Nonuniform Graph Partitioning with Unrelated Weights

0.16 MB·

by  Konstantin Makarychev

Nonuniform Graph Partitioning
with Unrelated Weights
KonstantinMakarychev Yury Makarychev∗
MicrosoftResearch ToyotaTechnologicalInstituteat Chicago
4
1
0
2
Abstract
r
p
We giveabi-criteriaapproximationalgorithmforthe MinimumNonuniformPartitioningproblem,
A
recentlyintroducedbyKrauthgamer,Naor,SchwartzandTalwar(2014). Inthisproblem,wearegiven

8 a graph G = (V,E) on n vertices and k numbersρ1,...,ρk. The goal is to partition the graph into
2 k disjoint sets P1,...,Pk satisfying Pi ρin so as to minimize the number of edges cut by the
| | ≤
partition.OuralgorithmhasanapproximationratioofO(√lognlogk)forgeneralgraphs,andanO(1)
]
S approximationforgraphswith excludedminors. ThisisanimprovementupontheO(logn)algorithm
D of Krauthgamer,Naor,SchwartzandTalwar (2014). Our approximationratio matches the best known
. ratiofortheMinimum(Uniform)k-Partitioningproblem.
s
c Weextendourresultstothecaseof“unrelatedweights”andtothecaseof“unrelatedd-dimensional
[ weights”. In the formercase, differentverticesmay have differentweightsand the weightof a vertex

3 maydependonthesetPi thevertexisassignedto. Inthelattercase,eachvertexuhasad-dimensional
v weightr(u,i)=(r1(u,i),...,rd(u,i))ifuisassignedtoPi. EachsetPi hasad-dimensionalcapacity
99 c(i)=(c1(i),...,cd(i)). ThegoalistofindapartitionsuchthatPu∈Pir(u,i)≤c(i)coordinate-wise.
6
0 1 Introduction
.
1
0
We study the Minimum Nonuniform Partitioning problem, which was recently proposed by Krauthgamer,
4
1 Naor, Schwartz and Talwar (2014). We are given a graph G = (V,E), parameter k and k numbers (ca-
: pacities) ρ ,...,ρ . Ourgoalistopartition thegraph Gintok pieces(bins)P ,...,P satisfying capacity
v 1 k 1 k
i constraints P ρ n so as to minimize the number of cut edges. The problem is a generalization of the
X | i| ≤ i
Minimum k-Partitioning problem studied by Krauthgamer, Naor,andSchwartz (2009), in which all bins
r
a haveequalcapacityρ = 1/k.
i
The problem has many applications (see Krauthgamer et al. 2014). Consider an example in cloud
computing: Imagine that we need to distribute n computational tasks – vertices of the graph – among k
machines, each with capacity ρ n. Different tasks communicate with each other. The amount of commu-
i
nication between tasks uand v equals theweight ofthe edges between the corresponding vertices uand v.
Ourgoal istodistribute tasks amongk machines subject to capacity constraints soastominimize thetotal
amountofcommunication betweenmachines.1
Theproblem isquitechallenging. Krauthgamer etal. (2014) notethatmanyexisting techniques donot
workforthisproblem. Particularly,itisnotclearhowtosolvethisproblemontreegraphs2andconsequently
∗SupportedbyNSFCAREERawardCCF-1150062andNSFgrantIIS-1302662.
1Inthisexample,weneedtosolveavariantoftheproblemwithedgeweights.
2Ouralgorithmgivesaconstantfactorbi-criteriaapproximationfortrees.
1
how to use Ra¨cke’s (2008) tree decomposition technique. Krauthgamer et al. (2014) give an O(logn)
bi-criteria approximation algorithm for the problem: the algorithm finds a partition P ,...,P such that
1 k
P O(ρ n) for every i and the number of cut edges is O(lognOPT). The algorithm first solves a
i i
| | ≤
configuration linearprogram andthenusesanewsophisticated methodtoroundthefractional solution.
