A New Approximation Technique for Resource-Allocation Problems

0.28 MB·English

by  Barna Saha

A New Approximation Technique for
Resource-Allocation Problems
BarnaSaha1 AravindSrinivasan1
∗ †
1UniversityofMaryland,CollegePark,MD20742,USA
[email protected] [email protected]
0 Abstract: We developa roundingmethod based on randomwalks in polytopes, which leads to im-
1 proved approximation algorithms and integrality gaps for several assignment problems that arise in
0 resourceallocation and scheduling. In particular, it generalizesthe work of Shmoys& Tardoson the
2
generalizedassignmentproblemintwo differentdirections, wherethe machineshavehardcapacities,

n and where some jobs can be dropped. We also outline possible applications and connections of this
a
methodologytodiscrepancytheoryanditeratedrounding.
J

0 Keywords: Scheduling;rounding;approximationalgorithms;integralitygap
1

] 1 Introduction inrandomizedrounding,weuserandomizationto
S
D The “relax-and-round” paradigm is a well- map x∗ = (x∗1,x∗2,...,x∗n) back to some x =
(x ,x ,...,x ) [39]. Typically, we choose a
. known approach in combinatorial optimization. 1 2 n
s value α that is problem-specific, and, indepen-
c Givenaninstanceofanoptimizationproblem,we
[ enlargethe setoffeasible solutionsI tosome set dently for each i, define xi to be 1 with proba-
v1 Ila′xa⊃tioIn–ofofttheenpthroebllienmea;r-wpreogthreanmmmianpga(LnP()efrfie-- bpirloitbyabαilxit∗iy,oafnd1−toαbxe∗i0. wInidtheptehnedceonmcepcleamn,ehnotawry-
ever, lead to noticeable deviationsfrom the mean
0 ciently computed, optimal) solution x I to
7 ∗ ∈ ′ for random variables that are required to be very
some “nearby” x I and prove that x is near-
4 ∈ closeto(orevenbeequalto)theirmean. Afruit-
optimal in I. This second “rounding” step is of-
1 fulideadevelopedin[27,32,45]istocarefullyin-
. ten a crucial ingredient, and many general tech-
1 troducedependenciesintotheroundingprocess:in
niqueshavebeendevelopedforit. Inthiswork,we
0 particular,somesumsofrandomvariablesareheld
presentanewroundingmethodologywhichleads
0 fixedwithprobabilityone,whilestillretainingran-
1 to several improved approximation algorithms in
domness in the individual variables and guaran-
: scheduling, and which, as we explain, appears to
v teeing certain types of negative-correlation prop-
i have connections and applications to other tech-
X ertiesamongthem. See[1]forarelateddetermin-
niquesandproblems,respectively.
istic approach that precedes these works. These
r We next present background on (randomized)
a dependent-roundingapproachesleadtonumerous
rounding and a fundamentalscheduling problem,
improvedapproximationalgorithmsinscheduling
beforedescribingourcontribution.
andpacket-routing[1,27,32,45].
Our work generalizes various dependent ran-
We now introduce a fundamental scheduling
domized rounding techniques that have been de-
model,whichhasspurredmanyadvancesandap-
veloped over the past decade or so. Recall that
plicationsincombinatorialoptimization,including
∗Dept.ofComputerScience,UniversityofMaryland,Col- linear-, quadratic- & convex-programmingrelax-
legePark,MD20742.ResearchsupportedbyNSFAwardNSF ationsandnewroundingapproaches[6,8,10,15,
CCF-0728839. 21, 29, 32, 34, 41, 43]. This model, scheduling
†Dept. of Computer Science and Institute for Advanced withunrelatedparallelmachines(UPM)–andits
ComputerStudies,UniversityofMaryland,CollegePark,MD
relatives–playakeyroleinthiswork.Herein,we
20742. Supported inpartbyNSFITRAwardCNS-0426683
andNSFAwardCNS-0626636. aregivenasetJ ofnjobs,asetM ofmmachines,
1
and non-negative values p (i M, j J): “Ax b”untilthenumberofconstraintsbecomes
i,j
∈ ∈ ≤
each job j has to be assigned to some machine, smaller than n, thus making the system linearly-
and assigning it to machine i will impose a pro- dependent – leading to the efficient computation
cessingtimeofp onmachinei. (Theword“un- of an r n that is in the nullspace of this re-
i,j
∈ ℜ
related” arises from the fact that there may be no ducedsystem. Wethencomputepositivescalarsα
pattern among the given numbers p .) Variants andβ suchthatx := x+αr andx := x βr
i,j 1 2
−
suchasthetypeofobjectivefunction(s)tobeop- bothliein[0,1]n,andbothhaveatleastonecom-
timizedinsuchanassignment,whetherthereisan ponentlyingin 0,1 ;wethenupdatextoaran-
{ }
additional“cost-function”,whetherafewjobscan domY as: Y := x withprobabilityβ/(α+β),
1
bedropped,andsituationswheretherearerelease andY := x withthecomplementaryprobability
2
dates for, and precedence constraints among, the α/(α + β). Thus we have rounded at least one
jobs,leadtoarichspectrumofproblemsandtech- further componentof x, and also have the useful
niques. We now briefly discuss two such highly- propertythatforallj,E[Y ]=x . Differentways
j j
impactfulresults[34, 41]. TheprimaryUPM ob- of conducting the “judicious” reduction lead to a
jectiveintheseworksistominimizethemakespan varietyofimprovedschedulingalgorithmsin[32].
