On some special cases of the restricted assignment problem

Information Processing Letters 116 (2016) 723–728

Contents lists available atScienceDirect

Information

Processing

Letters

www.elsevier.com/locate/ipl

On

some

special

cases

of

the

restricted

assignment

problem

Chao Wang

a

,

1

,

René Sitters

b

,

c

,∗

a_{EastChinaUniversityofScienceandTechnology,200237Shanghai,China} b_{VUUniversityAmsterdam,1081HVAmsterdam,TheNetherlands}

c_{CentrumWiskunde&Informatica(CWI),1098XGAmsterdam,TheNetherlands}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received21April2015

Receivedinrevisedform17March2016 Accepted16June2016

Availableonline17June2016 CommunicatedbyM.Chrobak Keywords:

Scheduling

Restrictedassignmentproblem Designofalgorithms

Weconsider somespecialcasesoftherestrictedassignmentproblem. Inthisscheduling problemonparallelmachines,anyjob j canonlybeassignedtooneofthemachinesin its givensubset Mj ofmachines.We give an LP-formulationfor theproblem withtwo

jobsizesandshowthatithasanintegraloptimalsolution.WealsopresentaPTASforthe casethattheMj’sareintervalsofthesamelength.Further,wegiveanewandverysimple

algorithmforthecasethat|Mj|=2 (knownasthegraphbalancingproblem)withratio 11/6.

©2016ElsevierB.V.Allrightsreserved.

1. Introduction

A classic algorithm by Lenstra, Shmoy and Tardos [1]

givesa 2-approximationforminimizing themakespan on unrelated parallel machines(denoted by R

||

Cmax). Inthis problem,wearegivenn jobsandm machinesand process-ing times pi j which give the time neededto process job j on machinei. Thegoalisto assignthejobstothe ma-chinessuchthatthemaximumload(totalprocessingtime) over all machines is minimized. The same paper shows thatapproximatingtheproblemwithinafactor3

/

2 is NP-hard.Today,nobetterupperorlowerboundisknownand improvingoneofthetwoboundsisconsideredoneofthe mainopenproblemsintheschedulingtheory.Evenforthe socalledrestrictedassignmentproblem nobetterratios are known.Here,eachjob j hasaprocessingtimepjand

pro-*

Correspondingauthorat:VUUniversityAmsterdam,1081HV Amster-dam,TheNetherlands.

E-mailaddresses:[email protected](C. Wang),[email protected] (R. Sitters).

1 _{ThisresearchdonewhilevisitingtheVUUniversityAmsterdamand} was supported in part bythe China Scholarship Council under grant No. 201306740038.

cessingset Mj

⊆ {

1

,

2

,

. . . ,

m

}

and pi j

=

pj fori

∈

Mj and pi j

= ∞

otherwise.

AbreakthroughwasmadebySvensson[2],whoshowed thattheintegralitygapoftheconfigurationLPforthe re-stricted assignment problem is at most 1

.

942. However, the proof does not give a polynomial time approach for constructingacorrespondingschedule.Aninteresting spe-cial case in which

|

Mj

|



2 for all j was considered by Ebenlendr et al. [3] who called this the graph balanc-ing problem. An instance may be seen as a multigraph G

= (

V

,

E

)

whereeach edge ej

∈

E has aweight pj.The edges should be oriented such that the maximum total weightofincomingedgesisminimized.Theauthorsgivea 1.75-approximationbyLP-roundingandalsoshowthatthe integralitygapoftheirLPis1

.

75.

Based on the work ofEbenlendr etal. [3],some spe-cialcasesofgraphbalancinghavebeenstudied.Lee,Leung and Pinedo [4] gave an FPTAS if the graph structure is restrictedto a tree.Verschae andWiese [5] showedthat even the configuration-LP has an integrality gap of 1.75 forthe graph balancingproblem and2 forthe unrelated versionwhereeachjob mayhavedifferentjob processing timesonitstwomachines.

http://dx.doi.org/10.1016/j.ipl.2016.06.007 0020-0190/©2016ElsevierB.V.Allrightsreserved.