In this paper, we present a rather simple SDP based O(√lognlogk) bi-criteria approximation al-
gorithm for the problem. We note that our approximation guarantee matches that of the algorithm of
Krauthgamer, Naor,andSchwartz (2009) for the the Minimum k-Partitioning problem (which is a special
case of Minimum Nonuniform Partitioning, see above). Our algorithm uses a technique of “orthogonal
separators” developed by Chlamtac,Makarychev, andMakarychev (2006)and later used byBansal, Feige,
Krauthgamer, Makarychev, Nagarajan, Naor, and Schwartz (2011) for the Small Set Expansion problem.
Usingorthogonal separators, itisrelatively easytogetadistribution overpartitions P ,...,P suchthat
1 k
{ }
E[P ] O(ρ n) for all i and the expected number of cut edges is O( lognlog(1/ρ )OPT) where
i i min
| | ≤ p
ρ = min ρ . The problem is that for some i, P may be much larger than its expected size. The al-
min i i i
gorithm of Krauthgamer et al. (2014) solves a similar problem by first simplifying the instance and then
groupingpartsP into“mega-buckets”. Weproposeasimplerfix: Roughlyspeaking,ifasetP containstoo
i i
manyvertices, weremovesome of these vertices and re-partition the removed vertices into k pieces again.
Thusweensurethatallcapacity constraints are(approximately) satisfied. Itturnsoutthateveryvertexgets
removedaconstantnumberoftimesinexpectation. Hence,there-partitioning stepincreasesthenumberof
cut edges only by a constant factor. Another problem is that 1/ρ may be much larger than k. To deal
min
with this problem, we transform the SDP solution (eliminating “short” vectors) and redefine thresholds ρ
i
sothat1/ρ becomesO(k).
min
Our technique is quite robust and allows us to solve more general versions of the problem, Nonuni-
form Graph Partitioning with unrelated weights and Nonuniform Graph Partitioning with unrelated d-
dimensional weights.
Minimum Nonuniform Graph Partitioning with unrelated weights captures the variant of the problem
where we assign vertices (tasks/jobs) to unrelated machines and the weight of a vertex (the size of the
task/job) depends onthemachineitisassigned to.
Definition 1.1 (Minimum Nonuniform Graph Partitioning with unrelated weights). We are given a graph
G = (V,E) on nvertices and anatural number k 2. Additionally, wearegiven k normalized measures
≥
µ ,...,µ on V (satisfying µ (V) = 1) and k numbers ρ ,...,ρ (0,1). Our goal is to partition the
1 k i 1 k
∈
graph into k pieces (bins) P , ..., P such that µ (P ) ρ so as to minimize the number of cut edges.
1 k i i i
≤
SomepiecesP maybeempty.
i
Wewillonly consider instances of Minimum Nonuniform GraphPartitioning thathave afeasible solu-
tion. WegiveanO ( lognlogmin(1/ρ ,k))bi-criteria approximation algorithm fortheproblem.
ε min
p
Theorem1.2. Foreveryε >0,thereexistsarandomizedpolynomial-timealgorithmthatgivenaninstance
of Minimum Nonuniform Graph Partitioning with unrelated weights finds a partition P ,...,P satisfying
1 k
µ (P ) 5(1+ε)ρ . TheexpectedcostofthesolutionisatmostD OPT,whereOPT istheoptimalvalue,
i i i
≤ ×
D = O ( lognlogmin(1/ρ ,k))andρ = min ρ . Forgraphswithexcluded minorsD = O (1).
ε min min i i ε
p
Nonuniform Graph Partitioning with unrelated d-dimensional weights further generalizes the problem.
Inthisvariantoftheproblem,weassumethatwehavedresources(e.g. CPUspeed,randomaccessmemory,
disk space, network). Each piece P has c (i) units of resource j 1,...,d , and each vertex urequires
i j
∈ { }
r (u,i) units ofresource j 1,...,d when itis assigned topiece P . Weneed topartition the graph so
j i
∈ { }
that capacity constraints for all resources are satisfied. The d-dimensional version of Minimum (uniform)
2
k-Partitioning was previously studied by Amiretal. (2014). In their problem, all ρ = 1/k are the same,
i
andr ’sdonotdepend oni.
j
Definition 1.3 (Minimum Nonuniform Graph Partitioning with unrelated d-dimensional weights). We are
givenagraphG = (V,E)onnvertices. Additionally,wearegivennon-negativenumbersc (i)andr (u,i)
j j
for i 1,...,k , j 1,...,d , u V. Our goal is to find a partition of V into P ,...,P subject to
1 k
∈ { } ∈ { } ∈
capacity constraints r (u,i) c (i)foreveryiandj soastominimizethenumberofcutedges.