– the maximum total load on any machine. It is Thesettingof[27,45]onbipartiteb-matchingscan
shown in [34] that this problem can be approxi- beinterpretedinthisframework.
mated to within a factor of 2; furthermore, even
somenaturalspecialcasescannotbeapproximated We further generalize the above-sketched ap-
better than 1.5 unless P = NP [34]. Despite proach of [32]. Suppose we are given a poly-
mucheffort,theseboundshavenotbeenimproved. tope in n dimensions, and a non-vertex point
P
The work of [41] builds on the upper-bound of x belonging to . An appropriate basic-feasible
P
[34]toconsiderthegeneralizedassignmentprob- solution will of course lead us to a vertex of ,
P
lem(GAP)whereweincuracostc ifwesched- but we approach (not necessarily reach) a vertex
i,j
ule job j on machine i; a simultaneous (2,1)– of by a random walk as follows. Let de-
P C
approximationfor the (makespan, total cost)-pair note the set of constraints defining which are
P
isdevelopedin[41],leadingtonumerousapplica- satisfied tightly (i.e., with equality) by x. Then,
tions(see,e.g.,[2,17]). note that there is a non-empty linear subspace S
of n such that for any nonzero r S, we can
Wegeneralizethemethodsof[1,27,31,32,45], ℜ ∈
travel up to some strictly-positive distance f(r)
via a type of random walk toward a vertex of
along r starting from x, while staying in and
the underlyingpolytopethatwe outline next. We P
continuing to satisfy all constraints in tightly.
thenpresentseveralapplicationsinschedulingand C
Our broad approach to conduct a random move
bipartite matching through problem-specific spe-
Y :=x+Rbychoosinganappropriatelyrandom
cializations of this approach, and discuss further
R from S, such that the property “E[Y ] = x ”
prospectsforthismethodology. j j
ofthepreviousparagraphstillholds. Inparticular,
The roundingapproachesof [1, 27, 31, 45] are let RandMove(x, ) – or simply RandMove(x)
P
generalized to linear systems as follows in [32]. if is understood – be as follows. Choose a
P
Supposewehaveann-dimensionalconstraintsys- nonzeror S arbitrarily,andsetY :=x+f(r)r
∈
tem Ax b with the additional constraints that withprobabilityf( r)/(f(r)+f( r)),andY :=
≤ − −
x [0,1]n. This will often be a LP-relaxation, x f( r)r with the complementary probability
∈ − −
whichweaimtoroundtosomey 0,1 n such of f(r)/(f(r) +f( r)). Note that if we repeat
∈ { } −
thatsomeconstraintsin“Ay b”holdwithprob- RandMove, weobtainarandomwalkthatfinally
≤
ability one, while the rest are violated “a little” leads us to a vertex of ; the high-level idea is
P
(with high probability). Given some x [0,1]n, to intersperse this walk with the idea of “judi-
∈
theroundingapproachof[32]isasfollows. First, ciously droppingsome constraints” from the pre-
we assume without loss of generality that x viousparagraph,aswellascombiningcertaincon-
∈
(0,1)n: those x that get rounded to 0 or 1 at straintstogetherintoone. Threemajordifferences
j
some point, are held fixed from then on. Next, from[32] are: (a) the care givento the tightcon-
we “judiciously” drop some of the constraints in straints , (b) the choice of which constraint to
C
2
dropbeingbasedon ,and(c)clubbingsomecon- structuredobjectivefunctionf with boundedco-
i
C
straintsintoone.Asdiscussednext,thisrecipeap- efficientsassociatedwitheachi M,theninfact
∈
pearsfruitfulinanumberofdirectionsinschedul- allthe f (X) f (x) canbeboundedindependent
i i
| − |
ing,andasanewroundingtechniqueingeneral. ofN. Thisappearstobethefirstsuchresulthere,
andhelpswith equitablemax-minfairallocations
Capacityconstraintsonmachines,randommatch-
asdiscussedbelow.
ings with sharp tail bounds. Handling “hard ca-
Schedulingwith outliers: makespan andfairness.
pacities” –those thatcannotbe violated– isgen-
Note that the (2,1) bicriteria approximation that
erallytrickyinvarioussettings,includingfacility-
we obtain for GAP above, generalizes the results
locationandothercoveringproblems[19,26,36].
of [41]. We now present such a generalization
Motivated by problems in crew-scheduling [22,
inanotherdirection: thatof“outliers”inschedul-
40] and by the fact that servers have a limit on
ing[29]. Forinstance,supposeinthe“processing
how many jobs can be assigned to them, the nat-
times p and costs c ” setting of GAP, we also
ural question of scheduling with a hard capacity- i,j i,j
haveaprofitπ forchoosingtoscheduleeachjob
constraint of “at most b jobs to be scheduled on j
i
j. Givena“hard”targetprofitΠ,targetmakespan
each machine i” has been studied in [18, 48, 50–
T and total cost C, the LP-rounding method of
52]. Most recently, the work of [18] has shown
[29]eitherprovesthatthesetargetsarenotsimul-
that this problem can be approximated to within
taneouslyachievable,orconstructsaschedulewith
a factor of 3 in the special case where the ma-
values(Π,3T,C(1+ǫ)) for any constantǫ > 0.
chinesare identical(job j hasprocessingtime p
j
Using ourroundingapproach,we improvethisto
on any machine). In 2, we use our random-
§ (Π,(2+ǫ)T,C(1+ǫ))in 3. (Thefactorsofǫin
walk approach to generalize this to the setting of §
thecostarerequiredduetothe hardnessofknap-
GAP and obtain the GAP bounds of [41] – i.e.,
sack [29].) Also, fairness is a fundamental issue
approximationratiosof2 and1forthe makespan
in dealing with outliers: e.g., in repeated runs of
and cost respectively, while satisfying the capac-
suchalgorithms,wemaynotdesirelongstarvation
ityconstraints: theimprovementsareinthemore-
of individualjob(s) in sacrifice to a globalobjec-
general scheduling model, adding the cost con-
tive function. Theorem 7 accommodates fairness
straint, and in the approximation ratio. We an-
intheformofscheduling-probabilitiesforthejobs
ticipate that such a capacity-sensitive generaliza-
thatcanbepartoftheinput.