CORE Metadata, citation and similar papers at core.ac.uk

Glass and Kellerer [6] explored the restricted assign-ment problem withonly two different processing times: 1 andk



2, and gave an approximation algorithm with factor 2

−

1

/

k. In the same paper, they considered the restricted assignment problem withthe additional struc-ture that the Mj’s are nested and gave a 2

−

1

/

m ap-proximation by list scheduling. For totally ordered sets a 3

/

2-approximation was designed. Huo and Leung [7]

improved the nested case by giving an algorithm with worst-case bound of 5

/

3, moreover they also studied so called tree-hierarchicalprocessingsets. Here, machines are thenodesofarootedtreeandprocessingsetscorrespond with a path fromsome vertexto the root. For the latter problem, they gave a 4

/

3-approximation algorithm. Mu-ratore et al. [8] improved this nested case by designing a polynomial time approximation scheme (PTAS). Epstein andLevin[9]alsostudiedthenestedandtree-hierarchical cases,andtheydesignedaPTASforthetwocases respec-tively.

For two arbitraryjob sizes, Kolliopoulos and Moysog-lou [10] gave a 1

.

883-approximation algorithm for com-putingtheoptimalvaluebasedonSvensson’s[2]result. If the jobs can onlybe assignedon atmost two machines, theratioreducesto1.652.

1.1. Ourresults

We give several results for special cases of the re-stricted assignment problem. First, we briefly discussthe 2-approximation algorithm for R

||

Cmax given by Shmoys and Tardos [11]. We show that this approach leads to a very simple 1

.

88-approximation algorithm for the graph balancing problem. Althoughthis ratiois worse than the 1.75-approximation given by Ebenlendr et al. [3], the al-gorithm andanalysis are very easy,given the framework of[11].Further,we studytherestrictedassignment prob-lemon intervals.Here,themachinescanbeordered such thateachprocessingsetisaconsecutivesetofmachines.

Weshowthatthespecialcaseoftwodifferent process-ing times s, b can be solved exactly in polynomial time, assuming that s



b and b

>

OPT

/

2. We obtain this by formulating an exactLP, i.e., withan integrality gapof 1. Interestingly, ourLPis strongerthan theconfigurationLP whichhasanintegralitygapofatleast3

/

2 inthiscase.

As mentioned above, the interval casehas been

well-studied and several polynomial time approximation

schemes are known. We extendtheseresults by givinga PTASfortheproblemofequallengthintervalsandby sim-plifying theanalysisofknown approximationschemes by usingtheframeworkof[11].

2. Preliminaries

The 2-approximationalgorithmfor R

||

Cmax byLenstra, Shmoys, and Tardos [1] is well-known. A slightly differ-entalgorithm andanalysiswasgivenafewyearslater by ShmoysandTardos[11].Theyintroducedanewrounding techniquewhichdoesnotrequirethefractionalsolutionto be a vertexofthe linearprogrammingrelaxation. An ad-vantageisthatitcanthereforebeappliedtoanyfractional assignmentandforvariantsoftheproblemasweshalldo

in this paper.Below, we give a shortexplanation of this roundingtechniquethatwewilluseseveraltimes.

The algorithms in [1] and[11] work as follows. First, a guess T for the optimal value is made. The algorithm

produces a schedule of length at most 2T if indeed

OPT



T . Using a binary search on T , this will give a 2-approximation algorithm. Denote by xi j



0 the frac-tion of job Jj assigned to machine Mi, i

∈ {

1

,

. . . ,

m

}

and j

∈ {

1

,

. . . ,

n

}

. If there exists a schedule of length at most T ,thenthefollowingLPhasafeasiblesolution.

