizetheearlywork ofthesmalljobsfora fixedassignmentofbigandhugejobs.Tosimplifyourproblem,we onlywanttomaximize thetotalprocessingtimeofthesmalljobs thatamachinecompletesbefored.Thismaydecreasetheobjective function value of the final schedule, but we will show that this errorisnegligible.

We can model the above problem with an integer program.

Weintroducen·(^m+1)^{binaryvariablesx}i,j(i=0,1,2,...,m, j= 1,2,...,n), where x_0,j=1 means that we do not schedule job j toanymachinebefored,whileincaseof1≤i≤m,x_i,j=1means thatjob jwill bescheduledon machinei,andwillbecompleted notlaterthand.

max m

i=1

j∈J^small

xi,jpj (3)

s.t.

j∈J^small

xi,jpj≤P_i^small, i=1,...,m, (4)

j∈J^small

xi,j≤n^small_i , i=1,...,m, (5)

m

i=0

x_i,j=1, j∈J^small, (6)

xi,j∈

{

⁰,1

}

, i=0,...,m, j∈J^small. (7) WegettheLP-relaxationoftheaboveintegerprogrambyreplacing x_i,j∈{0,1}withx_i,j≥0intheconstraints(7).

AlgorithmC

Input:partialscheduleofbigjobs

1.DeterminethevaluesP_i^small,n^small_i fori=1,...,m.

2. SolvetheLP-relaxationof(3)–(7),andlet x¯be abasicoptimal solution.

3. For i=1,...,m, ifx¯_i,j=1 fora job j,then assign that job to machinei.

4. Foreachmachine,scheduletheassignedjobsrightafterthebig jobswithoutidletimesinarbitraryorder.

Observethat fractional jobsofthe optimalLPsolutionarenot assignedtoanymachineby AlgorithmC,butthey willbe sched- uledbyStep6ofAlgorithmA.

The proofs ofthefollowing two claimseasily followfromthe definitions.

Proposition4. S_Aisfeasible.

Proposition5. The timecomplexity ofAlgorithm Bis polynomially boundedinthesizeoftheinput.

Proposition6. The timecomplexity ofAlgorithm Cis polynomially boundedinthesizeoftheinput.

Proof.We can determine a basic solution of a linear program withnm variablesand n+2m constraints intwo steps. First,ap- ply a polynomial time interior-point algorithm to find a pair of primal-dual optimal solutions, and then, we can use Megiddo’s methodto determine a basic solution x¯ for the primal program,

seee.g.,Wright(1997).TheotherstepsofAlgorithmCrequirelin- eartime.

Proposition7. Thetime complexity of Algorithm A is polynomially boundedinthesizeoftheinput.

Proof. Recallthatthenumberofthefeasiblelayoutsispolynomial (atmost

m+k₂

k₂

).EachoftheSteps3–6requiresO(nm)time,except Step5 ifitinvokes AlgorithmC,but itis alsopolynomial dueto Proposition6.

Without loss of generality, we assume that in S^∗ the huge and big jobs precede the small jobs on each machine, and the big jobs are scheduled in non-decreasing processing time order on each machine. We introduce an intermediate scheduleS_int: it is the same as S^∗ except that the processing time of each big job isrounded asinAlgorithmA.Thatis, theprocessingtime of eachbigjobisroundeddowntothegreatestnumberoftheform

ε

^2d(¹+

ε

)^k

,(k∈Z),andafterroundingwere-schedulethejobs on each machine in the same order as in S^∗, but with the de- creasedprocessingtimesofthebigjobs.Byconsideringthosebig jobson themachinesthat start beforedin S_int,we can uniquely identifyanassignmentofbigjobsforeachmachine.Therefore,we can determinethelayout t^∗ ofthe bigjobsthat start beforedin S_int.Nowwestateandprovethemainresultofthissection.

Theorem3. IfX(S^∗)≥

ε

·m·d,thenAlgorithmAis afactor (¹−4

ε

) approximationalgorithmforP

|

^dj=d,n_i≤N

|

^X.

Proof. Recallthat S_intisthescheduleobtainedfromS^∗ byround- ingdowntheprocessingtimeofeachbigjob,andshiftingthejobs to the left, if necessary, to eliminate any idle times (created by rounding) on the machines. Since p_j/(¹+

ε

)<p_j≤p_j, we have X(^Sint)≥X(^S∗)/(¹+

ε

)≥(¹−

ε

)^X(^S∗)^{.Lett∗ be thelayout ofbig} jobs corresponding to S_int. Algorithm A will consider the layout t^∗ at some iteration, and let S be the schedule created from t^∗. SinceX(S_A)≥X(S),itsufficestoprovethatX(^S)≥(¹−4

ε

)^X(^S∗)^.To achievethis,weproceedbyprovingaseriesoflemmas.

Lemma1. IfN≥nandX(S^∗)≥

ε

·m·d,thenX(^S)≥(¹−

ε

)^X(^S∗)^. Proof. IfAlgorithm B schedulesall thesmall jobswhen creating schedule S, then the only jobs finishing after d can be big and huge jobs.Since theset of bigand hugejobs that start befored inscheduleScontainsallthebigandhugejobsthatstartbefored inscheduleS_int,wegetX(S)≥X(S_int).

Ifthereis atleastonesmall jobthat remains unscheduled by AlgorithmB,thenconsidertheearlyworkinS.Weknowthatthe total processing time on each machine is at least d(¹−

ε

²) ^due theconditionofStep2ofAlgorithmB.Hence,X(^S)≥md(¹−

ε

²)^. SinceX(S)≤X(S^∗)≤m·d,andX(S^∗)≥

ε

·m·dbyassumption,wede- rive

X

(

^S

)

^≥

(

¹⁻

ε

²

)

^d·m≥

(

¹⁻

ε )

^X

(

^S∗

)

^,

asclaimed.

Proposition8. IfN<n,thenX(^S)≥X(^Sint)−3

ε

²·d·m.

Proof. Consider Algorithm C, when it creates S. It solves (3)–(7) andx¯istheoptimalbasicsolutionthatwegetfromthealgorithm.

Recall that ifi≥1then x¯_i,j=1 ifandonly ifjob j is assignedto machine i byAlgorithm C.We introduceanother integersolution x of (3)–(7). Letx_i,j:=1, ifa smalljob j completes before don machinei inS_int,otherwise,x_i,j:=0.Notethat xisafeasibleso- lution,becauseS_int isafeasibleschedule.

Let

v

(^x) ^{denotetheobjective functionvalue ofa solutionxof} (3)–(7),OPTIPtheoptimumvalueof(3)–(7)andOPTLPtheoptimum valueofitslinearrelaxation.Foranyfeasiblesolutionxof(3)–(7), we haveOPT_LP≥OPT_IP≥

v

(^x)^.LetX_int^{small denotetheearly workof}