HAL Id: hal-02545572
https://hal.archives-ouvertes.fr/hal-02545572
Submitted on 17 Apr 2020
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
Minimizing the makespan for a UET bipartite graph on
a single processor with an integer precedence delay
Alix Munier-Kordon
To cite this version:
Alix Munier-Kordon. Minimizing the makespan for a UET bipartite graph on a single processor with
an integer precedence delay. [Research Report] lip6.2001.016, LIP6. 2001. �hal-02545572�
graph on a single processor with an integer
precedence delay
AlixMUNIERKORDON
Lab oratoire LIP6,
4 place Jussieu,75 252 Paris cedex 05
Alix.Munier@lip6.fr
Abstract
We consider aset oftasksof unitexecutiontimes andabipartite precedence delaysgraph with ap ositive precedence delay d : an arc (i;j)ofthisgraph meansthat j canb eexecuted at leastdtimeunits after thecompletiontimeofi. The problemis to sequence the tasks inorderto minimizethemakespan.
Firstly,weprovethattheasso ciateddecisionproblemis NP-comp-lete. Then,weprovideanontrivialp olynomialtimealgorithmifthe degreeofevery tasksfromone ofthetwosets is2. Lastly,we give an
approximationalgorithmwithratio 3 2 .
1 Introduction
Single andmultipro cessorsscheduling problems haveb eenextensively stud-iedin theliterature [16]. Scheduling problems withprecedencedelays arise indep endently in severalimp ortant applicationsand manytheoretical
stud-ies were devotedto these problems : this class of problems was considered forresource-constrained scheduling problem [3,13]. It wasalso studied asa relaxation for thejob-shop problem [1,8]. For computer systems, it
corre-sp ondstothebasic pip elines scheduling problems [15,20].
An instance of a scheduling problem with precedence delays is usually denedbyasetoftasksT =f1;:::;ngwithdurationsp
i
NP-HardProblem Reference 1jchains;d ij =djC max Wikumetal.[23] 1jprec;d ij =d;p i =1jC max Leungetal.[18]
every arc(i;j)2E,taskj can b eexecuted atleastd ij
timeunits after the completion timeof i. The numb er of pro cessors islimited. The problem is tondascheduleminimizing themakespan,orotherregularcriteria. Using
standard notations [16], the minimization of the makespan is denoted by Pjprec;d
ij jC
max .
In this pap er, we supp ose that the graph G is bipartite : T is split
into two setsX and Y and every arc (i;j) 2 E veries i 2 X and j 2 Y. We also consider that there is only one pro cessor, the duration of tasks is one and that the delay is the same for every arc. This problem is noted
1jbipartite;d ij = d;p i =1jC max
. The decision problem asso ciated is called SEQUENCING WITHDELAYS and is dened as:
Instance: AbipartiteorientedgraphG=(X[Y;E),ap ositivedelay danda deadline D .
Question : isthere asolution tothesequencing problem withadelay danda makespan smallerthanorequal to D ?
Weprove in section 2that 1jbipartite;d ij =d;p i =1jC max isNP-Hard. The complexity of this problem was a challenging question since several
authorsproved theNP-Hardness of moregeneral instances of this problem as shown in the table 1. In section 3, we prove that if the degree of every task in X is 2, then the problem is p olynomial and we provide a greedy algorithmtosolveit.
Several authorshave adaptedthe classical p olynomial algorithmsform pro cessors and particular graphs structures to a sequencing problem with
a unique delay as shownin the table 2. Note that Bampis [2] proved that Pjbipartite ;p
i
= 1jC max
is NP-Hard, but his transformation do esn't seem tob eeasily extended toourproblem.
Wikum et al. [23] also proved several complexity results, p olynomial
sp ecial cases and approximation algorithms for unusual particular classes of graphs (in fact, sub classes of trees). Munier and Sourd proved that 1jchains;d ij = d;p i = pjC max
Polynomial Problem Reference Comments 1jtree;d ij =d;p i =1jC max
Bruno etal.[6] Based on [14]
1jprec;d ij =1;p i =1jC max
Leung etal. [18] Based on [7]
1jintervalorders;d ij =d;p i =1jC max
Leung etal.[18] Based on [21]
develop ed a p olynomial algorithm forPjtree;d ij D ;p i =1jC max if D is a constant value.