m



i=1 xi j

=

1 for all j n



j=1 xi jpi j



T for all i xi j

=

0 if pi j

>

T

,for all i,

j xi j



0 for all i,j

In[1],theauthorsconcludethat anyextremeoptimal LP-solutionhasatmostm fractional jobsandthereisa per-fect matching of these jobs to machines in the support graph.Hence, thisgivesascheduleof lengthatmost2T . In[11]theapproachisasfollows.Foreachmachinei, de-fine a completeordering

≺

i such that j

≺

ik if pi j



pik. Givena feasiblefractional solutionx,let yi

= 



jxi j



be the number ofjobs (rounded up) assignedto machine i. For each machine, order the fractions assigned to it by

≺

i and call this the LP-schedule. Then split this sched-ule up intoslots such that thefractions ineach slot (ex-cept forthe last)addup toexactly1.Thus,a jobfraction maybe splitandbeassignedtotwocontiguousslots,and there are yi slotsfor machine i. Definea bipartite graph G

= ({

W

,

V

},

E

)

asfollows.Foreachjob j thereisavertex wj andforeach machine i thereare vertices vi1

,

. . . ,

viyi.

Thereisanedgebetween wj andviz ifjob j isprocessed (partially)inthez-th slotofmachine i.Itfollowsdirectly fromHall’s theorem that G has a matchingthat contains allverticesW ,i.e.,alljobs.

It is easy to see that this matching yields a

2-approximation for R

||

Cmax. Consider an arbitrary ma-chine i.Foranyslotz,let pi_z be themaximumprocessing time pi jamongalljobs j scheduledinslotz onmachine i. (Inotherwords,letpi_z

=

max

{

pi j

| (

wj

,

viz

)

∈

E

}

.)Then,the jobmatched tovi

z hasprocessingtimeatmostpiz.LetLiz be the length of slot z on machine i in the LP-schedule. Since jobsare ordered by

≺

i, thejob matched to viz has processingtimeatmost

pi_z



Li_z−1for all z



2. (1) Hence,the totalloadofmachine i inthe schedule de-finedbythematchingisatmost

pi₁

+

y



i−1 z=1 Li_z



pi₁

+

yi



z=1 Li_z

=

pi₁

+

n



j=1 xi jpi j



2T

.

3. A simple 11

/

6-approximation for graph balancing

WeshowhowtheapproachbyShmoysandTardos[11], shown above, can be used to give a simple approxima-tionalgorithmforthegraphbalancingproblemwithratio

strictlylessthan2.Ourratioislargerthanthe1

.

75 given byEbenlendr etal.[3]butouralgorithmandanalysisare extremelyeasy,giventheroundingtechniqueofthe previ-oussection.

Withoutlossofgeneralityweassume thattheoptimal makespanis1 andisknowntothealgorithm,i.e.,the tar-get value is T

=

1. Let B be the set of big jobs, which are the jobs with processing time pj

>

1

/

2. Then, each machine can contain atmost one big job in the optimal schedule.Thefollowingsetoflinearconstraintsisequalto (LP3)in[3].Insteadofusingthegraphnotationfromthat paperwedefine Mj tobethesetofmachinesonwhich j canbeprocessedandassume

|

Mj

|

∈ {

1

,

2

}

foralljobs j. (LP)



i∈Mj

xi j

=

1, for all jobs j (2)



j

xi jpj



1, for all machines i (3)



j∈B

xi j



1, for all machines i (4)

xi j

=

0, for all i

∈

/

Mj (5)

xi j



0, for all i,j

.

(6)

Graph Balancing Algorithm (0

.

5

< β <

1)

Step 1. Findafeasiblesolutionfor(LP);

Step 2. If j isabigjobandxi j

 β

,thenassign j toi; Step 3. Fortheremainingjobs,findaperfectmatching

fol-lowingtheapproachofSection2.

Theorem 1. The algorithm above (with

β

=

2

/

3)gives an 11

/

6

≈

1

.

833-approximationforthegraphbalancingproblem.

