1
Cha pt er
Appr o xima tion f or S c he d ulin g on
P ar allel Ma c hine s with F ix ed J obs
or U na v ailabilit y P eriods
Lil ia naGr ig or i u
Abst ract
W e sur v e y re sul ts that addr ess the pr oblem of non-pr eem ptiv e sched uling
on parallel ma chine s with fix ed jobs or una vailability periods with the purpose
o f minimizing the ma xim um comp letion time. W e consid er both ide ntical an d
unif orm proc essor s, an d also addr ess the special case of sch edulin g on nonsimul-
tane ous parallel ma chine s, w hic h ma y star t proce ssing a t diff ere nt times . T he
discussed r esults inc lud e polynomial-time appro ximation alg orithms that achie v e
the best possible w orst -case appr oxima tion bound of 1.5in the class of polynomial
alg orithms unless P=NP for sched uling on iden tical proc essor s with at most one
fix ed job on eac h mac hine and on unif orm mac hines with at most one fixe d job on
ea ch mac hine . T he prese nt ed heuristics ha v e similarities with the LP T alg orithm or
the MUL TIFIT alg or ith m and the y are fast and easy to imple ment. F or sche duling
on nonsimultaneous mac hines , experime nts sugge st that they w ould perf orm w ell
in practice . W e also inclu de re fe ren ces t o the re levan t w ork in this area that cont ains
mor e comp lex alg orithms . W e then discuss the main methods of argumen t used in
the appro ximation bound pr oof s f or the simp le heuris tics , and comme nt upon cur-
r ent challeng es in this area b y describin g aspec ts of r elat ed pra c tical pr oble ms from
the au tomo tiv e industry .
K eyw or ds: multiproc essor sche duling , a vailabilit y constrain ts , fix ed jobs ,
unif orm proc essor s, w orst -case appro ximation, non simultaneous mac hines ,
mak espan, maximum com pletion time , una vailabilit y
1. Intr odu c tion
The nece ssity to assign re source s suc h as mac hines to jobs that need t o be
perf ormed withou t inte r ruption, whe re the time r equir ed f or a machine t o per-
f orm a ce r tain job is known in ad vanc e, is a wid ely enc ount ered pr oble m. It can
occur f or e xample in prod uction plan ning or w hen assigning landin g and tak e-off
stripes to plane s in airpor ts . S ometimes these r esour ce s become una vailable f or
pr edet ermined periods of time , f or example beca use o f nece ssary maint enance .
Minimizing th e ma ximum com pletion time o f all tasks is ofte n consid ere d as a goal,
f or examp le such that the w ork ers w ho opera te the machine s can unde r take othe r
a ctivities af t er war d or go home early . As a conseq uen ce , the pro blem o f sched ul-
ing on multiple mac hines with pre define d una vailabilit y periods ( do wntimes) t o
minimize the maximum com pletion time , that is, the lat est comp letion time of a job
in a sched ule, has been consid ere d. A closel y rela te d proble m, of sche duling with

Sche du l in g Pro blem - N ew A ppl ica tions a nd T rends
2
fix ed jobs , w her e on eac h mac hine certain jobs ha v e to be perf ormed at pre define d
times , has also been consid er ed. The diff ere nce betw een these tw o prob lems is
in the meaning o f the objec ti v e f unction: w hen sched uling with fixe d jobs, th e
maximum completion time of the jobs must be at least the lat est comp letion time of
a fixe d job , w her eas the maximum com pletion time w hen sche duling in the pres-
en ce of una vaila bility periods can occur bef ore the end o f an una vailabilit y period.
W e conside r th e s t ati c nonr esumable v ar iant of the prob lem of sch eduling with
una vailabilit y periods , w here the do wntimes are kno wn f or ea ch mac hine bef ore the
sche dule needs to be mad e, and w her e jobs that star t ex ecutin g bef ore a do wntime
canno t re sume ex ecution af t er it.
I n these prob lems , the job ex ecution times are usually assumed t o be giv en as an
int eg er number of com puting units suc h as cloc k cy cles or of othe r suit able units
su ch as time units . Similarly , the star ttimes and endtim es of una vaila bility periods
or of fix ed jobs are assumed t o be giv en as int ege r multiples o f ad equat ely chosen
time units . W e not e that an y prob lem with rational num bers as job dura tions and
star ttime s and endtime s of do wntime s or of fix ed jobs can be transf ormed in t o an
eq uivale nt pro blem w here these entitie s are int ege rs b y multiplying the m with an
a de qua te fact or , thus there is arg uabl y no loss of g eneralit y in this assumption w hen
con side r in g the repr esen tation of an y pra c tical pr oblem.
Both the pr oble m of multiproc essor sched uling on fixe d jobs and that of multi-
pr ocessor sched uling with una vailabilit y periods ar e s tr ong ly NP -har d as the y are
mor e ge neral than the stron gl y NP -har d multiproc essor sched uling pr oblem (MSP),
w hich has no do wntime s or fixe d jobs.
F or sched uling with do wntime s, it is NP -har d to find a sche dule that ends within
a giv en constan t multiple of an optimal sched ule ev en w hen sched uling on identi cal
ma chine s with at most one do wntime on eac h mac hine . W e discuss this in more detail
in S ection 4.2. T o obtain appr oxima tion re sul ts f or sched uling with una vailabilit y
periods in this cont ext, assumptions about the do wntime s w ere mad e suc h as the
assum ption that only a fra ction of the proce ssors can be una vailab le at the same time
[ 1 , 2 ], or b y comp aring the ge nerat ed sched ule t o the lat est among th e end of an opti -
mal sche dule or th e lat est end o f a do wntime , thus essen ti ally consid ering sc hedulin g
with fixe d jobs [ 3 , 4 ].
T o describe the perf ormanc e of an appr oxima tion alg orithm, w e use the notion
o f a wo rst-cas e a ppro xi mat ion bo und . I n this w ork, w e call w orst -case appro xima-
tion bound of an alg orithm A w hen applied t o a sched uling pr oblem SP the larg est
ra tio betw een the maximum com pletion time of a sched ule prod uc ed b y A and the
maximum completion time of an optimal sche dule f or a pro blem instan ce of S P .
F or the pro blem of multiproc essor sched uling with fix ed jobs to minimize the
maximum completion time , ev en in the case w here ther e is at most one fix ed job on
ea ch mac hine , it has been sho wn in [ 5] that no polynomial alg or ith m can ac hiev e a
w orst -case appro ximation bound that is less than 1.5 unles s P = NP . Sharbr odt etal.
[ 5] also giv e a polynomial-time appr oxima tion schem e (P T AS) f or sched uling on a
con stant numbe r o f unif orm proc es sors with fix ed jobs . P ol y nomial-time appr oxi-
mation alg orithms f or this pr oble m that achie v e the w orst -case appr oxima tion
bound o f 1.5 w ere giv en f or the ge neral pr oblem in [ 6]. F or the case w her e ther e is
at most one fixe d job on eac h mac hine, signifi canl ty simp ler heuris tics with low er
time comp lexities re semblin g the larg est pr oce ssing time alg orithm (LP T) [7 ] fo r
ide ntical pr ocessor s and the MUL T IFIT alg orithm [8 ] f or unif orm proc es sors also
a chie v e this bound [3 , 4 ].
The case w her e all do wntimes ar e at the beginning o f the proc essing tim e on all
ma chine s is called sch ed uling with nonsimultan eous mac hine a vaila ble times , as the
ma chine s star t proce ssing jobs nonsim ul tane ously . F or this proble m, polynomial-
time algorith ms with constan t w orst case appro ximation bounds exist.