At last,there aresomeapproximation algorithmsforproblems with
de-lays: Graham'slistschedulingalgorithm[11]wasextendedtoPjprec. delays ; d ij =k ;p j =1jC max
togive aworst-casep erformance ratioof 2 1=(m(k+ 1))[15,20]. ThisresultwasextendedbyMunieretal. [19]toPjprec.delays ;
d ij
jC max
. Bernstein and Gerner [5] study the p erformance ratio of the Coman-Graham algorithm for Pjprec. delays;d
ij = d;p i = 1jC max and slightly improve it in [4]. Schuurman [22]develop ed ap olynomial
approxi-mationscheme fora particularclassofprecedence constraints. Weprovein section 4 that the b ound 2 of Graham's list algorithm may b e achieved in
theworstcase for1jbipartite ;d ij
=d;p i
=1jC max
and we developa simple algorithmwith worstcasep erformance ratioequal to 3=2forthis problem.
2 Complexity of the problem
Let us consider a non orientedgraph G =(V;E)and an ordering L of the verticesofG(ie,aone-to-onefunctionL:V !f1;:::;jVjg). Forallinteger
i2f1;:::;jVjg,theset V L
(i)V is:
V L
(i)=fv2V;L(v)iand 9u2V;fv;ug2E and L(u)>ig
VERTEX SEPARATION is thendened as:
Instance: A nonorientedgraphG=(V;E)and ap ositive integerK.
Question : Is there an ordering L of the vertices of G such that, for
alli2f1;:::;jVjg,jV L
(i)jK?
This problem is proved to b e NP-complete in [17]. For the following, our pro ofswill b e moreelegant if we consider the converse ordering of the
tasks. Let n=jVj. If weset, 8v2V,L(v)=n L(v),j =n i+1and B
L
0(j)=V L
(i),wegetfor every value j2f1;:::;ng:
B L 0 (j)=fv2V;L 0 (v)>j and 9u2V;fv;ug2E and L 0 (u)jg
So,theequivalent INVERSE VERTEXSEPARATION problemmayb e dened as:
Instance: A nonorientedgraphG=(V;E)and ap ositive integerK.
Question : Is there an ordering L of the vertices of G such that, for alli2f1;:::;jVjg,jB
L
(i)jK ?
Weprove thefollowing theorem:
Theorem 2.1. Thereexistsapolynomialtransformationf fromINVERSE
VERTEX SEPARATIONto SEQUENCINGWITHDELAYS.
Proof. Let I b e an instance of INVERSE VERTEX SEPARATION. The
asso ciated instance f(I) is given by a bipartite graph G 0
= (X[Y;E 0
), a delayd anda deadline D dened as:
1. To any vertex v 2 V is asso ciated two elements x v 2 X and y v 2 Y andan arc(x v ;y v )2E 0 .
2. To anyedge fu;vg2E is asso ciated the arcs (x u ;y v ) and (x v ;y u ) in E 0 .
3. Thedelayis d=n 1 K and thedeadline D=2n.
f canb eclearlycomputedinp olynomial time(seeanexample gure1).
Let us supp ose thatL is asolution totheinstance I. Then,webuild a solution tof(I)asfollows :
1. Tasksfrom Y are executedb etweentime nand 2nfollowing L : they areexecuted fromy
L 1 (1) toy L 1 (n) .
2. Letus dene thepartition P i ;i=1:::nof X as: P i =fx L 1 (i) g[fx u ;u2B L (i)g i 1 [ j=1 P j
Tasksfrom X areexecuted b etween 0 and nfollowing P 1
:::P n
a
b
c
d
e
G = (V, E)
K = 2
x
a
x
b
x
c
x
d
x
e
y
e
y
d
y
c
y
b
y
a
d = 2
D = 10
f
Figure 1: Example oftransformationf
x
b
x
a
x
d
x
c
x
e
y
a
y
b
y
c
y
d
y
e
P
1
P
2
P
4
Figure 2: The schedule asso ciated withL
For example, if we consider the order dened by L(a) = 1, L(b) = 2, L(c) = 3, L(d) = 4 and L(e) = 5, the sets P
i , i = 1:::5, are dened by P 1 =fx a ;x b g,P 2 =fx c ;x d g,P 3 =;,P 4 =fx e gandP 5 =;. Figure2shows thecorresp onding solution forf(I)for ourexample.