Proof. In Step 2,at most one big job is assigned to any machine. Further, if a big job is assigned to i in Step 2, then any other big job j with xi j

>

0 must satisfy xi j



1

− β

andxij

 β

forsomeother machinei(sinceitcan only be scheduled on two machines) and it will be as-signed to i in Step 2. Hence, after Step 2, any machine i either contains no fractional big job and at most one complete big job (call this Case 1) or it hasat leastone fractional big job and no complete big job and any frac-tionalbigjob j onitsatisfiesxi j

>

1

− β

(Case2).

Case 1. Assumebigjob j isassignedtomachinei inStep 2. This causesthe load toincrease by atmost

(

1

− β)

pi j

≤

1

− β

.ThematchingprocedureofStep3addsatmostthe largest value pi j among the fractional jobs. Since all re-mainingfractionaljobsoni aresmall,theincreasedueto Step 3 isat most0.5. Therefore, the totalload of i is at most1

+ (

1

− β)

+

0

.

5

=

2

.

5

− β

.

Case 2. Followingthenotation ofSection 2,let pi₁ and pi₂ bethelargestprocessingtimeinslot1and2onmachine i. Then, pi

1



1 and, byconstraint (4), pi2



0

.

5. Further,by theconditionsof

Case

2,thelargestjob inslot1isa big job j with xi j

>

1

− β

.Hence, Li1

 (

1

− β)

pi1

+ β

pi2. The loadL

ˆ

i _{ofmachinei afterStep3satisfies}

ˆ

Li



pi₁

+

pi₂

+

y



i−1 z=2 Li_z



pi₁

+

pi₂

+

1

−

Li₁



pi₁

+

pi₂

+

1

− (

1

− β)

pi₁

− β

pi₂

=

1

+ β

pi₁

+ (

1

− β)

p₂i



1

+ β + (

1

− β)/

2

=

3/2

+ β/

2.

Combiningboth cases, wesee that for

β

=

2

/

3, thetotal loadofmachinei isatmost

max

{

2.5

− β,

1.5

+ β/

2

} =

11/6.

2

4. Restricted assignment on intervals

The restricted assignment problem on intervals has beendiscussedinseveralpapers,[7–9,12].Here,each pro-cessing set forms a consecutive set of machines. These papersonlyconsidered theversionswithnested,laminar, ortree-hierarchical processingsets.Forarbitraryintervals, no better approximation factor than 2 is known. In this section, we show how to solve the interval problem for general intervals ifthere are only two different process-ingtimes s



b andforwhichOPT

<

2b. Althoughthisis alsoa ratherrestrictedversion, itholds forgeneral inter-valsandcannotbesolvedeasilybydynamicprogramming asholdsforthepreviously consideredversions.Also,note that thegeneralrestricted assignmentproblem with pro-cessingtimes

{



,

1

}

andOPT

=

1 isnon-trivialand2isthe best approximation ratioknown. (Approximating justthe optimalvaluecanbedonewithinafactor5/3inthiscase, aswas shown by Svensson [2].) Our algorithm solves an LPsolely for the big jobs, wherethe constraints are im-pliedbythesmalljobs.Weshowthatanintegralsolution canbefoundinpolynomialtime.Then,thesmalljobsare addedina greedyway.Further,we giveinSection4.2an easydynamicprogrammingapproachforthecaseofequal length intervals, a specialcase not yet considered in the literature.

4.1.Intervalsandtwoprocessingtimes

Weassumeherethatan instanceonlyconsistsofsmall jobs with processing time pj

=

s and bigjobs with pro-cessing time pj

=

b



s. If the optimal value T is at least2b,thenweimmediatelygeta3

/

2-approximation us-ingtheoriginal2-approximationforR

||

Cmax.Interestingly, the problemcan be solved exactly in polynomial time if OPT

<

2b.

Theorem 2. Therestrictedassignmentproblemonintervalsand withprocessingtimespj

∈ {

s

,

b

}

(s



b)canbesolvedexactly inpolynomialtimeifOPT

<

2b.