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A ppro xi mati on f or Sc hedu li ng on P ar all el Mac hi nes wit h Fix ed J ob s or U na va ilability P er iods
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F or sched uling on iden tical nonsimultane ous parallel ma chine s, MUL T IFIT
a chie v es a tig ht w orst -case appr oxima tion bound of 2 4/19 (~1.2 632) [ 9 ] and anoth er
alg orithm achie v es a bound of 5/ 4 [ 10 ], w hile in the case o f sched uling on unif orm
nonsim ul tane ous paralle l machine s , a MUL T IFIT v ariant has a w orst -case appro xi-
mation bound o f 1.382 [ 11 ], w hich w as sho wn b y ge neralizin g the bound obtaine d
f or MUL TIFI T w hen sche duling on unif orm proc essor s in [ 12 ]. Experimen tal re sul ts
sugg est that f or sched uling on nonsim ul tane ous unif orm machine s , the MUL T IFIT
variant from [ 11 ] is ad equa t e f or use in pra c tic e, as w e discuss in S ection [ 4 ].
I n S ection 2, w e de scr ibe the w a ys in w hich the cont ent of this w ork can be used.
I n S ection 3 , w e intr odu ce the algorith ms LP T and MUL T IFIT .I n S ection 4, w e
con side r sc hedulin g with una vaila bility periods , w hile first f ocussing on sched ul-
ing with nonsimultane ous machin e a vaila ble times in S ection 4 . 1, and on the more
g eneral case w her e do wntime s do not ha v e to occur at the beginning o f the sched ule
in S ection 4.2. I n S ection 5, w e pre sent re sul ts on sche duling with fix ed jobs . S ection
6 con tains the description of main tec hniq ue s used in the w orst -case appro ximation
bound pr oof s and S ection 7 cont ains con cludin g remar ks and a discussion of some
o f the challen ge s in this area.
2. Con t ribution s of this w or k
W e next pre sent w a y s to use the cont ent of this w ork.
2. 1 A deeper under standing
This w ork aims to pro vide a deepe r unde rstandin g of multiple rela te d proble ms
that in v olv e sche duling on paralle l mac hines with fix ed jobs or una vailabilit y periods
t o minimize the ma ximum com pletion time . W e explain w h y multiproc essor sched ul -
ing with at most one una vailabilit y period on eac h mac hine does not ha v e a polyno -
mial-time appr oxima tion alg orithm with a constan t w orst -case appr oxima tion bound
unles s P=NP , w hich is the main reason w h y results on this topic are har d to obtain.
Fur therm ore , w e obser v e that most re sults in this area in v olv e variant s of LP T
and MUL T IFIT , and com ment on the othe r r esults obtaine d. W e also hope that this
w ork will incr ease a war ene ss of the se result s and of ho w they re lat e to eac h other .
2. 2 Pra ct ic al use of the heuristic s
The heuristics pre sent ed and r ef er ence d in this w ork can be used dir ec tl y in
pra c tic e or for re searc h purposes t o solv e the pro blems the y addr ess . T he heuristics
based on LP T and MUL TIFI T are fast and easy to imple ment and some o f them
ha v e best possible w orst -case appr oxima tion bounds in the class of pol ynomial
alg orithms unless P=NP for the pr oble ms the y addr ess . I n addition to w orst -case
appr oxima tion re sul t s, this w ork also highlig ht s f or some cases experime ntal
insig hts int o how the heuris tics w ould perf orm in practice based on how the y
perf orm f or rand omly g enera te d instan ces . As expected, th ey perf orm mu ch better
f or suc h instanc es than in the w orst case. Also , f or some cases , re f er enc es to more
com plex methods ar e pro vided in case the user pref ers t o use those .
2.3 Pr oof t echni ques
This w ork pre sent s the main proo f t ech niqu es used t o obtain w orst -case appro xi -
mation bounds for LP T - and MUL TIFI T -lik e heuristics w hen the aim is to minimize
the ma ximum com pletion time . T hus , the int er este d rea der is pro vided with a

Sche du l in g Pro blem - N ew A ppl ica tions a nd T rends
4
con cise description of the too ls that can be used to de v elop suc h proo fs , and he or she
ma y not ha v e t o rea d hundreds o f pag es in or der t o become a war e of all o f them or
w ork with an exper t in the ar ea w hen de v eloping su ch a proo f . E v en f or people with
expe r tise in the area, one or mor e of the ideas pre sent ed ma y be new and help f ul.
3. The alg or ithms LPT and MUL TIFIT
The alg orithms LP T and MUL TIFI T ar e among the most st u died appro ximation
alg orithms f or multiproc es sor sche dulin g with or without una vaila bility periods
or fix ed jobs . In this sec tion, w e describe the basic v er sions of these alg orithms f or
MSP , w hich can be stat ed as f ollo w s: giv en a set of m ma chines P and n jobs J find a
non-pr eem ptiv e sched ule that ends at the earliest possible time. A n on-pre emptime
sche dule is an assignme nt of jobs to proc essor s , t ogethe r with an ord er in w hich the
jobs on eac h proc essor are pr ocesse d.
The alg orithm LP T [7 ] w orks as f ollo w s:
The alg orithm MUL T IFIT w as first intr odu ced b y Coffmann Gar e y and J ohn son
in 1978 [ 8]. I t uses binar y searc h f or th e end of its r esulting sc hedule and r ecei v es as
inpu t an accur acy ε with w hich it det ermines this sche dule end. I n eac h iter ation it
assigns a dea dline and att em pts t o creat e a sched ule that cont ains all tasks that ends
at or bef or e that dea dline b y using the bin pa cking al gorith m first fit de cr easing
(FFD ). If a f easible sche dule is creat ed, it decr eases the dea dline and othe r wise it
incr eases the dea dline . T his proce ss is repeat ed until the diff er enc e betw een the cur-
r ent deadlin e and the pre viously consid ered d eadline is les s than ε . Mor e f ormally ,
the algorith m is de scr ibed as Alg orithm 2.
The MUL TIFI T algorith m re sul ts in a sche dule with a ma ximum com pletion
time that is within ε of the maximum com pletion time of the sche dule that w ould
r esult if the binary searc h f or the dea dline w ould be contin ued.
An example of a LP T -sc hedule and a MUL T IFIT sche dule f or the same pro blem
instan ce ar e prese nt ed in Fi g ur es 1 and 2 r espectiv el y .
F ig ur e 1 .
A LPT sc hedu le . T he jo bs a re n umber ed ac cor di ng to t he or der in w hic h t hey a re cons ider ed. A t s t art, w he n al l
pr oces sors a re av aila ble a t the s ame ti me , they a re c onsidere d in t he or der p1, p 2, p 3in t his exa mp le .
Algorithm 1 T he larg est proc essing time algorithm (LP T)
1: Ord er jo bs in nonincreasin g order of their proce ssing time .
2: I n this orde r assign each job at the ear liest possible time allow ed b y th e sc hedule that exists whe n the job is
assigned.