Wehavetoprovenowthatthis schedulefulll all theprecedence delays of G
0
. Let us consider the task y L
1 (i)
;i= 1:::n. We must show that all
its predecessors in G 0
are completed attime(n+i 1) d=K+i.
1. We claim that all the predecessors of y L 1 (i) in G 0 are in S i j=1 P j . Indeed,x L 1 (i) 2P j ;jibyconstruction.
The other predecessors of y L 1 (i) are vertices x v with v adjacent to u=L 1
(i)in G. Now, if L(v) <L(u), then x v 2P k with k L(v). Otherwise,v 2B L (i)sox v 2P k withkL(u). 2. Weshowthatj S i j=1 P j
jK+i. Indeed,thissetiscomp osedby: [1]i
tasksx L
1 (j)
,j=1:::i, and[2]tasksx u
withL(u)>i, sou2B L
(i).
G 0
is bipartite, we can exchange the tasks such that tasks from X are all
completedb eforethersttaskfromY. WebuildanorderLfromtasksinY suchthat,8i2f1;:::;ng,L
1
(i)isthetasku2V suchthaty u
isexecuted attimen+i 1. Then, we mustprovethat,8i2f1;:::ng,jB
L
(i)jK.
Letconsideri2f1;:::;ng. Tasksexecutedduringtheinterval[0;K+i) can b e decomp osed into [1]x
L 1 (1) :::x L 1 (i) and [2] A set Q i of K other tasksfrom X[Y. Letb e v2B L
(i). Weclaim thatx v
2Q i
. Indeed, we getthatL(v)>i and there exists u 2V with L(u)i and fu;vg2 E. By denition of G
0 , we have then(x v ;y u )2E,sox v 2Q i . Wededuce thatjB L (i)jjQ i j=K. Corollary 2.2. 1jbipartite;d ij =d;p i =1jC max isNP-Hard.
3 A polynomial special case
Let us consider a non oriented connected graph G =(V;E) without lo ops (i.e. withoutedgesfu;ug,u2V)andanordering Lofthevertices. We set jVj=n. 8i2f1;:::;ng,we dene thesequences E
L
(i)by :
E L
(i)=ffu;vg2E; L(u)ig
E L
(i) is the set of edges adjacent to at least one vertices in fL 1
(1);:::; L
1 (i)g.
We dene the problem MIN ADJACENT SET LINEAR ORDERING by:
Instance : A non oriented graph G = (V;E) without lo ops and a p ositive integerK.
Question : Is there an ordering L of the vertices of G such that, for alli2f1;:::;jVjg,jE
L
(i)jK+i?
NoticethattheformulationofthisproblemisquitesimilartoMIN-CUT LINEAR ARRANGEMENT [10], which is NP-complete. In thefollowing,
weconsider thesubproblem ofSEQUENCING WITHDELAYS withthe restrictionthatthe degree ofevery vertexfrom X isexactly 2.
Theorem 3.1. There exists a polynomial transformation from to MIN
(X[Y;E),a delaydand adeadline D . We build an instancef(I) ofMIN
ADJACENTSET LINEAR ORDERING asfollows :
G 0
=(Y;E 0
). Foreveryx2Xwith(x;y 1 )and(x;y 2 )2Eisasso ciated anedgee x =fy 1 ;y 2 gin E 0 . thevalue K =D d jYj 1.
f canb ecomputedinp olynomialtime. Weprovenowthatfisap olynomial
transformation(see gure 3foran example)
f
1
2
3
4
a
b
c
d
d = 2
D = 8
a
b
c
d
K = 1
Figure 3: Example oftransformationf
Let us supp ose that a solution to I is given. Then, without lo osing generality, we can supp ose that the tasks from X are p erformed during
[0;:::;jXj)and tasks from Y during [D jYj;:::;D ). We build a linear ordering L following the sequencing order of tasks Y : 8i 2 f1;:::;jYjg, L(i)istheith taskofY in theschedule.
8i2f1;:::;jYjg,let b et=D jYj+(i 1)=K+i+dthestarting
timeofthetaskL 1
(i)fromY. At timet d=K+i, allthepredecessors of L
1
(1);:::;L 1
(i)must b e completed. Now, forevery edge e x
2 E L
(i)
isasso ciated exactlyone of those predecessors. So, jE L
(i)jK+i.