tion of [41] would lead to improved approxima-
tion algorithms for several applications of GAP, Max-Min Fair Allocation. This problem, also
andpresentonesuchinSection5. known as the Santa Claus problem, is the max-
min versionof UPM, where we aim to maximize
Theorem 1 generalizes such capacitated prob-
theminimum“load”(viewedasutility)onthema-
lemstorandombipartite(b-)matchingswithtarget
chines;ithasreceivedagooddealofattentionre-
degreeboundsandsharptailboundsforgivenlin-
cently[4,5,8,10,15,24]. Weareabletoemploy
ear functions; see [23] to applications to models
dependentrandomizedroundingtonear-optimally
forcomplexnetworks. Recall thata (b)-matching
determinetheintegralitygapofawell-studiedLP
isasubgraphinwhicheveryvertexvhasdegreeat
relaxation. Also, Theorem 1 lets us generalize a
mostb(v). Givenafractional(b)-matchingxina
resultof[14]onmax-minfairnesstothesettingof
bipartitegraphG = (J,M,E) of N verticesand
equitablepartitioningofthejobs;see 4.
acollectionofk linearfunctions f ofx, many
i §
{ }
works have considered the problem of construct- Directions for the future: some potential con-
ing (b-)matchings X such that f (X) is “close” nectionsandapplications.Distributionsonstruc-
i
tof (x)simultaneouslyforeachi[3,27,28,38]. tured matchingsin bipartite graphsis a topic that
i
The works [28, 38] focus on the case of constant models many scenarios in discrete optimization,
k;thoseof[3,27]considergeneralk,andrequire and we view our work as a useful contribution
theusual“discrepancy”termofΩ( f (x)logN) to it. We explore further applications and con-
i
in f (X) f (x) for most/all i; in a few cases, nections in 6. A general question involving
| i − i | p §
o(N)verticeswillhavetoremainunmatchedalso. “roundingwell”isthelatticeapproximationprob-
In contrast, Theorem 1 shows that if there is one lem [39]: givenA 0,1 m n and p [0,1]n,
×
∈ { } ∈
3
we wanta q 0,1 n such that A (q p) the maximum number of jobs that can be sched-
is “small”; th∈e l{inear}discrepancykof A· is−definke∞d uled on any machine, and is a hard constraint.
tobelindisc(A) = max min A Formally the problem is as follows, where x
p [0,1]n q 0,1 n i,j
(q p) . The field of∈combinator∈i{al d}isckrep-· is the indicator variable for job j being sched-
− k∞
ancy theory [13] has developed several classical uled on machine i. Given m machines and n
results that bound lindisc(A) for various matrix jobs, where job j requires a processing time of
familiesA; column-sparsematriceshavereceived p in machine i and incurs a cost of c if as-
i,j i,j
muchattentioninthisregard. Section6discusses signedto i, the goalis to minimizethe makespan
a concrete approach to use our method for the T = max x p , subject to the constraint
i j i,j i,j
famous Beck-Fiala conjecture on the discrepancy that the total cost x c is at most C and
P i,j i,j i,j
of column-sparse matrices [12], in the setting of foreachmachinei, x b . C isthe given
P j i,j ≤ i
random matrices. 6 also suggests that there upperboundontotalcostandb isthecapacityof
i
§ P
may be deeper connections to iterated rounding, machinei,thatmustbeobeyed.
a fruitful approach in approximation algorithms Ourmaincontributionhereisanefficientalgo-
[25, 30, 33, 42, 49]. We view our approach as rithmSched-Capthathasthefollowingguarantee,
having broader connections/applications (e.g., to generalizingtheGAPboundsof[41]:
openproblemsincludingcapacitatedfacilityloca-
tion[36]),andarestudyingthesedirections. THEOREM2 There is an efficient algorithm
Sched-Capthatreturnsaschedulemaintainingall
2 Random Matchings with Linear thecapacityconstraints, ofcostatmost C and
Constraints, and GAP with Capac- makespanat most 2T, where T is the optimal
ityConstraints makespanwithcostC thatsatisfiesthecapacity
constraints.
We develop an efficient scheme to generate
random subgraphs of bipartite graphs that satisfy We guess the optimum makespan T by binary
harddegree-constraintsandnear-optimallysatisfy searchasin[34]. Ifp > T,x issetto0. The
i,j i,j
acollectionoflinearconstraints: solutiontothefollowingintegerprogramgivesthe
optimumschedule:
THEOREM1 LetG = (J,M,E) beabipartite
graphwith“jobs”J and“machines”M. Let c x C (Cost)
bethecollectionofedge-indexedvectorsy (wiFth i,j i,j ≤
i,j
X
y denotingy wheree = (i,j) E).Suppose
i,j e ∈ x =1 j (Assign)
wearegiven:(i)anintegerrequirementrjforeach i,j ∀
j J andanintegercapacityb foreachi M; Xi,j
i
∈ ∈
(ii)foreachi M,alinearobjectivefunction p x T i (Load)
i,j i,j
∈ ≤ ∀
fi : F → ℜgivenbyfi(y) = j:(i,j) Epi,jyi,j Xj
suchthat0 p ℓ foreachj,and∈(iii)avec-
≤ i,j ≤ i P xi,j bi i (Capacity)
torx withx [0,1]foreache. Then,we ≤ ∀
canef∈ficiFentlyconest∈ructarandomsubgraphofG Xj
x 0,1 i,j
givenbyabinaryvectorX ,suchthat: (a) i,j ∈{ }∀
withprobabilityone,eachj∈ FJ hasdegreeat xi,j =0 ifpi,j >T
∈
leastr ,eachi M hasdegreeatmostb,and
j ∈ i We relax the constraint “x 0,1 (i,j)”
|EfiX(Xe)=−xfei(.x)| < ℓi ∀i;and(b)foralle ∈ E, taotio“nxiL,jP-∈Ca[0p,.1W] ∀e(sio,ljv)e”tthoeoiL,bjPta∈tion{othbetaiL}nP∀threeloapx--
(cid:2)We(cid:3)willnowproveanimportantspecialcaseof timumLPsolutionx ;wenextshowhowSched-
∗
Theorem 1: GAP with individual capacity con- Caproundsx toobtainanintegralsolutionwithin
∗
straints on each machine. This special case cap- theapproximationguarantee.
tures much of the essence of Theorem 1; the full Notethatx [0,1]denotesthe“fraction”of
∗i,j ∈
proof of Theorem 1 is deferred to the final ver- job j assigned to machine i. Initialize X = x .