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The MUL TIFI T algorith m te nds to prod uc e more balanc ed sched ules than LP T ,
and, as a conseq uen ce it te nds to perf orm bett er w hen the aim is t o minimize the
maximum completion time . I t also has a high er time comp lexity , as it tr ie s to make a
sche dule about log 2
(
ub − lb
)
times , w her eas the LP T alg orithm only doe s that onc e.
When the instan ces are prohibiti v ely big and the sched ule needs to be made in a
short time, it ma y be indicat ed to use LP T or anothe r list sched uling alg orithm to
sche dule the jobs . T his is beca use in practical situations , there ma y not be enoug h
time to go thr ough the lis t of jobs more than onc e w hile sched uling thousands of
jobs and making sur e that all req uired c onstraint s are obe y ed b y the sched ule.
The reason w h y suc h big sched ules are mad e is that the com panies aim to estima te
de liv ery times for their ord ers .
3. 1 Time c omplexity of MUL TIFIT
I f the par amet er ε of MUL T IFIT is ad equa te ly set, f or example as a comput er
cloc k cy cle , the alg orithm returns the best possible MUL TIFI T sche dule , that is,
running it f ur the r w ould not re sul t in a bette r sched ule, as w as comm ent ed
upon in [ 4 ].
The binary searc h f or the deadline within the MUL T IFIT alg orithm happens
within log 2
[

(
ub − lb
)
/ ε
]
time , w hich is at most log 2
(
ub / ε
)
, w hich is the size of ub i n
binary , assuming that the number s f or the upper bound and the lo w er bound do not
chan ge their repr esen tation durin g the ex ecution of the algorith m and that they allo w
within their repr esent ation for an ac curacy of ε . I n S ection 1, w e mentione d that the
job dura tions are usually giv en as int ege r multiple s of an ad equa te ly chose n (time)
F ig ur e 2 .
A MUL T IFIT sc hed ule toget her with a p ossib le de adli ne . T he jo bs a re n umber ed accor di ng to t he or der i n whic h
t hey a re cons ider ed. The proc es sors a re c onsidere d in t he or der p1, p 2, p 3 whe n gener ati ng the s che dule .
Algorithm 2 T he algorith m MUL TIFI T
1: Ord er th e jobs in non-increasin g order of their d uration.
2: Assign upper bound (ub ) and low er bound (lb ) for the e nd of sched ule; (for example , lb = sum of job
dur ations/ number of proc essors , ub = sum of job durations).
3: Assign b = (ub + lb ) / 2 as deadline .
4: FFD: Assign tasks in non-increasin g order on the first proce ssor on whi ch they fit w hile r especting the
dea dline (the proc essor s are consider ed in each iter ation in the same or der).
5: If all tasks ar e succ essfully schedule d decrease the u pper bound: ub = b .
6: Else increase the low er bound: lb = b .
7: If ub − lb ≥ ε loop ba ck to Step3 .

Sche du l in g Pro blem - N ew A ppl ica tions a nd T rends
6
unit; ther ef ore , the end of an y sched ule is an int ege r , and thus , ther e is no point in
making ε les s than 1, in w hich case the MUL T IFIT loop is repea te d log
2
(
ub
)
times .
As a conseq uen ce , the numbe r o f times the MUL TIFI T loop is called is polyno-
mial in the size of the input, as an y reasona ble upper bound is at most the sum o f
the proc essing tim es of all jobs , w hic h can be repr esen ted within at most the t otal
num ber of bits used to r epr ese nt all jobs . In [ 4], G rigoriu and Frie sen also comme nt
that if the upper bound is 2y ears , the lo w er bound is 0 and the dea dline is det er-
mined with an ac curacy of 1 0 − 13 s, the MUL T IFIT loop is called at most 4 0 times .
The time com plexit y of MUL TIFI T is thus O
(
n log n + nm log 2
(
ub
)

)
, as the jobs
need t o be sor te d acc ordin g to their ex ecution times in a non-incr easing or der in
St ep ( 1), and as in eac h ite ration of the MUL T IFIT loop , the algorith m look s f or
ea ch job for the first proc essor on w hich it fits; thus , it will ha v e t o look a t most at m
pr ocessor s . R ecall that n is the numbe r o f jobs in the conside red pr oblem instan ce .
4. Sch eduling w ith una vaila bilit y per i ods
I n this section, w e first pr esent re sul t s for the case w here all una vailabilit y
periods are at the beginning o f the sche dule . T hen, w e pre sent re sul ts f or the mor e
g eneral case w her e the una vailabilit y periods can occur an y w here in the sched ule.
4. 1 Sche duling w ith n onsimultane ous ma chine a vaila ble times
This sec tion addr esse s the case w her e the pr ocessor s ma y ha v e una vailabilit y
periods at the star t of their proc es sing time . T his sit uation is mor e ge neral than the
multipr ocessor sched uling pr oblem (w here ther e are no fixe d jobs or do wntimes)
and les s ge neral than the prob lems of sched uling with fixe d jobs or with una vail-
abilit y periods. As the less ge neral multipr oces sor sc hed uling pr oblem is NP -har d,
so are the prob lems of sched uling on iden tical mac hines with nonsim ul tane ous
ma chine a vailab le times (NMSP : nonsim ul tane ous multiproc essor sche duling
pr oblem) and sc hedulin g on unif orm proce ssors with nonsimultan eous mac hine
a vailab le times (UNMSP ) w hen minimizing the ma ximum com pletion time . Due
t o the NP - hardne ss of these pr oble ms, po lynomial-time appro ximation algorith ms
lik e LP T and MUL TIFI T and their variant s ha v e been st u died f or th eir so lution. As
bef ore , w e will c ontinu e to den ote with the number of proc essor s in the pro blem
instan ce being c onsider ed with m .
4.1.1 S c hed ul in g on ide nt ical nons im ult a neous pr oce ssor s
F or NMSP , w orst -case appro ximation bounds f or LP T of 3 / 2 − 1 /
(
2 m
)
and f or a
modified v ersion of LP T (MLP T) of 4/ 3 ha v e been obtained b y Lee [ 13 ]. T he bound
obtain ed b y L ee in [ 13 ] w as impr o v ed u pon by K eller er in [ 10 ], w her e a dual appr o x -
imation alg orithm with a w orst -case appr oxima tion bound o f 5/ 4 was pr esent ed.
When MUL T IFIT is used f or MSP , a dea dline re sults in periods of eq ual dura tion
in w hich jobs can be sche duled on eac h proc es sor ; thus the sched ules resultin g from
using an y ord ering o f pr oce ssors in step ( 4) o f MUL T IFIT ha v e the same ma xim um
com pletion time . When con side r in g NMSP , thus allowin g for nonsimultane ous
ma chine s, th e ord er in w hich proc essor s are consid ered be comes r ele vant, as the
period in w hich jobs can be ex ecut ed on eac h proc essor corre sponding t o a dea dline
depe nds on the time the proc essor bec omes a vailab le. MUL TIFI T variant s that
a ddr es s such situations usually orde r the proc essor s in non- de creasing or de r o f their
periods in w hich jobs can be sched uled. Thus, in this case , the ord ering is in non-
incr easing or der of the times at w hic h the proc essor s become a vailable .

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A ppro xi mati on f or Sc hedu li ng on P ar all el Mac hi nes wit h Fix ed J ob s or U na va ilability P er iods
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A bound o f 9 / 7 (about 1.28 5 7 ) w as obtained f or MUL T IFIT b y Chang and
H wang [ 14 ]. I n [ 10 ], K eller er giv es a proble m instanc e of NMSP f or w hich the
appr oxima tion fact or of its MUL T IFIT sche dule is 2 4/19 ( about 1.2632). M ore
r ecen tly , 2 4/19 was sho wn to be the exa c t w orst -case appro ximation bound w hen
using MUL TIFI T f or NMSP b y H wan g and Lim [ 9 ]. B y com parison, a tig ht w orst -
case appro ximation bound o f 13/11 (abou t 1. 18 182) was sho wn b y Y ue [ 15 ] fo r
MUTLIFI T w hen applie d to MSP .
4.1.2 S c hed ul in g on un if or m nonsi mu lt aneo us pr oces sors
F or the unif orm multiproc essor sched uling pr oblem with simultaneous
ma chine a vailab le times (UMSP ), that is , w her e proc essor s ex ecut e jobs at dif -
f er ent speeds , the amount of jobs that fit on a proc essor corre sponding t o a giv en
MUL T IFIT dea dline depends on the speed o f that proc essor . U sually , the slo w est
pr ocessor is consid ere d to ha v e a speed o f 1, and f or ea ch job j , the time it w ould
tak e to pr ocess it on this proce ssor l j is giv en. T hus , on a mac hine with speed 5 , a
j o b j n e e d s a t i m e o f l j / 5 to be proce ssed. As a conseq uen ce , the ord ering in w hich
the proc essor s are conside red in St ep ( 4 ) of MUL T IFIT is in most cases re levan t
t o the ma xim um comp letion time of the resultin g sched ule. MUL TIFI T for UMSP
or der s proce ssors in each it eration bef ore its St ep ( 4) in non-decr easing or der
o f the dur ation of the proc essin g time on that proc essor times the speed of tha t
pr ocessor [ 12 , 16 ].
F or UMSP , appro ximation bounds of 1. 4 and 1. 382 w ere obtain ed f or MUL T IFIT
b y F riesen and L an gst on [ 16 ] and b y Chen [ 12 ] respe c ti v el y . In [ 17 ], Burkar d and
H e deri v e a w orst -case appro ximation bound of
√