Pu∈V j ≤ j
Wepresentabi-criteria approximation algorithm forthisproblem.
Theorem 1.4. For every ε > 0, there exists a randomized polynomial-time algorithm that given an in-
stance of Minimum Nonuniform GraphPartitioning withunrelated d-dimensional weights finds a partition
P ,...,P satisfying
1 k
r (v,i) 5d(1+ε)c (i) foreveryiandj.
X j ≤ j
v∈V
TheexpectedcostofthesolutionisatmostD OPT,whereOPT istheoptimalvalue,D = O (√lognlogk).
ε
×
Forgraphswithexcluded minorsD = O (1).
ε
We note that this result is a simple corollary of Theorem 1.2 welet µ′(u) = max (r (u,i)/c (i)) and
i j j j
thenapplyourresulttomeasuresµ (u) = µ′(u)/µ′(V)(wedescribethedetailsinAppendixC).
i i i
Weremarkthatouralgorithms workifedgesinthegraphhavearbitrary positive weights. However,for
simplicity ofexposition, wedescribe thealgorithms forthesetting wherealledgeweightsareequaltoone.
Todealwitharbitrary edgeweights,weonlyneedtochangetheSDPobjectivefunction.
OurpaperstrengthenstheresultofKrauthgameretal. (2014)intwoways. First,itimprovestheapprox-
imation factor from O(logn)to O(√lognlogk). Second, it studies considerably moregeneral variants of
the problem, Minimum Nonuniform Partitioning with unrelated weights and Minimum Nonuniform Parti-
tioning with unrelated d-dimensional weights. Webelieve that these variants are very natural. Indeed, one
ofthemainmotivationsfortheMinimumNonuniformPartitioningproblemisitsapplicationstoscheduling
andloadbalancing: intheseapplications, thegoalistoassign taskstomachinessoastominimizethetotal
amountofcommunication betweendifferent machines, subject tocapacity constraints. Theconstraints that
we study in the paper are very general and analogous to those that are often considered in the scheduling
literature. We note that the method developed in Krauthgamer et al. (2014) does not handle these more
generalvariants oftheproblem.
2 Algorithm
SDPRelaxation. OurrelaxationfortheproblemisbasedontheSDPrelaxationfortheSmallSetExpansion
(SSE) problem of Bansaletal. (2011). We write the SSE relaxation for every cluster P and then add
i
consistency constraints similar to constraints used in Unique Games. For every vertex u and index i
∈
1,...,k ,weintroduce avectoru¯ . Intheintegralsolution, thisvectorissimplytheindicator variablefor
i
{ }
the event “u P ”. It is easy to see that in the integral case, the number of cut edges equals (1). Indeed,
i
∈
if u and v lie in the same Pj, then u¯i = v¯i for all i; if u lies in Pj′ and v lies in Pj′′ (for j′ = j′′) then
6
u¯ v¯ 2 = 1fori j′,j′′ and u¯ v¯ 2 = 0fori / j′,j′′ . TheSDPobjectiveistominimize(1).
i i i i
k − k ∈{ } k − k ∈ { }
Weaddconstraint(2)sayingthatµ (P ) ρ . Wefurtheraddspreadingconstraints(4)fromBansaletal.
i i i
≤
(2011)(seealsoLouisandMakarychev(2014)). Thespreadingconstraintsabovearesatisfiedintheintegral
solution: If u / P , then u¯ = 0and both sides of theinequality equal 0. Ifu P , then the left hand side
i i i
∈ ∈
equalsµ (P ),andtherighthandsideequalsρ .
i i i
3
We write standard ℓ2-triangle inequalities (6) and (7). Finally, we add consistency constraints. Every
2
vertexumustbeassigned tooneandonlyoneP ,henceconstraint (5)issatisfied. Weobtain thefollowing
i
SDPrelaxation.