Conversely, let us supp ose that a solution to f() is given. Then, we p erformtasksfromY following Lduring theinterval [D jYj;:::;D ). We
dene thenthefollowing sequence X i X : 1. X 1 =fx2X ;e x 2E L (1)g, S i
Noticethat,byconstructionthat,8i2f1;:::;ng, i j=1 X i =fe x 2E L (i)g. TasksofXarep erformedduring[0;:::;jX)followingX
1 ;X 2 :::X n . Every task from S i j=1 X i
is then completed at time K +i (see gure 4 for the corresp onding schedule).
X
1
3
1
2
4
c
a
d
b
X
2
X
3
Figure 4: A corresp onding schedule
We mustprove thatthe delays constraints arefullled : let us consider thetask y=(L
1
(i)). Forevery taskx 2 1
(y) is asso ciated e x 2E L (i). So, x 2 S i j=1 X i
and is completed at time K +i. Since y is p erformed at timet=D jYj+i 1,we get:
t (K+i)=D jYj+i 1 (K+i)=d
So,thedelaysare fullled.
Theorem 3.2. Let us consider an instance I of MIN ADJACENT SET LINEAR ORDERING given by a graph G=(V;E)and an integerK >0. A necessaryand suÆcient condition for the existenceof a solution isthat
jEjK+jVj 1
Proof. The conditionis necessary: since thegraphG isconnectedwithout
lo ops, every linear ordering L veries E L
(n 1)=E. So, if L veries the condition,we getthe condition ofthetheorem.
The condition is suÆcient : let us consider a linear ordering L and a
family ofgraph G i
,i=0;:::;ndened such that,
G 0
=G,
8i = 1;:::;n, we cho ose a vertex u in the subgraph G i 1 = (V fL 1 (1);:::;L 1
(i 1)g;E)with aminimum degree in G i 1 and we setL(u)=i. G n =;. We note E i the edges of G i
. Noticethat,8i=1;:::;n, the twosets E L
(i) and E
i
area partitionof E.
We prove by contradiction that the linear ordering L is a solution to
L
in G is greater than or equal to K +2. So, 2jEj jVj(K+2). By
hyp othesis, we get2K+2jVj 2KjVj+2jVj, soK(2 jVj)2.
Since K > 0, we get that jVj < 2, so jVj = 1. In this case, we get
jE L
(1)j=jEj=0,which contradictsjEjK+2.
Now,letussupp osethat,fori<n 2,8j2f1;:::;ig,jE L (j)jK+j and that jE L (i+1)j(i+1)+K+1. Forevery vertexu 2G i , we setd G i
(u) thedegree ofu in G i
.
Thetotalnumb er of edgesveriesjEj=jE L (i+1)j+jE i+1 j. 1. Byhyp othesis, jE L (i+1)j(i+1)+K+1.
2. Bydenitionofthesequences G i ,jE i+1 j=jE i j d G i (L 1 (i+1)). Sinceu=L 1 (i+1)is thevertexofG i
witha minimum degree,
thenumb er ofarcs ofG i veries 2jE i j(n i)d G i (L 1 (i+1)) So, jE i+1 j 1 2 (n i)d Gi (L 1 (i+1)) d Gi (L 1 (i+1)) We show that d G i (L 1
(i+1)) 2. Indeed, let us denote by
e(k )=fL 1 (i+1);L 1 (k )g anedgeofGadjacenttoL 1 (i+1). Then,we geteasily that E
L (i+1) E L (i)=fe(k )2G i g,so d G i (L 1 (i + 1))=jE L (i + 1)j jE L (i)j(i + 1) + K+ 1 (K+ i)=2 We deducethat jE i+1 j n i 2 2 d G i (L 1 (i+1))n i 2
So,thetotalnumb er ofedges of Gveries :
jEj=jE L
(i+1)j+jE i+1
j(i+1)+K+1+n i 2=jVj+K
whichcontradictsthehyp othesis of thetheorem.
Noticethatthis pro ofisconstructive: if thecondition ofthetheorem is
fullled, one can easily implements agreedy p olynomial algorithm tobuild alinear ordering.
Corollary 3.3. is polynomial.
Inthis section,we considertheanalysis ofthep erformancesof two approx-imation algorithms.