∗
sionofthiswork.Thecapacityconstraintspecifies The algorithm is composed of several iterations.
4
The random value of the assignment-vectorX at allowableifY isindeednotavertexof .
P
theendofiterationhoftheoverallalgorithmisde- Theabovethreestepscompleteiteration(h+1).
notedbyXh. Eachiterationhconductsarandom-
It is not hard to verify that the invariants (I1)-
izedupdateusingtheRandMoveonthepolytope
(I4) hold true (though the fact that we drop the
ofalinearsystemconstructedfromasubsetofthe
all-importantcapacityconstraintformachinesi
constraintsofLP-Cap.Therefore,byinductionon ∈
M may look bothersome, a moment’s reflection
h,wewillhaveforall(i,j,h)thatE Xh =x . 1
i,j ∗i,j shows that such a machine cannot have a tight
Let J and M denote the set of jobs and ma-
(cid:2) (cid:3) capacity-constraintsinceitssolerelevantjobjhas
chines,respectively. Supposeweareatthebegin-
value Y (0,1)). Since we make at least one
ningofsomeiteration(h+1)oftheoverallalgo- i,j ∈
furtherconstrainttight via RandMove in each it-
rithm: wearecurrentlylookingatthevaluesXh .
i,j eration,invariant(I4)showsthatweterminate,and
Wewillmaintainfourinvariants:
that the number of iterations is at most the ini-
(I1) Onceavariablex getsassignedto0or1,it
i,j tial number of constraints. Let us next present
isneverchanged;
Lemma3,akeylemma:
(I2) Theconstraints(Assign)alwayshold;and
(I3) Once a constraint in (Capacity) becomes
LEMMA 3 InnoiterationisY avertexofthecur-
tight,itremainstight.
rentpolytope .
(IV) Once a constraint is dropped in some itera- P
tion,itisneverreinstated.
Proof. Suppose that in a particular iteration, Y
Iteration (h+1) of Sched-Cap consists of three
isavertexof . Fixthenotationv,M etc.w.r.t.
mainsteps: P k
this iteration; let m = M , and let n denote
k k ′
1. Since we aim to maintain (I1), let us remove theremainingnumberofj|obst|hatareyettobeas-
allXh 0,1 ;i.e.,weprojectXh tothoseco- signed permanently to a machine. Let us lower-
i,j ∈ { }
ordinates(i,j)for whichXh (0,1), to obtain and upper-bound the number of variables v. On
i,j ∈
thecurrentvectorY of“floating”(to-be-rounded) theonehand,wehavev = k m ,bydef-
k 1 · k
variables;let (A Y =u )denotethecurrent initionofthesetsM ;sinceeac≥hremainingjobj
S ≡ h h k P
linearsystemthatrepresentsLP-Cap.(A issome contributesatleasttwovariables(co-ordinatesfor
h
matrixandu isavector;weavoidusing“ ”to Y),wealsohavev 2n. Thusweget
h h ′
S ≥
simplifynotation.) Inparticular,the“capacity”of
machineiin isitsresidualcapacityb ,i.e.,b mi- v n + (k/2) m . (1)
S ′i i ≥ ′ · k
nusthenumberofjobsthathavebeenpermanently k 1
X≥
assignedtoithusfar.
On the other hand, since Y has been assumed to
2. LetY v forsomev;notethatY (0,1)v.
∈ ℜ ∈ be a vertex of , the number t of constraints in
LetM denotethesetofallmachinesiforwhich P
k that are satisfied tightly by Y, must be at least
exactly k of the values Y are positive. We will P
i,j v. How large can t be? Each current job con-
nowdropsomeoftheconstraintsin :
S tributesone(Assign)constrainttot;byour“drop-
(D1) foreachi M ,wedropitsloadandcapac-
∈ 1 pingconstraints”steps(D1),(D2)and(D3)above,
ityconstraintsfrom ;
S thenumberoftightconstraints(“load”and/or“ca-
(D2) foreachi M ,wedropitsloadconstraint
∈ 2 pacity”) contributed by the machines is at most
and rewrite its capacity constraintas x +
x Xh +Xh ,wherej ,j air,ej1the m2+m3+ k 42mk. Thuswehave
i,j2 ≤ ⌈ i,j1 i,j2⌉ 1 2 ≥
twojobsfractionallyassignedtoi. P
v t n +m +m + 2m . (2)
(D3) for each i (M3) for which both its load ≤ ≤ ′ 2 3 k
and capacity∈constraints are tight in , we kX≥4
S
dropitsloadconstraintfrom .
Comparison of (1) and (2) and a moment’s re-
S
3. Let denote the polytope defined by this re- flectionshowsthatsuchasituationispossibleonly
P
duced system of constraints. A key claim that is if:(i)m =m =0andm =m = =0;(ii)
1 3 5 6
···
proveninLemma3belowisthatY isnotavertex the capacity constraintsare tight for all machines
of . We now invoke RandMove(Y, ); this is inM M –i.e.,forallmachines;and(iii)t=v.