Asbefore,weassume w.l.o.g.thattheoptimalvalueis guessedcorrectlyandthat OPT

=

1.Forintegersi



i,let

[

i

,

i

]

be the set

{

i

,

. . . ,

i

}

. We compute an upper bound U

(

i

,

i

)

onthenumberofbigjobsthatcanbescheduledon

themachineset

[

i

,

i

]

,foreachsuchinterval

[

i

,

i

]

.Denote theinterval ofjob j byMj

= [

lj

,

rj

]

.Let S

(

i

,

i

)

betheset ofsmalljobs j forwhich Mj

⊆ [

i

,

i

]

.Now,aneasy upper boundonthenumberofbigjobsthatcanbescheduledin the interval

[

i

,

i

]

is



i

−

i

+

1

−

s

|

S

(

i

,

i

)

|



/

b



. However, thisboundturnsouttobenotstrongenough.Toimprove this, notethatanymachineeithercontainsonebigjobor none. In both cases, we know the number ofsmall jobs thatcanbeaddedtothemachine.Definek0andk1asthe numberofsmalljobsthatcanbescheduledonamachine ifthereis,respectively,nooronebigjob.Thenk0

= 

1

/

s

and k1

= (

1

−

b

)/

s

. Now let U

(

i

,

i

)

be the maximum number ofbig jobsthat can be scheduled within the in-terval

[

i

,

i

]

such that the remaining capacity is enough to add all jobsfrom S

(

i

,

i

)

,assuming that each of these small jobs can be scheduled on any of the machines in

[

i

,

i

]

. This upper bound can easily be computed: Denote S

= |

S

(

i

,

i

)

|

and I

=

i

−

i

+

1.Then U

(

i

,

i

)

is the maxi-mumvalue z

∈ {

0

,

1

,

. . . ,

I

}

suchthatzk1

+ (

I

−

z

)

k0

≥

S.

This leads to thefollowing linearprogram, exclusively forthebigjobs. Here, xi j isthefractionof j

∈

B doneon machinei. (LP-B)



k∈[i,i]



j∈B xkj



U

(

i

,

i

)

1



i



i



m (7)



i∈Mj xi j

=

1 j

∈

B xi j

=

0 i

∈

/

Mj

,

j

∈

B xi j



0 i

∈

Mj

,

j

∈

B

.

Notethat theinequalities



j∈B

xi j



1 forallmachinesi areincludedbytakingi

=

iinthefirstconstraints.

Lemma 3. Thereisafeasible0

,

1-solutionto(LP-B)anditcan befoundinpolynomialtime.

Proof. Consider an arbitraryfeasible solution x to(LP-B). Foranyl

∈ {

1

,

. . . ,

|

B

|}

,letm

(

l

)

bethesmallestmachine in-dexi forwhich

i



k=1



j∈B xkj



l.LetNB

= {

m

(

1

),

. . . ,

m

(

|

B

|)}

. Notethatallm

(

l

)

aredifferentsince



j∈B

xi j



1 forany ma-chinei.Hence,

|

NB

|

= |

B

|

.WeshowthatthebigjobsB can bescheduledonthesetofmachinesNB andthatanysuch assignmentisfeasiblefor(LP-B).

For any B

⊆

B, let M

(

B

)

= ∪

j∈BMj. If M

(

B

)

is a consecutive set then by definitionof NB,

|

M

(

B

)

∩

NB

|







k∈M(B)



j∈B xkj





k∈M(B)



j∈B xkj

= |

B

|

.Hence,

|

M

(

B

)

∩

NB

|  |

B

|.

(8)

Ifontheotherhand,M

(

B

)

istheunionofnon-intersecting

intervals, then (9) follows by considering the

non-intersecting intervalsof M

(

B

)

separately. ByHall’s theo-rem,thebigjobscanbeassignedto NB.

Now assume that we do assign the big jobs to NB. Then, the numberofbig jobsassignedwithin an interval

[

i

,

i

]

is NB

∩ [

i

,

i

] 

⎡

⎢

⎢

⎢



k∈[i,i]



j∈B xkj

⎤

⎥

⎥

⎥



U

(

i

,

i

_).