SDPRelaxation
k
1
min u¯ v¯ 2 (1)
2X X k i − ik
i=1(u,v)∈E
subjectto
u¯ 2µ (u) ρ foralli [k] (2)
Xk ik i ≤ i ∈
u∈V
u¯ ,v¯ µ (v) u¯ 2ρ (3)
Xh i ii i ≤ k ik i
v∈V
forallu V, i [k] (4)
∈ ∈
k
u¯ 2 = 1 forallu V (5)
Xk ik ∈
i=1
u¯ v¯ 2+ v¯ w¯ 2 u¯ w¯ 2 forallu,v,w V, i [k] (6)
i i i i i i
k − k k − k ≥ k − k ∈ ∈
0 u¯ ,v¯ u¯ 2 forallu,v V, i [k] (7)
i i i
≤ h i ≤ k k ∈ ∈
SmallSetExpansionandOrthogonalSeparators. Ouralgorithmusesatechniquecalled“orthogonal
separators”. ThenotionoforthogonalseparatorswasintroducedinChlamtac,Makarychev, andMakarychev
(2006), where it was used in an algorithm for Unique Games. Later, Bansaletal. (2011) showed that the
following holds. If the SDP solution satisfies constraints (3), (4), (6), and (7), then for every ε (0,1),
∈
δ (0,1),andi [k],thereexistadistortionD = O ( lognlog(1/(δρ ))),andaprobabilitydistribution
i ε i
∈ ∈ p
over subsets of V such that for arandom set S V (“orthogonal separator”) distributed according to this
i
⊂
distribution, wehaveforα= 1/n,
µ (S ) (1+ε)ρ (always);
i i i
• ≤
Forallu,Pr(u S ) [(1 δ)α u¯ 2,α u¯ 2];
i i i
• ∈ ∈ − k k k k
Forall(u,v) E,Pr(u S ,v / S ) αD u¯ v¯ 2.
i i i i i
• ∈ ∈ ∈ ≤ ·k − k
WeletD = max D . Thisstatement wasprovedinBansaletal.(2011)implicitly, soforcompleteness we
i i
proveitintheAppendix—seeTheoremA.1. Forgraphswithexcludedminorsandbounded genusgraphs,
D = O (1).
ε
Algorithm. Letus examine a somewhat na¨ıve algorithm for the problem inspired by the algorithm of
Bansaletal. (2011) for Small Set Expansion. Weshall maintain the set of active (yet unassigned) vertices
A(t). Initially,allverticesareactive,i.e. A(0) = V. Ateverystept,wepickarandomindexi 1,...,k
∈{ }
andsampleanorthogonal separatorS (t)asdescribedabove. WeassignallactiveverticesfromS (t)tothe
i i
binnumberi:
P (t+1) = P (t) (S (t) A(t)),
i i i
∪ ∩
4
Si(t) A(t) partitioned partitioned
∩
vertices vertices
5(1+ε)ρ1 5(1+ε)ρ1
5(1+ε)ρi 5(1+ε)ρi
5(1+ε)ρ2 5(1+ε)ρk 5(1+ε)ρ2 5(1+ε)ρk
Si(t) A(t)
∩
P (t) P (t) ... P(t) ... P (t) P (t+1) P (t+1) ... P(t+1) ... P (t+1)
1 2 i k 1 2 i k
reactivated reactivated
reactivated reactivated
Figure1: ThefigureshowshowweupdatesetsP (t)initerationt. Inthisfigure,rectanglesrepresentlayers
i
of vertices in sets P (t) (on the left) and P (t +1) (on the right). All vertices in these layers are inactive
i i
(theyarealreadypartitioned). Bluehorizontal linesshowcapacity constraints. Intheexampleshowninthe
figure, weaddsetS (t) A(t)toP (t). Themeasure oftheobtained setisgreater than 5(1+ε)ρ ,andso
i i i
∩
weremovethe twobottom layers from P (t) (S (t) A(t)) (the removed layers areshown in blue). We
i i
∪ ∩
get aset of measure at most5(1+ε)ρ . Vertices inthe removed layers are reactivated after the iteration is
i
over.
andmarkallnewlyassigned vertices asinactive i.e.,weletA(t+1) = A(t) S (t). Westopwhentheset
i
\
ofactiveverticesA(t)isempty. Weoutputthepartition = P (T),...,P (T) ,whereT istheindexof
1 k
P { }
thelastiteration.