The rst one is the classical Grahamlist scheduling algorithm [12]. At eachtimet,aschedulabletaskischosentob ep erformedwithoutanypriority
rule. For the bipartite graph G = (X[Y;E), it consists on p erforming tasks from X in any order and tasks from Y as so on asp ossible. Several
authorsshowthatthep erformanceratioofthisalgorithmisupp erb ounded asymptotically by2 [15, 20,19]. Weprove here that this b ound isreached forbipartitegraphs:
Theorem 4.1. The performance ratio of a list scheduling for a bipartite
graph tends asymptotically to 2.
Proof. Letus consider avalue d>0 and a bipartitegraph G=(X[Y;E)
withX =fa 1
;:::;a d
g[fbg, Y =fcg and E =f(b;c)g. In the worstcase fortheGrahamlistscheduling algorithm,tasks fa
1 ;:::;a
d
gare p erformed
rst. We getthena schedule of length l 1
=2d+2.
Now,we can geta schedule without idle slotsif wep erformbrst. The length ofthis second schedule is thenl
2
=d+2. The p erformanceratiois thenb oundedby: r=
2d+2 d+2 =2 2 d+2 ! d!1 2.
We present nowa slightly b etter approximation algorithm : let us sup-p ose that G = (X [Y;E) with jXj = n, jYj = m and n m. In the
opp osite,wemo dify theorientationof theedgesand weconsider thegraph G
0
=(Y [X ;E 0
). We can get a feasible schedule for G by considering the
inverse order ofa schedule forG 0
. Let us consider theset X
1
of tasks from X with a strictly p ositive out-degree (i.e.,X
1
is theset of X with atleast one successorin Y). The idea istoapplya listscheduling algorithmwhich p erformstasksfromX
1 b efore thosefrom X 2 =X X 1 . We denote by C opt (resp. C H
) the makespan of an optimal schedule (resp. a schedule obtained using this algorithm). We set jX
i j=n i ;i=1;2 and p = max(0;d+1 n 2
m). We prove the following upp er b ound on C opt : Lemma 4.2. C opt n+m+p.
Proof. The last task of X 1
is p erformed at time t n 1
and has at least one successor in Y, so C opt n 1 +d+1. Now, if p = d+1 n 2 m,
2 1 Otherwise, p=0and wegetobviously C
opt
n+m.
Theorem 4.3. Theperformance ratio of thisalgorithm isbounded by 3 2 .
Proof. We denote by I the idle slots of theschedule obtainedby our algo-rithm. We get,usingthe previous lemma :
C H
=n+m+jIjC opt
+(jIj p)
1. IfjIjp,weget thetheorem.
2. LetusassumenowthatjIj>p. WebuildasubsetI p
Ibyremoving
fromI thepthrstidleslotsinourschedule. Letb eanelementk2I p andt(k )the timeof thisidle slot.
Clearly, by denition of I p
, t(k )p+n. Moreover, there is at least one taskfrom y 2Y p erformedafter t(k ) such thaty is notreadyat timet(k ), sot(k )n 1 +d. We get jIj p=jI p jn 1 +d (p+n) Then, jIj p=jI p jd n 2 max(0;d+1 n 2 m) We deducethat jI p jmin(d n 2 ;m 1) So,jI p jjYj.
Now,the inequalityb etween C H and C opt b ecomes : C H C opt +jI p jC opt +jYj
Since jYjjXj, we get that jYj 1 2 (jXj+jYj) 1 2 C opt and we get thetheorem.
We can prove that the b ound 3 2
is asymptotically tight : indeed, let us consideranintegern>0 andthebipartitegraphG=(X[Y;E)withX = fx 1 ;:::;x n g, Y =fy 1 ;:::;y n
g and the arcs E =f(x i
;y j
);1jing.
We setd=n 1. NotethatjXj=n=jYj. If we p erform task from X such that t(x
i
) = i 1;i = 1;:::;n, then tasks from Y can'tb e p erformed b eforen+d 1. So, we get a makespan
L 1
=3n 2.
Now,ifwep erformtaskfromfromXsuchthatt(x i
)=n i;i=1;:::;n, thenwegeta schedule without idle slotswith makespanL
2 =2n.
Several new questionsarisefrom theresults presentedhere:
In order to study the b orderline b etween NP-complete and p olyno-mialproblems, thecomplexity of the problem with a bipartite graph wherethedegreeofverticesfromXdo esnotexceed 3isaninteresting
problem.
Theexistenceofb etterapproximationalgorithmsisalsoaninteresting
question.