_
6 / 2 (abou t 1.2 2 4 7 ) o f MUL TIFI T
f or UMSP with at most tw o proc essor s, an d show a bett er bound of
(

√

_
2 + 1
)
/ 2
(a bout 1.2 0 7 1) f or the case w her e MUL T IFIT is combine d with LP T as an incum -
ben t algorith m.
In [ 11 ], Grigoriu and F riesen sho w that bounds that apply to the MUL T IFIT
variant s from pre vious w ork suc h as [ 12 , 16 , 17 ] w her e sched uling on tw o unif orm
pr ocessor s is consid er ed also apply t o a sligh tly diffe ren t proposed variant of
MUL T IFIT f or UNMSP , LMUL T IFIT , w hich w as first proposed in [ 4 ] i n a m o r e ge n -
eral f orm. The diff ere nc e betw een the MUL T IFIT variant s from [ 12 , 16 , 17 ] on the
one hand and LMUL T IFIT on the other hand is that in the latt er , the choic e of the
initial upper and low er bounds is not giv en expli citly within the alg orithm, and thus
the w orst -case appro ximation bound proo fs are more g eneral, as the y w ork f or an y
initial choic es of upper and lo w er bounds f or the dura tion of the re sul ting sc hedule .
A first step in the proo fs that the bounds hold in the more g eneral case , w here ther e
ar e uniform nonsim ul tane ous paralle l mac hines , w as t o show that LMUL T IFIT
obe y s the bounds of the earlier MUL TIFI T v ari ant s in the simultane ous machin es
case for the instan ces consid ere d in those w orks .
U sing LP T f or UNMSP has been consid ere d in [ 18 ], whe re w orst -case appro xi-
mation bound o f 5/3 w as sho wn in the gen eral case , as w ell as a better bound f or the
case w here ther e are only tw o ma chine s .
F or the case w her e the number of mac hines is constan t, a P T AS exists f or
UNMSP [ 11 ], w hich was de r i v ed from a P T AS f or sched uling on a constan t num-
ber of unif orm proc essor s with fixe d jobs from [5]. As the objectiv e f un c tion f or
sche duling with fix ed jobs that are at the beginning o f the sched ule and sched uling
with una vailabilit y periods that are at the beginnin g of the sched ule diff er , the
P T AS from [5] can not be used unalte red t o addr ess UNMSP .T o addr ess UNMSP ,
the P T AS from [5] is first run f or the transf ormed pr oblem instanc e w her e the
una vailabilit y periods becom e fixe d jobs , and then f or all prob lem instanc es re sul t -
ing fr om suc ces siv ely re movin g the mac hine with the lat est end of a fix ed job from

Sche du l in g Pro blem - N ew A ppl ica tions a nd T rends
8
the transf ormed instan ce [ 11 ]. T his acc ounts f or the cases w her e a number betw een
1 and m − 1 o f pr ocessor s star t proc es sing afte r th e end of the optimal sched ule .
In [ 19 ], a low er bound is deri v ed f or the end of an optimal sched ule of an
UNMSP instan ce , and using tha t bound appro ximation factor s f or LMUL T IFIT
sche dules of ran doml y ge nera t ed instanc es are det ermined. T he reasonab ly exte n-
si v e experime nts describe d in [ 19 ] sugge st that LMUL T IFIT is a g ood option for
solvin g UNMSP in practic e , not only beca use o f being fast and easy to implem ent,
but also beca use it has v ery good appr o ximation fact ors (les s than 1.0 3) f or the
g enera te d instanc es with an a v era ge o f at leas t fiv e jobs f or each mac hine . In ord er to
obtain the appro ximation fa c t ors , a low er bound f or th e end of the optimal sched ule
that can be calculat ed dir ec tl y f r om the pro blem instanc e was used.
4. 2 Multipr oc essor sc hed uli ng w ith a vailability con st raints
I n this section, w e conside r the multiproc essor sched uling pro blem w her e
do wntime s can occur at an y time during the sche duling horiz on.
Sur v e ys with f ocus on sched uling with a vailabilit y constrain ts are gi v en in
[ 20 – 24 ]. Be side s the make span, the a uthors of these w orks sur v ey w ork on v arious
othe r o bjec ti v e f unctions suc h as t otal com pletion time, and also addr es s additional
variant s of the prob lem, su ch as its r esuma ble v er sion.
U nless assum ptions about the una vailabilit y periods are mad e or unless P = NP ,
ther e is no polynomial-time appro ximation algorith m with a constan t w orst -case
appr oxima tion bound f or the pro blem of sched uling with una vailabilit y periods t o
minimize the maximum com pletion time , since ther e is a pol y nomial-time r ed uction
from the NP -har d pro blem o f 3- Partition to the prob lem of finding a sche dule that has
a ma ximum com pletion time that is at most α times the end of an optimal sched ule f or
this proble m. W e next outline suc h a red uction. Let X be an instan ce of 3- Partition, that
is , a set of 3 m positiv e int eger s , giv en with the purpose of finding ou t w hether there is a
partition of these numbe rs int o m set s, su ch that the sum of the number s in eac h set is
the same f or all set s. Let S be the sum of all numbe rs in X . T he instanc e Y of sched uling
with una vailabilit y periods that w e build is giv en as f ollo w s: ther e are m proce ssors , each
o f w hich has an una vailabilit y period of dur ation
(
α + 1
)
S that s tarts at time S / m , and
the job dur ations in Y are the numbe rs in X . X is a y es-inst ance of 3- Par tition if and only
if in instanc e Y
, the optimal sched ule ends at time S / m . I n suc h a sit uation, an y appr oxi -
mation alg orithm with w orst -case appr oxima tion fact or α f or sched uling with a vail -
abilit y constrain ts will find a sched ule that ends at or be f ore time 𝛼 S / m w hich is less than
(
α + 1
)
S . T hus , the f ound sche dule must end bef ore or w hen the una vailabilit y periods
star t, a t time S / m . I n suc h a sched ule, th e sets of d urations of jobs on eac h proc essor
ar e a 3- Partition of X . T her ef ore , an y polynomial-time appro ximation alg orithm for
sche duling with una vailabilit y periods with a w orst -case appro ximation factor α can be
used t o solv e 3- P ar tition in polynomial time .
4.2.1 S c hed ul in g on ide nt ical m ac hi nes wit h un a v aila bility p er iods
F or re sumable sched uling , w her e the ex ecution of jobs ma y continu e af t er
a down time that int err u pted the m, but w her e jobs cannot be pre empt ed b y the
sche duling al gorithm, an d w here one mac hine does not shut do wn and all other
ma chine s shut do wn at most once , L ee sho w s that the make span of LP T is in the
w orst case
(
m + 1
)
/ 2 times the optimal make span [ 25 ].
In [ 1], H wan g and Chang mak e the assum ption that at most half of the mac hines
ar e una vailab le at an y time, an d sho w f or this sit uation that the w orst -case appro xi-
mation bound o f LP T is 2. In [ 3], it is sho wn that no polynomial alg orithm can