₍₉₎

The firstinequalityin(9)followsby definitionof NB and the second inequality follows from (7) and the fact that U

(

i

,

i

)

isinteger.By(9),theroundedsolutionremains fea-siblefor(LP-B).

2

Lemma 4. Anyintegralsolutionof(LP-B)canbecompletedby addingthesmalljobsinagreedyway.

Proof. Given anintegral solutionof(LP-B),thesmalljobs are scheduled on the machines in the order 1

,

2

,

. . . ,

m as follows. For machine i, consider all yet unscheduled smalljobs j forwhich i

∈

Mj andadd thesejobsin non-decreasingorderoftheirrightsiderjuntiltheloadofthe machinebecomesmorethan1

−

s orallsmalljobs j with i

∈

Mj areassigned.

Assume some smalljob k doesnot get scheduledthis way.Then, anymachine inthe set

[

lk

,

rk

]

hasthe follow-ingtwo properties:(i)itsloadismorethan1

−

s and(ii) it only contains small jobs j with rj



rk. Let i



lk be thesmallestmachineindexsuchthatthesetwoproperties holdforallmachinesin

[

i

,

rk

]

.

Assumei

>

1 andconsideranarbitraryjob j scheduled in

[

i

,

rk

]

.Since j wasnot scheduled on machine i

−

1 it followsfromtheorderinwhichsmalljobsare scheduled thati



lj.Hence,foranyjob j scheduledonamachinein

[

i

,

rk

]

i



lj



rj



rk

.

(10)

Clearly,

(10)

holdsaswellfori

=

1.Weseethat allsmall jobsscheduledon oneofthe machinesin

[

i

,

rk

]

arefrom S

(

i

,

rk

)

butthereisnospacetoaddjobk in.Thisviolates constraint(7)fortheinterval

[

i

,

rk

]

.

2

Integralitygaps. Weshowedthattheintegralitygapforour LPis 1inthis restrictedversion. Onemight alsoadd the small jobs in the LP and get an LP for all jobs and still maintainanexactrelaxation.However,asweshowed,itis sufficienttoconsiderbigjobsonly.Interestingly,the inte-gralitygapofthe configurationLPisatleast 3

/

2 forthis version asshownbythefollowingexample.(See [2]fora definition of the configuration LP.) There are 4 machines and7jobs.Jobsa,b,c havemachineset

[

1

,

2

]

andjobsd, e, f havemachineset

[

3

,

4

]

.Allthesehavepj

=

1.Thelast job g hasset

[

1

,

4

]

andprocessingtime2.Forthe config-urationLPsolution,takex1

(

a

,

b

)

=

x1

(

b

,

c

)

=

x1

(

c

,

a

)

=

1

/

4 andx1

(

g

)

=

1

/

4.On machine2, take x2

(

a

,

b

)

=

x2

(

b

,

c

)

=

x2

(

c

,

a

)

=

1

/

4 andx2

(

g

)

=

1

/

4.Dothesameformachines 3 and 4. The LP-value is 2 while the optimal makespan is 3.

4.2. APTASforintervalsofequallength

A PTAS for laminar set systems was given by Mura-tore etal.[8]andEpsteinandLevin [9].Forlaminar sets we havethat for anytwo sets Mj, Mk, either Mj

⊆

Mk, Mj

⊇

Mk or Mj

∩

Mk

= ∅

. It is easy to see that laminar sets canberepresentedby acollectionofintervalsandit

isthereforeaspecialcaseoftheintervalproblem.Here,we give aPTAS foranotherspecialcase:that ofequallength intervals. Infact, it holdsforanyinstance where,forany two jobs j, k, the inequality lj



lk implies rj



rk. Our PTASusessimilarideasashasbeenusedforlaminarsets, suchaspartitioningjobsinsmallandlargejobs.However, forschedulingthesmalljobswedonotconstructa sepa-rategreedyprocedure(asin[8]and

[9]

) butsimply refer to the general 2-approximation for R

||

Cmax as described inSection2.Wenote thatthisideaappliestolaminarset systemsaswellandgivesasimplificationovertheanalysis in[8]and[9].