We can show that the number of edges cut by the algorithm is at most O(D OPT), where D is
×
the distortion of orthogonal separators. Furthermore, the expected weight of each P is O(ρ ). However,
i i
weights ofsomepieces maysignificantly deviate from theexpectation and maybemuchlarger thanρ . So
i
we need to alter the algorithm to guarantee that all sizes are bounded by O(ρ ) simultaneously. We face
i
a problem similar to the one Krauthgamer, Naor,SchwartzandTalwar (2014) had to solve in their paper.
Their solution is rather complex and does not seem towork in theweighted case. Here, wepropose avery
simplefixforthena¨ıvealgorithm wepresented above. Weshallstore vertices ineverybininlayers. When
weaddnewverticestoabinatsomeiteration, weputtheminanewlayerontopofalreadystoredvertices.
Now, if the weight of the bin number i is greater than 5(1+ε)ρ , we remove bottom layers from this bin
i
sothat its weight isatmost5(1+ε)ρ . Then wemarkthe removed vertices asactive and jumptothe next
i
iteration. It is clear that this algorithm always returns a solution satisfying µ (P ) 5(1 +ε)ρ for all i.
i i i
≤
But now we need to prove that the algorithm terminates, and that the expected number of cut edges is still
bounded byO(D OPT).
×
Beforeproceeding totheanalysis, wedescribe thealgorithm indetail.
AlgorithmforNonuniformPartitioningwithUnrelatedWeights
Input: a graph G = (V,E) on n vertices; a positive integer k n; a sequence of numbers ρ ,...,ρ
1 k
≤ ∈
(0,1)(withρ + +ρ 1);weightsµ : V R+ (withµ (V) = 1).
1 k i i
··· ≥ →
Output: apartitioning ofverticesintodisjoint setsP ,...,P suchthatµ (P ) 5(1+ε)ρ .
1 k i i i
≤
ThealgorithmmaintainsapartitioningofV intoasetofactiveverticesA(t)andksetsP (t),...P (t),
1 k
•
which wecall bins. For every inactive vertex u / A(t), we remember its depth in the bin it belongs
∈
5
to. Wedenotethedepthbydepth (t). Ifu A(t),thenweletdepth (t) = .
u ∈ u ⊥
Initially, setA(0) = V;andP (0) = ∅,depth (t)= foralli;t = 0.
• i u ⊥
whileA(t) = ∅
• 6
1. Pickanindexi 1,...,k uniformly atrandom.
∈ { }
2. Sampleanorthogonal separator S (t) V withδ = ε/4asdescribed inSection2.
i
⊂
3. Store allactive vertices from the setS (t)inthebin number i. Ifµ (P (t) (S (t) A(t)))
i i i i
∪ ∩ ≤
5(1+ε)ρ ,thensimplyaddtheseverticestoP (t+1):
i i
P (t+1) = P (t) (S (t) A(t)).
i i i
∪ ∩
Otherwise,findthelargest depthdsuchthatµ (P (t+1)) 5(1+ε)ρ ,where
i i i
≤
P (t+1) = u P (t) :depth (t) d (S (t) A(t)).
i { ∈ i u ≤ }∪ i ∩
Inotherwords,addtothebinnumberiverticesfromS (t) A(t)andremoveverticesfromthe
i
∩
bottom layerssothattheweightofthebinisatmost5(1+ε)ρ .
i
4. Ifweputatleastonenewvertexinthebiniatthecurrentiteration, thatis,ifA(t) S (t) = ∅,
i
∩ 6
thensetthedepthofallnewlystoredvertices to1;increase thedepthofallothervertices inthe
biniby1.
5. Update the set of active vertices: letA(t+1) = V P (t+1) and depth (t+1) = for
\Sj j u ⊥
u A(t+1). Lett = t+1.
∈
SetT = tandreturnthepartitioning P (T),...,P (T).
1 k
•
Note that Step 3 is well defined. We can always find an index d such that µ (P (t + 1)) 5(1 + ε)ρ ,
i i i
≤
becauseford = 0,wehaveP (t+1) = S (t) A(t)andthus
i i
∩
µ(P (t+1)) = µ (S (t) A(t)) µ (S (t)) (1+ε)ρ < 5(1+ε)ρ ,
i i i i i i i
∩ ≤ ≤
bythefirstpropertyoforthogonal separators.