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ha v e a bette r boun d than 2 f or this pro blem unless P = NP . The re sul t f r om [ 1 ] is
g enerali zed in [ 2] w her e it is shown that if at most λ ∈
{
1, … , m − 1
}
machin es ma y be
una vailab le at an y time, LPT has a w orst -case appro ximation bound of
1 + 1
_

2 [ m /
(
m − λ
)
] , and that this bound is tigh t f or LPT .
4.2.2 S c hed ul in g on un if or m mac hi nes wit h u na va ilabil ity periods
In [ 26 ], sched uling with at most one una vailabilit y period on eac h mac hine
is conside red and e xact alg orithms are giv en f or small proble m instanc es . T he
a uthors con side r sep arat el y the case of ide ntical jobs , and also conside r tot al
com pletion time beside the mak espan as an objectiv e f unction. F or large r
instan ces , the y propose an LP T -lik e alg orithm, w hich assigns jobs in nonincr easing
or der of their proc essing tim e to the faste st mac hine on w hich the y w ould finish
being pr oce ssed at the earlie st time. They also do experime nts on a to tal numbe r of
68 g enera te d instanc es w her e error margins of at most 5 . 6% are observ ed. T he y do
not com par e their heuristic t o the pre viously proposed h e uristic from [ 4 ], w hich
w e discuss in S ection 5. 1.2, w hich w as also proposed f or this prob lem, ev en though
its w orst -case appr oxima tion bound w as sho wn f or th e objectiv e f un c tion o f
sche duling with fix ed jobs .
5. Sch eduling w ith fi x ed jo bs
The pro blem of sc hedulin g with fix ed jobs is giv en as a numbe r of proc essor s,
w her e each pr oce ssor ma y ha v e jobs that must be ex ecut ed durin g certain giv en
periods on it, tog ether with a number of othe r jobs w hich can be ex ecut ed b y an y
pr ocessor . As not ed in S ection 1, job dura tions or re quire d ex ecution times are
expr esse d f or examp le as a number of significant units suc h as cloc k cy cle s. F or
unif orm proc essor s this numbe r r epr ese nts the time need ed b y a job to be ex ecut ed
on the slow est proc essor . In case ther e are unif orm proc essor s, ea ch proc essor also
has a speed fa c t or , b y w hich the time need ed b y a job on the slow est proc essor is
di vide d in orde r t o obtain the time need ed f or the proc essor to ex ecut e the job .
As not ed bef ore , the pro blem o f sched uling with fix ed jobs diff ers from the
pr oblem of sche dulin g with una vaila bility periods in that its maximum com pletion
time cannot be earlie r than the lat est com pletion time o f a fix ed job .
In [ 5 ], Scharbr odt etal. gi v e a pol ynomial-time appro ximation scheme f or sched -
uling on a const ant numbe r o f unif orm machin es with fix ed jobs . T he y also sho w that
it is NP -har d t o obtain a sched ule that ends within a fact or of less than 1.5 whe n sched -
uling on iden tical proc essor s with at most one fix ed job on each mac hine . E v en thoug h
the y do not specif y their re sult in this w a y , their proo f that no pol y nomial-time
appr oxima tion alg orithm can ha v e a bette r w orst -case appro ximation bound than 1.5
f or multiproc essor sched uling with fix ed jobs does not use the fa c t that ther e can be
mor e than one fix ed job on eac h machin e, whic h implie s the pr evious stat eme nt.
I f all fix ed jobs ar e at the beginning o f the sched ule, th e results pr esent ed in
S ection 4. 1 app ly , as the optimal sched ule of ea ch pro blem instanc e of sched uling on
nonsim ul tane ous machin es can only pot entially g et w orse w hen sched uling with fix ed
jobs is consid ere d instea d, and since the re sulting sch ed ule of an appr oxima tion alg o -
rithm ends lat er f or sche duling with fix ed jobs only if its maximum com pletion time is
the comp letion time of a fixe d job , w hich the optimal sched ule also needs to ex ecut e .
W e next consid er the case w here ther e is at most one fix ed job on eac h mac hine in
S ection 5. 1, and the case w her e ther e can be multiple fix ed jobs on eac h mac hine in
S ection 5.2.

Sche du l in g Pro blem - N ew A ppl ica tions a nd T rends
10
5. 1 Sche duling w ith a t most on e fi x ed job on ea c h ma chine
When sched uling on multiple proc es sors with at most one fix ed job on eac h
ma chine , LP T and MUL TIFI T v ari ant s ha v e been sho wn to a chie v e a w orst -case
appr oxima tion bound o f 1.5 , w hich is best possible for this pro blem unless P = NP .
5.1.1 Same-sp ee d pr oce ssor s
F or sched uling on iden tical mac hines with at most one fixe d job on eac h
ma chine , an LP T - like alg orithm, LP TX , w as giv en in [ 3 ], f or whi ch a w orst -case
appr oxima tion bound o f 1.5 w as sho wn. Bef ore r unnin g LP T , LP T X cr eat es an
or der of pr oces sors , w hich is then used b y LP T to br eak tie s in case tw o proc essor s
bec ome a vaila ble at the same time . T he ord ere d lis t of proc essor s crea te d bef ore
appl y ing LPT is buil t in tw o st eps:
1. All pr oce ssors that ha v e an una vailabilit y period that is not at the beginnin g
o f the sche dule ar e assigned t o this list in non- d ecreasin g ord er of the star t of
their do wntime .
2. All other proc essor s (that is, those w hic h ha v e the do wntime s at the beginning
o f the sche duling pe riod or ha v e no do wntime at all) are append ed t o the list
built in the pr evious step in non-decr easing or der of the times at w hich the y
can s tart ex ecutin g jobs.
5.1.2 U nif orm proc es sors
I n the mor e ge ner al case of sched uling on unif orm mac hines with at most one
fix ed job on eac h mac hine , a MUL T IFIT - like alg orithm, LMUL T IFIT , w as giv en in [ 4 ],
w hich ac hiev es the w orst -case appr oxima tion bound of 1. 5 . F or a MUL T IFIT variant t o
w ork in the pr ese nce of do wntime s, it must be specified ho w it deals with the fact that
ther e are mor e than one time inte r val in w hich jobs can be sched uled on one proc essor .
After a MUL TIFI T deadlin e is assigned and be f ore appl y ing the bin pac king
alg orithm FFD (se e S ection 3), LMUL T IFIT ord ers all time int er v als in w hich jobs
can be sched uled in non-decr easing or der o f their l e n g th . H er e, the len gth of a time
int er v al is the dura tion of the time int er val m ul tiplie d by the speed fa c t or o f the
pr ocessor on w hich the inte r val occur s .
5. 2 Sche duling w ith multipl e fi x ed job s on eac h ma c hine
F or sched uling on iden tical proc essor s with fix ed jobs , w her e the num ber of
fix ed jobs on a mac hine can be arbitrary , appro ximation algorith ms that are mu ch
mor e comp lex than LP T and MUL T IFIT w er e giv en in [ 27 ], with a w orst -case
appr oxima tion bound o f 1.5 + ε , w h e r e ε is the paramet er of a f ully polynomial
time appro ximation sche me (FP T AS) f or the multiple subset sum proble m, and in
[ 6] with a w orst -case appro ximation bound o f 1.5 , w her e a FP T AS f or the multiple
knapsa ck prob lem is used as a subroutin e.
In [ 28 ], a v ery long pr oof is outlined tha t LMUL TIFI T achie v es a w orst -case
appr oxima tion bound o f 1.5 w hen sche duling on iden tical proc essor s with at most
tw o fix ed jobs on eac h mac hine . In [29 ], an algorith m using tw o MUL TIFI T -lik e
alg orithms is sho wn to ha v e a w orst -case appro ximation bound of 1. 62 5 , w hich
lik ely can be impr ov ed t o 1. 6 without ex cessi v e eff or t.
The time com plexit y of the MUL T IFIT -lik e and LP T -lik e algorith ms is significanlty
lo w er than that of the algorith ms from [ 6 , 27 ], and the y are also significantl y easier t o