4.2.1. Thedynamicprogram

Let K be the number of different job sizes. Jobs of the same size are said to be of the same type. Let nk be the number of jobs of type k (k

=

1

,

. . . ,

K ). For any type k labelthe jobsbynon-decreasing valuesrj.Denote the jobs in this order by J(₁k)

,

J₂(k)

,

. . .

Jn(kk). Polynomiality

of the DP follows from the observation that there is an optimal schedulein which, foreach type k, the jobs ap-pear in order J₁(k)

,

J₂(k)

,

. . .

. (If two jobs are in reversed order they can be swapped.) For every machine i, we define vectors

(

b1

,

b2

,

. . . ,

bK

)

to record the number of jobsfrom each type assigned to machines1

,

2

,

. . . ,

i. Say Fi

(

b1

,

b2

,

. . . ,

bK

)

=

1 ifit isfeasible toschedule thejobs

∪

k

{

J(₁k)

,

. . . ,

J_b(k_k)

}

withinamakespanofT onthefirsti ma-chines andlet it be 0 otherwise. Define F0

(

0

,

. . . ,

0

)

=

1. Then Fi

(

b1

,

b2

,

. . . ,

bK

)

=

1 if andonly ifthere are num-bersak

≤

bk(k

=

1

...

K )suchthat

•

Fi−1

(

a1

,

a2

,

. . . ,

aK

)

=

1.

•

For each k, all jobs J(_ak)

k+1

,

. . . ,

J

(k)

bk contain i in their

interval.

•

Thetotallengthofthejobsin



k

{

J(_ak) k+1

,

. . . ,

J (k) bk

}

isat mostT .

AfeasiblescheduleexistsifandonlyifFm

(

n1

,

...,

nK

)

=

1. The sizeof theDP table is O

(

mnK

₎

_{andthecomputation} ofeachvaluetakesO

(

nK

₎

_time.

4.2.2. Roundingtheinstance

LetT

=

1 bethetargetvalueandassumethatO P T



1 (hence,pj



1 forall j).Let1

/



bealargepositiveinteger. Callajob j big ifpj

>



andcallitsmall otherwise.Round theprocessingtimeofeverybigjobdowntop_j

= 

pj

2



2.

Consequently,thenumberofdifferentprocessingtimesof theroundedbigjobsisatmost1

/



2_{.Foreverysmalljob,} round its processing time down to p_j

=

_np j 



 n. Next, spliteachroundedsmalljobinto



npj



tinyjobswith pro-cessing time _n for each of them. The intervals forthese tiny jobs are unchanged. So in the rounded instance,we onlyhavetheroundedbigjobsandthetinyjobsoffixed size _n. The total number ofjob sizes is K

=

O

(

1

/



2

₎

_{. If} OPT



1,then thereisfeasiblescheduleoflengthatmost 1fortheroundedinstanceaswell.WecanusetheDP to findanoptimalsolutionfortheroundedinstancein poly-nomialtime.

Theorem 5. Givenan optimalschedule forthe rounded in-stance,wecanfinda schedulefortheoriginalinstancewith makespanCmax



1

+

4

.

Proof. We keep the assignmentof thebig jobsthesame as in the optimal schedule for the rounded instance as givenbytheDP.Theroundedprocessingtimeofabigjob isatleast





2



2

>



−



2

>



/

2 forsmallenough



,e.g.