2 4
P P ∪
5
However,insuchasituation,thetconstraintsin c = 2; the case of c = 1 is similar to the case
P
constitute the tight assignmentconstraints for the of ℓ = 2 with y + y 1. Here again, sim-
1 2
≤
jobsandthetightcapacityconstraintsforthema- ple algebrayields thatif 0 p ,p ,p T and
1 2 3
≤ ≤
chines,andarehencelinearlydependent(sincethe 0 < y ,y ,y < 1 with y +y +y = c = 2,
1 2 3 1 2 3
total assignment “emanating from” the jobs must thenfor anybinaryvector(X ,X ,X ) of Ham-
1 2 3
equalthe totalassignment“arrivinginto”the ma- ming weight c = 2, p X + p X + p X <
1 1 2 2 3 3
chines). Thuswereachacontradiction,andhence p y +p y +p y +max p ,p ,p . 2
1 1 2 2 3 3 1 2 3
Y isnotavertexof . 2 { }
P Finallywehavethefollowinglemma.
Wenextshowthatthefinalmakespanisatmost
2T withprobabilityone: LEMMA 5 AlgorithmSched-Capcanbederan-
domizedtocreateascheduleofcostatmostC.
LEMMA4 LetX denotethefinalroundedvec-
tor. Algorithm Sched-Cap returns a schedule, Proof. Let Xih,j denote the value of xi,j at iter-
where with probability one: (i) all capacity- ation h. We know for all i,j,h, E[Xh ] = x ,
i,j ∗i,j
constraints on the machines are satisfied, and where x is solution of LP-Cap. Therefore, at
∗i,j
(ii) for all i, X p < x p + the end, we have that the total expected cost in-
j J i.j i,j j ∗i,j i,j
maxj J:x∗ (0,1)p∈i,j. curredis C. The procedurecan be derandomized
∈ i,j∈ P P directlybythemethodofconditionalexpectation,
Proof. Part(i) mostlyfollowsfromthe factthat givingan1-approximationtocost. 2
we never drop any capacity constraint; the only
Lemmas4and5yieldTheorem2.
caretobetakenisformachinesithatendupinM
1
andhencehavetheir capacity-constraintdropped.
3 Schedulingwith Outliers
However, as argued soon after the description of
thethreestepsofaniteration,notethatsuchama- In this section, we consider GAP with outliers
chinecannothaveatightcapacity-constraintwhen and with a hard profit constraint [29]. Formally,
such a constraintwas dropped; hence, even if the theproblemisasfollows,wherex istheindica-
i,j
remainingjobjgotassignedfinallytoi,itscapac- tor variablefor job j to be scheduledon machine
ityconstraintcannotbeviolated. i. Givenm machinesandn jobs, wherejob j re-
Letusnowprove(ii). Fixamachinei. Ifatall quiresprocessingtimeofp inmachinei,incurs
i,j
itsload-constraintwasdropped,itmustbewheni acostofc ifassignedtoiandprovidesaprofitof
i,j
endedupinM1,M2orM3. ThecaseofM1isar- πj ifscheduled,thegoalistominimizethemake-
guedasinthepreviousparagraph.Sosupposei span, T = max x p , subject to the con-
∈ i j i,j i,j
Mgoℓtdfororpspoemde. Lℓe∈tu{s2fi,r3s}tcwohnesnidietrstlhoeadcacsoenℓst=rain2t. astnrdaitnottathlaptrtohfiettotaPlπcost xi,jxisi,jactil,ejaisstaΠt.mostC
Letthe two jobsfractionallyassignedon i atthat OurmaincontrijbujtionPiheir,ejisthefollowing:
pointhaveprocessingtimes(p ,p )andfractional P P
1 2
assignments(y1,y2)oni,where0 ≤ p1,p2 ≤ T, THEOREM6 Foranyconstantǫ > 0,thereisan
and0<y1,y2 <1. Ify1+y2 1,weknowthat efficientalgorithmSched-Outlierthatreturnsa
≤
at the end, the assignment vector X will have at scheduleofprofitatleastΠ,costatmostC(1+ǫ)
mostoneofX1andX2beingone.Simplealgebra andmakespanatmost(2 + ǫ)T,whereT isthe
now shows thatp1X1 +p2X2 < p1y1 +p2y2 + optimalmakespanwithcostCandprofitΠ.
max p ,p as required. If 1 < y + y 2,
1 2 1 2
{ } ≤
then both X and X can be assigned and again, Notethatthisisanimprovementoverthework
1 2
p X +p X <p y +p y +max p ,p . For of[29], thatconstructsa schedulewith makespan
1 1 2 2 1 1 2 2 1 2
{ }
the case ℓ = 3, we knowfrom(I3) and(D3)that 3T withprofitΠandcostC(1+ǫ).Inaddition,our
its capacity-constraintmustbe tight at some inte- approachalso accommodatesfairness, a basic re-
gral value u at that point, and that this capacity- quirementindealingwithoutliers,especiallywhen
constraint was preserved until the end. We must problems have to be run repeatedly. We formu-
havec =1or2here. Letusjustconsiderthecase late fairness via stochastic programs that specify
6
foreachjobj,alower-boundr ontheprobability Notethatx [0,1]denotesthefractionofjob
j ∗i,j ∈
that it gets scheduled. We adapt our approach to j assignedtomachineiinx . Initially, x =
∗ i ∗i,j
honorsuchrequirements: y . Initialize X = x . The algorithm is com-
j∗ ∗ P
posed of several iterations; the random values at
the end of iteration h of the overallalgorithmare
THEOREM7 Thereisanefficientrandomizedal-
denoted by Xh. (Since y is given by the equal-
gorithmthatreturnsascheduleofprofitatleastΠ, j
ity x ,Xh iseffectivelythesetofvariables.)