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imp lement; ho w ev er , ther e is little hope in our opinion that bounds bette r than 1.55
can be shown in the ge ner al case for suc h alg or ith ms with proo fs of reasona ble length.
Gi v en suc h pro blems , the user must decid e w hat is best suit ed f or his or her needs .
6. Pr oof t ec hniqu es
W orst -case appro ximation re sults in this r esearc h area about LP T and
MUL T IFIT varian ts mainly use proo fs b y contr adiction in w hic h some proo f
t ech niqu es appear v ery of t en. W e next de scr ibe tw o t ech niqu es that w e conside r to
be the most re levan t, and then comme nt upon some othe r methods that ar e used
r elati v ely often. I n the f ollowin g , w e call job lengths the dura tions o f the jobs on the
slo w est proc es sor .
6. 1 Mi nimal c ount er examp le
A v ery w ell-kn own proo f method is that of assuming tha t ther e exists a minimal
coun ter exam ple t o a theore m T , that is, a pro blem instanc e for w hich the algorith m ’ s
sche dule does not obe y that theor em, with a minimal numbe r of proc essor s, o f jobs,
o f do wntimes ( or of fix ed jobs) tha t do not star t at the beginning o f the sched ule , and /
or of othe r quantitie s that can be minimized w hich are chosen t o fit the stat emen ts
t o pro v e. A minimal c ount ere xample exists w hene v er there is a count er example , and
thus , sho wing that it doe s not exist pro v es T .In ord er t o define minimality among
coun ter exam ples , the auth or o f the proo f first choose s a par tial or der among th e prob -
lem instanc es . An instanc e w hich is minimal ac cor ding t o this par tial ord er among th e
instan ces f or w hich a theor em T doe s not apply is called a minimal count er example .
This method can be v ery pow erful, beca use af te r as suming that a minimal
coun ter exam ple doe s not obey a theor em T , man y usef ul pr oper ties of a minimal
coun ter exam ple can be de r i v ed from the fact that no l esser coun te rexam ple exists ,
w hich can ultimat el y lead t o a con tradiction.
H ere , a lesser count ere xample is a count er examp le with les s proce ssors , or les s
tasks , or less do wntimes , or with a job with a smaller length, dependin g on ho w the
or der of instan ces w as de fined. Sho wing that a minimal coun ter exam ple doe s not
exist is usually significantly easier than dev elopin g a dire c t pr oo f f or T .
The theor em t o pro v e could be that the w orst -case appro ximation bound holds ,
but it can also be an int ermediary result that is lat er used t o pro v e the w orst -case
appr oxima tion bound. One could addr es s instance s that ha v e a cer tain prope r ty
first, and th en sho w that the w orst -case appro ximation bound holds for these , f or
exam ple b y using a minimal count er examp le among the se instanc es , and then do
the same f or all o ther instan ce s.
S ometime s it is enoug h to de fine the parti al or der only using the number of
pr ocessor s [ 11 ], w hile in most cases , it is useful to inclu de multiple chara c t eristics
o f prob lem instan ce s, su ch as all or a par t of the chara c t eristics enume rat ed abo v e.
I n one situation, a minimal coun ter exam ple w as de fined t o also ha v e minimal
job lengths , meaning tha t if in a minimal count ere xample the length of one job is
r edu ced, the re sul ting in s tan ce is not a count er examp le [ 28 ].
6. 2 W eig hing ar g umen ts
F or a pro blem instan ce that doe s not obey a w orst -case appr oxima tion bound,
t h e r e i s a j o b ( X ) that is sched uled suc h that it cros ses the bound to pro v e times the
end o f an optimal sched ule ( B ) f or LP T - like alg orithms , or that can not be sched-
uled whe n the dea dline is at B f or MUL T IFIT -lik e alg orithms . If the ord er define d on

Sche du l in g Pro blem - N ew A ppl ica tions a nd T rends
12
instan ces inc lude s the numbe r o f jobs , there is only one such job in a minimal coun-
t ere xample to (f or examp le) a theore m that the analyzed al g orithm ( A) g ene ra te s
only sche dules that obe y the bound. O the r wise , all jobs that w ould be sched uled
af t er war d can be re mov ed and a lesse r count er example could be obtaine d, r esult -
ing in a con tradiction. As a conseq uen ce of ho w LP T and MUL TIFI T w ork, X i s t h e
smallest job o f the minimal count ere xample in both cases .
The sched ule S A gen erat ed b y an LP T - like alg orithm A until a job w ould cross B
and the sched ule S A gen erat ed b y a MUL T IFIT -lik e alg orithm A if the MUL TIFI T
dea dline is at B d o n o t c o n t a i n t h e j o b X . An optimal sched ule , how ev er , cont ains all
jobs , inc luding X .
Ther ef ore , the optimal sched ule has more t otal ex ecution time than S A . Also , if
nonze r o w eight s are assigned t o eac h job , the optimal sched ule has a to tal w eig ht of
all its jobs that is gr eat er than that of S A , the diff ere nce being the w eigh t of X . An
a de qua te assignm ent o f w eigh ts t o jobs can lead t o the conc lusion that the sum of
w eight s of all jobs cont ained in the sched ule S A is at least the sum of the w eigh ts of
all jobs in the consid er ed optimal sc hed ule, a con tradiction.
Ther e are infinite w a y s of assigning w eigh ts , and ther e is no uniqu e strat egy
that leads t o succ ess . U sually , the w eigh t f unction is mono tonic with reg ard t o
job lengths , and, as X has the smallest job length, its w eight can be set to 1. I n the
f ollo wing , w e deno te both the job X and the time X w ould need t o be ex ecut ed
on the slow est proc essor by X . T he other w eight s can be assigned f or in te r vals
o f job len g ths , f or e xample , a job with length within
[
X ,1 . 5 X
)
could be assigned
w eight 1. W eigh ts can also be assigned oth er wise: for examp le, a job with a
ce r tain prope r ty can be chosen t o be the end o f a w eigh t int er val. J obs that ha v e
a cer tain prope r ty can also be assigned a specific w eight that corr esponds to that
pr oper ty . Of course , w hile pro ving diff er ent theor ems that lead t o the proo f of a
w orst -case appro ximation bound, a new w eight f un c tion can be chosen f or each
s t a te m e n t t o p r ove.
6.3 N ormalizi n g t ime int er v als and jo b ex ecut ion time s
I n ord er to reason easier about time , one can divid e all dur ations o f time int er -
vals in w hic h jobs can be sche dule d by X if sched uling on identi cal proc essor s .
F or unif orm proc essor s, su ch int er vals can be divid ed b y the time it w ould take
X t o ex ecut e on the proc essor on w hich the int er val oc curs , in or der t o deri v e the
le n g th of the inte r val [ 4 ]. Also , all job dur ations can be divid ed b y X and these
normalize d dura tions can be used in the proo fs . F or examp le , a theor em could be
pr ov ed tha t there are no jobs that ha v e a long er dura tion than 5, or it can be stat ed
that the unused time inte r vals on proc essor s bef ore the MUL T IFIT dea dline B all
ha v e length less than 1, as other wise X w ould ha v e been sched uled in one of those
int er v als.
6. 4 T ask density
I n the case of unif orm proce ssors , the time int er vals can ha v e unbound ed
len g ths because the speed fa c t ors ma y be arbitraril y high. A w a y to de scribe the
amount o f jobs assigned within a sche dule in suc h an int er val is t o use tas k densi-
ties , whic h can be de fined f or eac h task as being the ratio betw een its w eigh t and its
len g th. Also , a t ask typ e de nsity can be defin ed as a low er bound f or all pos sible task
de nsities of tasks o f that type. T he con cept o f task densit y can be used in ord er to
r eason about time int er vals that ma y be v ery long . F or e xample , the to tal w eigh t of
all tasks in an int er val o f len g th t i s a t m o s t t times the maximum task type densit y
amon g all task types repr esent ed within that int er v al.