References
[1] E.Balas,J.K.Lenstra,andA.Vazacop oulos.Theonemachineproblem withdelayedprecedence constraintsand itsuseinjob-shop scheduling. Management Science,41:94{109,1995.
[2] E.Bampis. Thecomplexityofshortschedules foruetbipartitegraphs. RAIROOperations Research,33:367 {370,1999.
[3] M. Bartusch, R. H. Mohring, and F. J. Radermacher. Scheduling project networks with resource constraints and time windows. Annals
of Operations Research,16:201{ 240,1988.
[4] D. Bernstein. An improved approximation algorithm for scheduling
pip elined machines. In Int. Conf. on Parallel Processing, volume 1, pages 430{ 433,1988.
[5] D. Bernstein and I. Gertner. Scheduling expressions on a pip elined pro cessor with a maximum delay of one cycle. ACMTransactions on ProgrammingLanguagesand Systems,11:57{ 66,1989.
[6] J. Bruno, J. W. Jones, and K. So. Deterministic scheduling with pip elined pro cessors. IEEE Transactions on Computers, C-29:308 { 316,1980.
[7] E.G.ComanandR.L.Graham. Optimalscheduling fortwopro cessor systems. Acta Informatica,1:200{213,1972.
[8] S. Dauzere-Peres and J.-B. Lasserre. A mo died shifting b ottleneck pro cedure forjob{shopscheduling. Int.J.ProductionResearch,31:923
scheduling withdelayconstraints. Rapp ortInterneMIT.
[10] M.R.GareyandD.S.Johnson.ComputersandIntractability: AGuide to the Theory of NP{Completeness. Freeman,San Francisco,1979.
[11] R. L. Graham. Bounds for certain multipro cessing anomalies. Bell System Tech.J., 45:1563{ 1581,1966.
[12] R. L. Graham. Bounds on the p erformance of scheduling algorithms.
In E. G. Coman,editor, Computer and Job-shop Scheduling Theory. JohnWiley Ltd.,1976.
[13] W.Herro elenand E.Demeulemeester. Recentadvancesin branch-and-b oundpro ceduresforresource-constrainedprojectschedulingproblems. InP.Chretienne,E.G.ComanJr.,J.K.Lenstra,andZ.Liu, editors,
Scheduling Theory and its Applications, chapter 12, pages 259 { 276. JohnWiley &Sons, 1995.
[14] T.C.Hu. Parallel sequencing andassemblylines problems. Operations Research,9:841{ 848,1961.
[15] E. Lawler, J. K. Lenstra, C. Martel, B. Simons, and L. Sto ckmeyer. Pip eline scheduling: Asurvey.TechnicalRep ortRJ5738(57717),IBM
ResearchDivision, San Jose,California, 1987.
[16] E.L.Lawler,J.K.Lenstra,A.H.G.Rinno oyKan,and D.B.Shmoys.
Sequencing and scheduling: Algorithms and complexity. In S. C. Graves, A. H. G. Rinno oy Kan, and P. H. Zipkin, editors, Logistics of ProductionandInventory,volume 4ofHandbooksin Operations Re-search and Management Science, chapter 9, pages 445 { 522. North{
Holland, Amsterdam, TheNetherlands, 1993.
[17] T. Lengauer. Black-white p ebbles and graph separation. Acta Infor-matica, 16:465{ 475,1981.
[18] J. Y.{T.Leung, 0. Vornb erger, and J. Wittho. On some variants of the bandwidth minimization problem. SIAM J. Computing, 13:650 { 667,1984.
[19] A.Munier,M.Queyranne,andA.S.Schulz.Approximationb oundsfor a general class of precedence constrained parallel machine scheduling
Conference.
[20] K. W.Palem and B. Simons. Scheduling time critical instructions on risc machines. In Proceedingsof the 17th Annual Symposium on
Prin-ciplesof Programming Languages,pages 270{280,1990.
[21] C.H Papadimitriou and M. Yannakakis. Schedulin interval-ordered
tasks. SIAM Journal onComputing, 8(3),1979.
[22] P.Schuurman. Afully p olynomial approximationschemefora schedul-ing problem with intree-typ e precedence delays. Operations Research Letters,23:9{11, 1998.
[23] E. D. Wikum, D. C. Llewellyn, and G. L. Nemhauser. One{machine generalized precedence constrained scheduling problems. Operations