expectedcostatmost2C andmakespanatmost i i,j
Eachiterationh(exceptperhapsthelastone)con-
3T andguaranteesthatforeachjobj,itissched-
P
ducts a randomized update using RandMove on
uledwithprobabilityr ,whereT istheoptimal
j
a suitable polytope constructed from a subset of
expectedmakespanwithexpectedcostC andex-
the constraints of LP-Out. Therefore, for all h
pectedprofitΠ. Ifthefairnessguaranteeonany
exceptperhapsthe last, we have E Xh = x .
onejobcanberelaxed,thenforeveryfixedǫ>0, i,j ∗i,j
thereisanefficientalgorithmtoconstructasched- A variable Xih,j is said to be floati(cid:2)ng if(cid:3)it lies in
ulethathasprofitatleastΠ,expectedcostatmost (0,1), and a job is floating if it is not yet finally
C(1+1/ǫ)andmakespanatmost(2+ǫ)T. assigned.Thesubgraphof(J,M,E)composedof
thefloatingedges(i,j),naturallysuggeststhefol-
We defer the proof of Theorem 7 to the full lowingnotationatanypointoftime: machinesof
version, and focus on Theorem 6. Guess the op- “degree”k inaniterationarethosewithexactlyk
timum makespan T by binary search as in [34]. floatingjobsassignedfractionally,andjobsof“de-
If p > T, x is set to 0. We guess all as- gree”karethoseassignedfractionallytoexactlyk
i,j i,j
signments (i,j) where c > ǫC, with ǫ = ǫ2. machinesin iterationh. Notethatsince we allow
i,j ′ ′
Anyvalidschedulecanscheduleatmost1/ǫ′pairs yj < 1, there can exist singleton (i.e., degree-1)
with assignment costs higher than ǫC; since ǫ jobswhicharefloating.
′ ′
is a constant, this guessing can be done in poly- Supposeweareatthebeginningofsomeitera-
nomial time. For all (i,j) with c > ǫC, let tion(h+1)oftheoverallalgorithm;sowearecur-
i,j ′
0,1 be a correct guessed assignment. rentlylookingatthevaluesXh . Wewillmaintain
i,j i,j
G ∈ { }
Thesolutiontothefollowingintegerprogramthen thefollowinginvariants:
givesanoptimalsolution: (I1’) Onceavariablex getsassignedto0or1,
i,j
itisneverchanged;
(I2’) Ifj isnotasingleton,then x remains
i i,j
c x C (Cost) atitsinitialvalue;
i,j i,j
≤ P
(I3’) Theconstraint(Profit)alwaysholds;
i,j
X
(I4’) Once a constraintis dropped, it is neverre-
x =y j (Assign)
i,j j ∀ instated.
i
X Algorithm Sched-Outlier starts by initializing
p x T i (Load)
i,j i,j with LP-Out. Iteration (h + 1) consists of four
≤ ∀
Xj majorsteps.
πjyj Π (Profit) 1. Since we aim to maintain(I1’), we removeall
≥
Xj Xih,j ∈ {0,1}; i.e., we project Xh to those co-
xi,j 0,1 i,j; yj 0,1 j ordinates (i,j) for which Xh (0,1), to ob-
∈{ } ∀ ∈{ } ∀ i,j ∈
x =0 ifp >T tainthecurrentvectorZof“floating”variables;let
i,j i,j
(A Z =u )denotethecurrentlinearsystem
xi,j = i,j (i,j)suchthatci,j >ǫ′C S ≡ h h
G ∀ that represents LP-Out. (A is some matrix and
h
u isavector.)
Werelaxtheconstraint“x 0,1 andy h
i,j j
∈ { } ∈
0,1 ” to “x [0,1] and y [0,1]” to ob- 2. LetZ v forsomev; notethatZ (0,1)v.
i,j j
{ } ∈ ∈ ∈ ℜ ∈
tain the LP relaxation LP-Out. We solve the LP Let M and N denote the set of degree-k ma-
k k
to obtain an optimal LP solution x ,y ; we next chinesanddegree-kjobsrespectively,withm =
∗ ∗ k
show how Sched-Outlier roundsx ,y to obtain M and n = N . We will now drop/replace
∗ ∗ k k k
| | | |
theclaimedapproximation. someoftheconstraintsin :
S
7
(D1’) foreachi M ,wedropitsloadconstraint v = km . Therefore,v = k(m +
from ; ∈ 1 n ) + km≥11. Zkbeing a vertex of k,≥v2 2 EkQ.
S k P 2 PP ≤
(D2’) for each i N , we drop its assignment Thus, we must have, n ,m = 0, k 3 and
1 k k
∈ ∀ ≥
constraintfrom ;weincludeoneprofitcon- m =0.Henceeveryfloatingmachinehasexactly
1
S
straint, Z π = Xh π that two floatingjobsassigned to itand everyfloating
replaces thj∈eNc1onsi,tjraijnt (Projfi∈tN).1 (Ni,jotej that job is assigned exactly to two floating machines
at this pPoint, the values XPh are some con- (Config-1ofFigure1).
i,j
stants.) Case 2: Thereareatleast3singletonjobs: We
Thus, the assignment constraints of the single- have n 3. Then the number of linear con-
1
≥
tonjobsarereplacedbyoneprofitconstraint. straintsisEQ = m + n +1. The
k 2 k k 2 k
3. If Z is a vertex of then define fractional last “1” comes from≥considering ≥one profit con-
S straintforthesingPletonjobs. TPhenumberofvari-
assignment of a machine i by h = Z .
1D.efiDneropa ajollbasjsigtonmbeentticgohnt,strifaiintsi∈oPMf jtZh∈eJi,jnoi,n=j- a32b+les, kv 2=k2(nm21k++Pnkk)≥+2mk221(m. Hke+ncentkh)e+sysmt2e1m≥is
tight jobs and maintain a single prPofit constraint, alwaysu≥nderdeterminedandZ cannotbeavertex
P
Z π = Xh π , where of .