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6.5 Pr oce ssor with mor e ex ecution ti me in th e opt imal sch edule
W e use the nota tions from the pr evious subsection. Sinc e X is not cont ained
i n S A , the re must be a pr ocessor p * in the optimal sche dule that has a tot al ex ecu-
tion time that is grea t er than that of S A o n p *. Su ch a proce ssor can be analyze d in
det ail and ma y be shown t o ha v e certain pr oper ties that, in conjunction with other
methods , re sul t in proo fs of useful theor ems . F or example , the exist en ce of p * ma y
imp ly a ce r tain minimum dura tion of the optimal sched ule, in conjun c tion with the
observation that X could no t be fitted b y A o n p * without causin g the maximum
com pletion time of S A t o exc eed B .
7 . C onc lusion s and futur e challen ge s
I n this w ork, w e consid ere d w orst -case appro ximation f or sched uling on mul-
tiple mac hines with a vailabilit y constrain ts or with fix ed jobs in or der t o minimize
the ma ximum com pletion time . W e sur v ey ed re sul ts obtaine d in this re search ar ea
and com ment ed upon the alg or ith ms used.
Pr ominent among the se alg orithms ar e LP T and MUL T IFIT and their v ari ant s,
w her eas for multiproc essor sched uling with fix ed jobs , mor e comp lica t ed alg o-
rithms w er e used t o achie v e best possible w orst -case appr oxima tion bounds in the
clas s of polynomial alg orithms assuming tha t P ≠ NP in the g eneral case w her e ther e
can be an y number of fix ed jobs on eac h machin e.
The pro blem of sc hedulin g with a vailabilit y constrain ts canno t be appro ximat ed
b y a pol ynomial-time alg orithm with a constan t w orst -case appro ximation bound,
ev en if there is at most one do wntime on eac h mac hine , unles s assumptions about
the do wntimes are mad e. The re sul ts w e prese nt ed in this area addr ess the prob lem
o f sched uling on iden tical proc essor s with at most one do wntime on each mac hine ,
with various assumptions .
Du e to its diffe ren t objectiv e f unction, the pro blem of sched uling on iden tical
par allel mac hines with fixe d jobs allow ed f or the dev elopment of a polynomial-time
appr oxima tion alg orithm with a w orst -case appr oxima tion bound o f 1.5 , and the
de v elopment o f a P T AS f or sched uling on a constan t number of unif orm mac hines
with fixe d jobs w as also possible .
The MUL TIFI T and LP T variant s dev eloped f or the discussed variants of the se
pr oblems could be useful in practice , as their time com plexity is lo w and thus the y
should be able to addr ess v ery larg e prob lem instanc es , as the y are easy to imple-
men t, and beca use in some cases their w orst -case appro ximation bounds could be
con side red t o be g ood enough. I n the case of sch edulin g on uniform nonsim ul tane-
ous machine s , the a v era ge perf ormanc e of a MUL T IFIT variant w as st u died, and
sho wn t o be v ery g ood, as the experime nts sugge s t that in ge neral, f or instanc es
that can be r elev ant in pra c tice and f or w hich exha ustiv e search is not an option, the
alg orithm returns sched ules with a maximum com pletion time that is within 3% of
that of an optimal sched ule.
W e also elabora te d on the most enc ount ere d pr oo f te chniq ues in w orst -case
appr oxima tion bound pr oo fs f or LP T and MUL T IFIT variant s.
The limitation s of the pr ese nte d w orks re sul t mainly f r om the difficulty of the
pr oblem of sche dulin g with una vaila bility periods w hen consid ering the subject o f
appr oxima tion. T o asses s how w ell the propose d heur istics f or this proble m perf orm
unde r such condition s is difficul t, as it is hard t o ha v e a good e stimat e of the opti-
mal sche dule unless it is compu ted b y an exa c t algorith m as w as done in [ 26 ]. This
pr oblem has ther ef ore been addr essed onl y b y consid ering the special case w her e
ther e is at most one do wntime on eac h mac hine .

Sche du l in g Pro blem - N ew A ppl ica tions a nd T rends
14
I n the f utur e , it ma y be int ere s tin g to comp are the heuristics proposed f or th e
same proble ms experime ntall y .
Anoth er limitation of the discussed w orks and of man y othe r r esear ch w orks
on sched uling re sults from the fa c t the y att empt to unde rstand pr oblem s with
one , tw o , or at most thr ee aspects at one time, w her eas in man y pra c tical pr ob -
lems suc h as some pr oduction plannin g pr oblems , man y aspects occur at once .
F or examp le, a vaila bility constraint s can appear alongsid e a multit u de o f other
con straint s that ha v e to be conside red sim ul tane ously . These can be prec eden ce
con straint s, tha t certain jobs ha v e t o be assigned t o certain mac hines , or pre f er -
en ces of the manufacture r that the mac hines should not ha v e more than a 60%
or anothe r pr ed efine d load f or examp le in ord er to lea v e room f or un expected
ev ents . Fur thermor e, or der s ofte n come online, an d if an urg ent ord er from an
important clien t needs to be giv en priority , this can alte r the de liv ery times of
othe r or de rs . Also , deli v ery times and de la y s ha v e a big re levan ce in practic e,
as not de liv ering on time can cause fines . Such pra c tical pr oblem s can also ha v e
seq uen ce-depend ent set up times , the nece ssity f or set up opera tor s to be pr esent t o
perf orm set ups , the pre f er enc e that set up times are k ept low b y putting jobs from
the same family of types of jobs consecu tiv el y on mac hines w henev er possible ,
the nece ssity f or w ork er s to att end t o certain mac hines w hile prod uction tak es
pla ce , and w ork er breaks and holida y s. T he pre existing sche dule also has to be
k ept unchan ged f or a pred efined tim e period since mat erials are br oug ht to the
pr oduction plac e in prep aration f or the prod uction proc ess . I n addition, or der s
ma y ha v e pr ioritie s and deadlin es . F or su ch prob lems , giv en the time constrain ts
in w hich the sched ule needs t o be gen erat ed and tha t ther e can be thousands of
jobs , usually a heuristic is emp lo ye d that first ord ers the jobs f or e xample b y using
priorities assigned t o them and / or their dea dlines and then sched ules them on the
ma chine s in that ord er w hile also obe y ing all c onstraint s and att emptin g to f ulfill
all pr ef ere nce s. Diffi cul ties in r esearc hing su ch pr oblems inc lude that pro babl y f or
diff er ent sets o f ord ers diff ere nt sche duling stra tegie s ma y be bette r , and that an
optimal sche dule ma y be v ery hard t o find and thus it is hard t o quan tif y ho w w ell
a heur istic perf orms .
A ck no wledgements
This publica tion w as suppor t ed b y the Open Ac ces s P ub lication Fund of
T ec hnische U niv ersitä t Berlin.
Confli c t of int er est
The a uthor dec lares no conflict of int er est.
Ab br ev ia tions
MSP multipr ocessor sched uling pr oblem
NMSP nonsim ul tane ous multiproc essor sche duling pr oble m
UNMSP unif orm nonsim ul tane ous multiproc es sor sche dulin g prob lem
LP T larg est proc essing tim e
FFD first fit decr easing

15
A ppro xi mati on f or Sc hedu li ng on P ar all el Mac hi nes wit h Fix ed J ob s or U na va ilability P er iods
DO I: http:// dx . doi . org / 1 0 . 57 72 / in tec hopen.8 9 694
A uthor details
LilianaGrig oriu 1,2
1 Department of Compu ter S cien ce and Engine ering , F aculty of Contr ol and
Compu ter s , P olit ehnica U niv ersit y Buchar es , Bu char est, R omani a
2 Department of Mathe matics , T ech nical U niv ersit y of Ber lin, Berlin, German y
* Addr ess all corr esponde nce to: liliana.grig oriu@cs .pub .r o
© 2 0 19 The A uthor( s). Lic ense e In tec hOpen. This chapt er is distribut ed under the t erms
o f the Crea tiv e Commons A ttribution Lice nse (http:// crea tiv ec ommons . org/lic ense s /
b y/3 .0), w hic h permits unr estrict ed use, distribution, an d r eprodu ction in an y medium,
pr ovide d the original w ork is prope rly cit ed.