Analysis. We will first prove Theorem 2.1 that states that the algorithm has approximation factor
D = O ( lognlog(1/ρ ))onarbitrary graphs, andD = O (1)ongraphs excluding aminor. Thenwe
ε min ε
p
willshowhowtoobtainD = O (√lognlogk)approximationonarbitrarygraphs(seeAppendixB).Tothis
ε
end,wewilltransformtheSDPsolutionandredefinemeasuresµ andcapacitiesρ sothatρ δ/k,then
i i min
≥
applyTheorem2.1. ThenewSDPsolutionwillsatisfyallSDPconstraintsexceptpossiblyforconstraint(5);
itwillhoweversatisfyarelaxedconstraint
k
u¯ 2 [1 δ,1] forallu V. (5′)
Xk ik ∈ − ∈
i=1
ThusinTheorem 2.1,wewillassumeonly that thesolution satisfies theSDPrelaxation withconstraint (5)
replaced byconstraint (5′).
6
Theorem2.1. Thealgorithm returnsapartitioning P (T),...,P (T)satisfying µ (P ) 5(1+ε)ρ . The
1 k i i i
≤
expected numberofiterations ofthealgorithm isatmostE[T] 4n2k+1andtheexpected numberofcut
≤
edgesisatmostO(D SDP)= O(D OPT),whereD = O ( lognlog(1/ρ ))isthedistortion of
ε min
× × p
orthogonal separators; ρ = min ρ . Ifthegraphhasanexcluded minor,thenD = O (1)(theconstant
min i i ε
depends ontheexcluded minor).
WeassumeonlythattheSDPsolutiongiventothealgorithmsatisfiestheSDPrelaxationwithconstraint
(5)replacedbyconstraint (5′).
As we mentioned earlier, the algorithm always returns a valid partitioning. We need to verify that the
algorithm terminates in expected polynomial time, and that itproduces cuts of cost at most O(D OPT)
×
(seealsoRemarkC.1).
Thestate of thealgorithm atiteration t isdetermined bythe sets A(t), P (t),...,P (t)and the depths
1 k
of the elements. We denote the state by (t) = A(t),P (t),...,P (t),depth(t) . Observe that the
1 k
C { }
probability that the algorithm is in the state ∗ at iteration (t + 1) is determined only by the state of the
C
algorithm atiteration t. Itdoes notdepend on t(given (t)). Sothestates ofthealgorithm form aMarkov
C
random chain. Thenumber ofpossible states isfinite(since thedepth ofeveryvertexisbounded byn). To
simplify the notation, weassume that for t T, (t) = (T). Thisisconsistent withthe definition of the
algorithm — if we did not stop the algorith≥m atCtime T,Cit would simply idle, since A(t) = ∅, and thus
S (t) A(t) = ∅fort T.
i
∩ ≥
Weareinterestedintheprobabilitythataninactivevertexuwhichliesinthetoplayerofoneofthebins
(i.e.,u / A(t)anddepth (t)= 1)isremovedfromthatbinwithinmiterations. Welet
∈ u
f(m,u, ∗) = Pr( t [t ,t +m] s.t. u A(t) (t )= ∗,depth (t ) = 1).
C ∃ ∈ 0 0 ∈ | C 0 C u 0
Thatis,f(m,u, ∗)istheprobabilitythatuisremovedfromthebiniatoneoftheiterationst [t ,t +m]
0 0
C ∈
given that at iteration t the state of the algorithm is ∗ and uis in the top layer of the bin i. Note that the
0
C
probability abovedoesnotdependont andthusf(m,u, ∗)iswelldefined. Welet
0
C
f(m)= maxmaxf(m,u, ∗).
u∈V C∗ C
Ourfistlemmagivesaboundontheexpected number ofstepsonwhichavertexuisactiveintermsof
f(m).