J j∈aNre1t∪hJeNnoni,tjighjtjobs. Ijf∈ZN1is∪JnNotyei,tjajvertexof CPase 3: Thereareexactly2singletonjobs: We
N
P P
themodified , thenwhilethereexistsa machine have n = 2. Following similar counting argu-
1
S
i′whosedegreedsatisfieshi′ (d 1 ǫ),drop ments,wecanshowthateachmachinemusthave
≥ − −
theloadconstraintonmachinei. exactly two floating jobs assigned to it and each
′
jobexcepttwoisassignedtoexactlytwomachines
4. Let denote the polytope defined by this re-
P fractionally(Config-2ofFigure1).
duced system of constraints. If Z is not a vertex
Case 4: There is exactly 1 singleton job: We
of , invoke RandMove(Z, ). Else, if there is
P P haven =1.ThenEQ= m + n +
a degree-3 machine with 1 singleton job, assign 1 k 2 k k 2 k
thesingletonjobandthecheaper(lessprocessing 1andv ≥ 12 +n2 + 32n3P+ m≥21 +m2P+≥23m3+
time)ofthenon-singletonjobstoit. Ifthereexist k(m + n ). If Z is a vertex of , then
k 4 2 k k P
two singletonjobs, thendiscard the one with less v ≥EQ. There are few possible configurations
P≤
profit. Assign remaining of the jobs in a way, so thatmightariseinthiscase.
thateachmachinegetsatmostoneextrajob. (i)Onlyonejobofdegree3andonejobofde-
Wenowproveakeylemma,whichshowswhen gree1. Alltheotherjobshavedegree2andallthe
in step 4, Z can possibly be a vertex of the poly- machineshavedegree2. WecallthisConfig-3.
topeunderconsideration. Recallthatinthebipar- (ii)Onlyonemachineofdegree3andonejobof
tite graph G = (J,M,E), we have in iteration degree1. The restof the jobsandmachineshave
(h + 1) that (i,j) E iff Xh (0,1). Any degree2. WecallthisConfig-4.
∈ i,j ∈
jobormachinehavingdegree0isnotpartofG. (iii)Onlyonemachineofdegree1andonejob
ofdegree1.Therestofthejobsandmachineshave
LEMMA8 Letmdenotethenumberofmachine- degree2. WecallthisConfig-5.
nodesinG.Ifm 1,thenZisnotavertexofthe TheseconfigurationsareshowninFig.1. Each
≥ ǫ
currentpolytope. configurationcanhavearbitrarynumberofdisjoint
cycles.
Proof. Let us consider the different possible Inanyconfiguration,if thereisa cyclewithall
configurationsofG, whenZ becomesavertexof tightjobs,thentherealwaysexistsamachinewith
the polytope at step 3. There are several cases total fractional assignment 1 and hence its load
P
toconsiderdependingonthenumberofsingleton constraintcanalwaysbedroppedtomakethesys-
floatingjobsinGinthatiteration. tem underdetermined. So we assume there is no
Case 1: There is no singleton job: We have such cycle in any configurations. Now suppose
n = 0. Then, the number of constraints in is the algorithm reaches Config-1. If there are two
1
S
EQ= m + n . Alsothenumberof non-tightjobs,thenthesystembecomesunderde-
k 2 k k 2 k
floatingvar≥iablesisv =≥ kn . Alternatively, termined.Therefore,therecanbeatmostonenon-
P P k≥2 k
P
8
Figure1:Differentconfigurationsofmachine-jobbipartitegraphatStep3and4
tightjobandonlyonecycle(sayC)withthatnon- constraintcanbedroppedtomakethesystemun-
tightjob.LetChavemmachinesandthusmjobs. derdetermined. Otherwise,thetotalassignmentto
Therefore, x m 1.Thusthereexists thedegree-2machinesfromallthejobsinthecy-
i,j C i,j ≥ −
amachine,such∈thatthetotalfractionalassignment cle is at least m 2+ǫ. Therefore, there exists
ofjobsontPhatmachineis m 1 = 1 1/m. If at least one degre−e-2 machine with fractional as-
≥ m− −
m 1,thenthereexistsamachinewithtotalfrac- signmentatleast m 2+ǫ = 1 1 ǫ 1 ǫ, if
≥ ǫ m− 1 − m−1 ≥ −
tional assignment (1 ǫ). Dropping the load m 1. Thiscomplet−estheproof. − 2
constraint on that m≥achi−ne makes the system un- ≥ ǫ
We next show that the final profit is at least Π
derdetermined.
andthefinalmakespanisatmost(2+ǫ)T:
If the algorithm reaches Config-2, then all the
jobs must be tight for Z to be a vertex. If there
LEMMA 9 LetX denotethefinalroundedvec-
aremmachines,thenthenumberofnon-singleton
tor.AlgorithmSched-Outlierreturnsaschedule,
jobs is m 1. If x + x 1, then fol-
− i1,j1 i2,j2 ≥ wherewithprobabilityone,(i)profitisatleastΠ,
lowing similar averaging argument as in Config-
(ii)foralli, X p < x p +(1+
1, we canshowthemachinewith maximumfrac- ǫ)maxj J:x∗ j∈0J,1 pii,,jj.i,j j ∗i,j i,j
tionaljobassignment,musthaveatotalfractional ∈ iP,j∈{ } P
assignmentatleast1. Otherwise,ifm 1,again
≥ ǫ Proof. (i) This essentially followsfrom the fact
themachinewithmaximumfractionaljobassign-
that whenever assignment constraint on any job
ment, must have a total fractional assignment at
is dropped, its profit constraint is included in the
least 1 ǫ. For Config-3 and 5, if Z is a ver-
global profit constraint of the system. At step 4,
−
tex of , then all jobs must be tight and using
exceptforonecase(Config-2),allthejobsareal-
P
same argument, there exists a machine with frac-
waysassigned,soprofitcannotdecreaseinthose
tionalassignmentatleast(1 ǫ)if thealgorithm
cases. A singleton job (say j ) is dropped, only
− 1
reachesConfig-3andthereexistsamachinewith
when G has two singleton jobs j ,j fraction-
1 2
fractional assignment 1, if the algorithm reaches
ally assigned to i and i respectively, with to-
1 2
Config-5.
tal assignment x + x < 1. Otherwise
i1,j1 i2,j2
IfthealgorithmreachesConfig-4,thenagainall thesystemremainsunderdeterminedfromLemma
jobs must be tight. If the degree-3 machine has 8. Since the job with higher profit is retained,
fractionalassignment at least 2 ǫ, then its load π x +π x max π ,π .