16
Sche du l in g Pro blem - N ew A ppl ica tions a nd T rends
[1] H wang H -C, Chan g SY .P arallel
ma chine s sched uling with machin e
shut do wns . Comput er s and
Mathe matics with Application s . J une
1998; 36 :21-31
[2] H wang H -C, Lee K, Chang SY .T he
eff ec t of ma chine a vailabilit y on
the w orst -case perf ormanc e of
LP T .Discr et e Applied Math ematics .
April 2 00 5; 14 8 :49-6 1
[3] G rig or iu L , F r ie sen DK.S ch ed uling
on s ame-spee d proc essor s with at
most one do wntime on eac h machine .
Discr ete Optimiz a tion. N ov ember
2 0 10 ; 7 :212-2 21
[4 ] Grig or iu L , F r ie sen DK.S che duling
on uniform proc essor s with at most one
do wntime on eac h mac hine. Discr et e
Optimization. N ov ember 2 0 1 5; 17 :1 4-2 4
[ 5] S charbr odt M, St eg er A , W eisser H.
Appr oxima bility of sc hed uling with
fix ed jobs . J ournal of S ch ed uling .
N ov ember 19 99; 2 ( 6):26 7-28 4
[ 6] J ansen K, Pra del L, S ch warz UM,
Sv ensson O .F as t er appr oxima tion
alg orithms f or sched uling with fix ed
jobs . I n: 17th Conf er enc e of Computin g:
The A us tralasian T he or y S ymposium
( CA TS 2 0 1 1), P er th, A ustralia, J anuary .
201 1
[7 ] Graham RL .Bounds on
multipr ocessin g timing anomalie s.
SIAM J ournal of Applied Ma themati cs.
Mar ch 1969; 17 :4 1 6-429
[8] Co ffman EG J r , Gare y MR ,
J ohnson DS.An appli cation of bin-
pa ckin g to multiproc essor sched uling .
SIAM J ournal on Computin g. F ebruar y
1978; 7 :1-1 7
[9] H wang HC, Lim K.Exa c t
perf ormance of MUL T IFIT f or
nonsim ul tane ous machin es . Discret e
Applied Ma themati cs. 2 0 1 4; 16 7 :1 7 2-1 87
[ 10 ] K eller er H.Alg orithms f or
multipr ocessor sc hedulin g with
ma chine r elease times . IIE T ransactions .
1998; 30 :9 91-999
[1 1 ] Grig oriu L , F riesen DK.
Appr oxima tion f or sc hed uling on
unif orm nonsim ul tane ous parallel
ma chine s. J ournal of S che duling .
Dec ember 2 01 7; 20 :5 93-6 0 0
[1 2 ] Chen B.Tigh te r boun d for multifit
sche duling on unif orm proc essor s.
Discr ete Applied Math ematics . Ma y
1991; 31 :227-26 0
[ 13] Lee CY .P arallel ma chine sched uling
with nonsimultaneous mac hine
a vailab le time. Discr et e Applied
Mathe matics . J anuary 19 91; 30 :5 3-61
[14] Chang SY ,H wan g HC.
The w orst -case analy sis o f the
MUL T IFIT alg orithm f or sched uling
nonsim ul tane ous paralle l machine s .
Discr ete Applied Math ematics . J une
199 9; 92 :135-1 4 7
[ 1 5] Y ue M.On the exa c t uppe r bound
o f the MUL T IFIT proc essor sche duling
alg orithm. Annals o f Opera tions
R esearc h. Dece mber 19 90; 24 :2 33-2 5 9
[ 1 6] F riesen DK, L angst on MA .Bounds
f or multifit sche dulin g on unif orm
pr ocessor s . SIAM J ournal on
Compu ting . F ebruary 198 3; 12 :6 0-69
[17 ] Burkar d RE , H e Y .A not e on
MUL T IFIT sched uling f or unif orm
ma chine s. Com puting . 19 98; 61 :27 7-28 3
[18] He Y .U nif orm machin e sched uling
with machine a vailab le constr a i nts . A c t a
Mat ema ticae Applicat ae Sinica (Eng lish
S erie s). 2 00 0; 16 :1 22-1 29
[1 9] Grig oriu L , Frie sen DK.S ch edulin g
on uniform nonsim ul tane ous parallel
ma chine s. I n: Fink A , Fiig enschuh A ,
Geig er M, edit ors . Operation s R esear ch
R ef eren c es

17
A ppro xi mati on f or Sc hedu li ng on P ar all el Mac hi nes wit h Fix ed J ob s or U na va ilability P er iods
DO I: http:// dx . doi . org / 1 0 . 57 72 / in tec hopen.8 9 694
Pr oceedin gs 2 01 6 S elect ed Pa per s of
the Annual I nt ernational Conf ere nce
o f the German Operation s R esear ch
S ociet y ( GOR), Hann ov er , A ug ust 30–
S ept embe r 2. 2 0 16 . pp .4 6 7-4 7 3
[20] L ee CY , Lei L , Pined o M.Curren t
tr ends in det erministic sched uling .
Annals of Oper ations R esear ch. April
1997; 70 :1-4 1
[2 1] S anla ville E , S c hmidt G.Machin e
sche duling with a vailabilit y constraint s.
A c ta I nforma tica. S ept embe r
1998; 35 :7 95-811
[2 2] S ch midt G.S ched uling with limited
ma chine a vailabilit y . European J ournal
o f Opera tional R esearc h. F ebruary
2000; 1 21 :1-1 5
[2 3] Lee C - Y .Mac hine sched uling with
a vailabilit y constrain ts . In: Leung
JY - T , edit or . Handbook of Sched uling:
Alg orithms , M odels and P erf ormanc e
Analy sis. Lond on: Chapman & Hall/
CR C ; 2 00 4. pp .2 2-1-22-13
[2 4] Ma Y , Chu C, Zuo C.A sur v ey of
sche duling with det erministic machin e
a vailabilit y constrain ts . Comput ers &
I ndustrial Enginee ring . 2 01 0; 58 :199-211
[ 2 5] Lee CY .Ma chine sched uling with
an a vailabilit y constrain t. J ournal
o f Global Optimization. De cembe r
1996; 9 :395-41 6
[ 26] Kaabi J , Harra th Y .Sched uling on
unif orm par allel mac hines with periodic
una vailabilit y constrain ts . In terna tional
J ournal of Prod uction R esearc h.
2 0 19; 57 ( 1):21 6-2 27
[ 27 ] Diedric h F , J ansen K.I mpr ov ed
appr oxima tion alg orithms f or
sche duling with fix ed jobs . Proc eedings
o f 2 0th A CM -SIAM S ymposium
on Discret e Alg or ith ms (SOD A ).
2 0 09:6 75-68 4
[28] Grig oriu L .M ultiproc essor
sche duling with a vailabilit y constraint s
[PhD thesis]. Co llege Sta tion, T X, U S A :
T exas A&M U niv er sity; 2 01 0
[ 29 ] Grig oriu L .S c hed uling on parallel
ma chine s with variable a vailabilit y
pa tte rns [PhD thesis]. Bu char est,
R omani a: P olit ehnica U niv ersit y
Bu char est; 2 0 1 2

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