<

1

/

2.Consequently, thereare atmost2

/



bigjobson a machine giving a total rounding error due to big jobs of at most 2

/



·



2

₌

₂

_

_{. Note that the DP gives us an} assignmentforthetinyjobswhichcanbeseen asa frac-tionalassignmentfortheroundedsmalljobs.Weapplythe matching procedure of Section 2 to these fractions. This procedure gives an integer assignment for the rounded smalljobs with an increase in the makespan of at most thelengthofoneroundedsmalljob,whichislessthan



. Finally,rounding uptheprocessingtimeofthesmalljobs givesanadditionalincreaseofatmostn

·

_n

=



.Hence,the totalincreaseinthemakespanisatmost4



.

2

5. Some open problems

The main subject of our research has been the re-stricted assignment problem on intervals. The problem turnedouttobemuchharderthanexpectedanditisnot clearif a PTAS exists.Even forthe restricted version de-scribedunder

Problem

1,nobetterapproximationthanthe general2-approximationforR

||

Cmaxisknown.

Problem 1. Givean approximation ratiobetter than2 for the restricted assignment problem with interval process-ingsets.Evenfortheprobleminwhichallintervalsshare onemachinenothingbetterisknown.Weconjecturethat a3

/

2-approximationispossible.

Problem 2. Find anapproximation ratiobetter than2 for therestrictedassignment problemwithprocessingsetsof sizeatmost3.

Problem 3. Whatisthecomplexityoftheintervalproblem withonlytwoprocessingtimess,b (asinSection4.1)and OPT



2b? For the very special casethat OPT

=

2b

=

3a andsmall jobscan go to anymachine, we can solve the problemby a similar technique asin Section 4.1. Tosee this,notethatintheoptimalsolution,amachinecanhold twobigjobsorthreesmalljobs.Ifthereisonlyonebigjob onamachine,thenonlyonemoresmalljobcanbe sched-uled in the spaceleft. If there are two big jobs orthree smalljobsonamachine,thenthemachinegetsfully occu-pied.Sotheproblemisequivalenttofindingthemaximal numberofpairedbigjobs.

References

[1]J. Lenstra, D. Shmoys, E. Tardos, Approximation algorithms for scheduling unrelated parallel machines, Math. Program. 46 (1–3) (1990)259–271.

[2]O.Svensson,SantaClausschedulesjobsonunrelatedmachines,in: STOC2011Proceedingsofthe43rdAnnualACMSymposiumon The-oryofComputing,2011,pp. 617–626.

[3]T.Ebenlendr,M.Krcál,J.Sgall, Graphbalancing:a specialcaseof schedulingunrelatedparallelmachines,Algorithmica68 (1)(2014) 62–80.

[4]K.Lee,J.Leung,M.Pinedo,Anoteongraphbalancingproblemswith restrictions,Inf.Process.Lett.110(2009)24–29.

[5]J.Verschae,A.Wieese,Ontheconfiguration-LPforschedulingon un-relatedmachines,in:Algorithms-ESA,2011,pp. 530–542.

[6]C.Glass,H.Kellerer,Parallelmachineschedulingwithjobassignment restrictions,Nav.Res.Logist.54 (3)(2007)250–257.

[7]Y. Huo, J. Leung, Fast approximation algorithms for job schedul-ingwithprocessingsetrestrictions,Theor.Comput.Sci.411(2010) 3947–3955.

[8]G.Muratore,U.Schwarz,G.Woeginger,Parallelmachinescheduling withnestedjobassignmentrestrictions,Oper.Res.Lett.38(2010) 47–50.

[9]L.Epstein,A.Levin,Schedulingwithprocessingsetrestrictions:PTAS resultsforseveralvariants,Int.J.Prod.Econ.133(2011)586–595. [10]S.Kolliopoulos,Y.Moysoglou,The2-valuedcaseofmakespan

min-imizationwithassignmentconstraints,Inf.Process.Lett.113(2013) 39–43.

[11]D.Shmoys,E.Tardos,Anapproximationalgorithmforthegeneralized assignmentproblem,Math.Program.62(1993)461–474.

[12]J.Leung,C.Li,Schedulingwithprocessingsetrestrictions:asurvey, Int.J.Prod.Econ.116(2008)251–262.