− j1 i1,j1 j2 i2,j2 ≤ { j1 j2}
9
(ii) From Lemma 8 and(D1’), load constraints 4 Max-Min Fair Allocation
are dropped from machines i M and might
∈ 1 Wenowpresentourresultsformax-minfairal-
be dropped from machine i M M . For
∈ 2 ∪ 3 location[4,5,8,14,15]. Therearemgoodsandk
i M , only the remaining job j with Xh >
∈ 1 i,j persons. Eachpersonihasanon-negativeinteger
0, can get fully assigned to it. Hence for i
∈ valuationui,j forgoodj. Thevaluationfunctions
M , its total load is bounded by x p +
1 j ∗i,j i,j arelinear,i.e. u = u foranysetofC
maxj∈J:x∗i,j∈{0,1}pi,j. ForanymachPinei ∈ M2∪ goods. Thegoali,iCstoallojc∈aCteei,ajchgoodtoaper-
M ,iftheirdegreed(2or3)issuchthat,itsfrac- P
3 son such that the “least happyperson is as happy
tional assignment is at least d 1 ǫ, then by
aspossible”: i.e.,min u ismaximized. Oural-
− − i i,C
simple algebra, it can be shown that for any such
gorithmis based uponroundingthe configuration
machine i, its total load is at most x p +
j ∗i,j i,j LPwhichisdescribedinSubsection4.1
(1 + ǫ)maxj J:x∗ 0,1 pi,j. For the remaining
machinescon∈sideir,jw∈{hat}happensatsPtep4. Except
THEOREM11 Givenanyfeasiblesolutiontothe
whenConfig-4isreached,anyremainingmachine
configurationLP,itcanberoundedtoafeasible
igetsatmostoneextrajob,andthusitstotalloadis
integersolutionsuchthateverypersongetsatleast
bWohuenndeCdobnyfig-4jixs∗ir,ejapci,hje+damtastxejp∈J4:,xif∗i,jth∈e{0d,1e}gprei,ej-. Θ( lokglologgkk)fractionoftheoptimumutilitywith
3 machine haPs a fractional assignment at most 1, higqhprobabilityinpolynomialtime.
thenforanyvalueofm,therewillexistadegree-2
machinewhosefractionalassignmentis1,givinga The approximation factor of O( klogk )
loglogk
contradiction.Hence,letj1,j2,j3bethethreejobs is an improvement of the previousqwork of
assignedfractionallytothedegree-3machineiand [5], that achieved an approximation factor of
letj bethe singletonjob, andx +x > 1. O(√klog3k); our bound is near-optimal since
3 i,j1 i,j2
If p p , then the degree-3 machine gets the integrality gap of the configuration LP is
i,j1 ≤ i,j2
j ,j . Elsethedegree-3machinegetsj ,j . The Ω(√k) [8]. However, note that the recent work
1 3 2 3
degree-3machinegets2jobs,butitsfractionalas- ofChakrabarty,ChuzhoyandKhanna[20]hasim-
signmentfromj andj isalreadyatleast1.Since provedtheboundtomǫ. (Alsonotethatm k.)
1 2
≥
thejobwithlessprocessingtimeamongj andj Ourmainpointistoshowtheapplicabilityofour
1 2
areassignedtoi,itsincreaseinloadcanbeatmost types of rounding approaches to this variation of
jx∗i,jpi,j +maxj∈J:x∗i,j∈{0,1}pi,j. 2 theproblemaswell.
In the context of fair allocation, an additional
PFinallywehavethefollowinglemma.
important criterion can be an equitable partition-
ingof goods: we mayimposean upperboundon
LEMMA10 AlgorithmSched-Outliercanbede-
the number of items a person might receive. For
randomizedtooutputascheduleofcostatmost
example, we may want each person to receive at
C(1+ǫ).
most m goods. Theorem1 leadsto the follow-
⌈k⌉
ing:
Proof. In all iterations h, except the last one,
for all i,j, E[Xh ] = x , where x is solu-
i,j ∗i,j ∗i,j
tion of LP-Out. Therefore, before the last itera- THEOREM12 Suppose, in max-min allocation,
tion,wehavethatthetotalexpectedcostincurred wearegivenupperboundsci onthenumberof
isC. Theprocedurecanbederandomizeddirectly itemsthateachpersoni canreceive,inaddition
by the method of conditional expectation, giving totheutilityvaluesui,j. LetT betheoptimum
an1-approximationtocost, justbeforethelastit- max-minallocationvaluethatsatisfiesciforalli.
eration. Nowat thelast iteration,since atmost 1 Then,wecanefficientlyconstructanallocationin
ǫ
jobsareassignedandeachassignmentrequiresat whichforeachpersonitheboundciholdsandshe
mostǫ′C = ǫ2C incost,thetotalincreaseincost receivesatotalutilityofatleastT −maxjui,j.
is atmost ǫC, givingthe requiredapproximation.
Thisgeneralizestheresultof[14],whichyields
2
the “T max u ” value when no boundssuch
j i,j
−
Lemmas9and10yieldTheorem6. asthec aregiven. Toourknowledge,theresults
i
10