Lemma2.2. Foreverypossible stateofthealgorithm ∗,everyvertexu,andnaturalnumbert ,
0
C
t0+m k
Pr(u A(t) (t ) = ∗) . (8)
X ∈ | C 0 C ≤ (1 2δ)α(1 f(m 1))
t=t0 − − −
Proof. Thelefthandsideofinequality(8)equalsexpectednumber(conditionedon (t ) = ∗)ofiterations
0
C C
tintheinterval[t ,t +m]atwhichuisactivei.e.,u A(t). Ourgoalistoupperboundthisquantity.
0 0
∈
Initially, at time t , u is active or inactive. At every time t when u is active, u is thrown in one of the
0
binsP withprobability atleast(here, weusethattheSDPsolutionsatisfiesconstraint (5′))
i
k
1 (1 2δ)α
(1 δ)α u¯ 2 − .
k X − k ik ≥ k
i=1
So the expected number of iterations passed since u becomes active till u is stored in one of the bins and
thusbecomesinactiveisatmostk/((1 2δ)α).
−
7
Suppose that uis stored in abin i atiteration t, then u P (t+1) and depth (t+1) = 1. Thus, the
∈ i u
probabilitythatuisreactivatedtilliterationt +mi.e.,theprobabilitythatforsomeτ [(t+1),t +m]
0 0
∈ ⊂
[(t+1),(t+1)+(m 1)], u A(τ)isatmostf(m 1). Consequently, theexpectednumberofiterations
− ∈ −
t [t ,t +m]atwhichuisactiveisbounded by
0 0
∈
k 1 k f(m) k f2(m) k
· + · + · + = .
(1 2δ)α (1 2δ)α (1 2δ)α ··· (1 2δ)α(1 f(m))
− − − − −
Wenowshowthatf(m) 1/2forallm.
≤
Lemma2.3. Forallnaturalm,f(m) 1/2.
≤
Proof. Weprovethislemmabyinduction onm. Form = 0,thestatementistrivialasf(0) = 0.
Consider an arbitrary state ∗, bin i∗, vertex u, and iteration t0. Suppose that (t0) = ∗, u Pi∗(t0)
C C C ∈
and depth (t ) = 1 i.e., u lies in the top layer in the bin i∗. We need to estimate the probability that u
u 0
is removed from the bin i∗ till iteration t +m. The vertex u is removed from the bin i∗ if and only if at
0
someiterationt t ,...,t +m 1 ,uis“pushedaway”fromthebinbynewvertices(seeStep2ofthe
0 0
∈ { − }
algorithm). Thishappensonlyiftheweightofverticesaddedtothebini∗ atiterations t ,...,t +m 1
0 0
{ − }
plus the weight of vertices in the first layer of the bin at iteration t exceeds 5(1+ε)ρ . Since the weight
0 i
of vertices in the first layer is at most (1 + ε)ρ , the weight of vertices added to the bin i∗ at iterations
i
t0,...,t0+m 1 mustbegreaterthan4(1+ε)ρi∗.
{ − }
Wecomputetheexpectedweightofverticesthrowninthebini∗ atiterations t t ,...,t +m 1 .
0 0
∈ { − }
Letus introduce some notation: M = t ,...,t +m 1 ; i(t) is the index i chosen by the algorithm at
0 0
{ − }
theiteration t. LetXM,i∗ betheweightofverticesthrowninthebini∗ atiterations t M. Then,
∈
E(cid:2)XM,i∗ |C(t0) = C∗(cid:3) = Eh X µi∗(cid:0)Si∗(t)∩A(t)(cid:1) |C(t0) = C∗i (9)
t∈M
s.t.i(t)=i∗
= X XPr(cid:0)i(t) = i∗ andv ∈ Si∗(t)∩A(t) | C(t0) = C∗(cid:1)µi∗(v).
t∈Mv∈V
Theevent“i(t) = i∗ andv Si∗(t)”isindependent fromtheevent“v A(t)and (t0)= ∗”. Thus,
∈ ∈ C C
Pr i(t) = i∗ andv Si∗(t) A(t) (t0)= ∗
(cid:0) ∈ ∩ | C C (cid:1)
= Pr i(t) = i∗ andv Si∗(t) Pr v A(t) (t0)= ∗ .
(cid:0) ∈ (cid:1)· (cid:0) ∈ | C C (cid:1)
Sincei(t)ischosen uniformlyatrandom in 1,...,k ,wehavePr(i(t) = i∗) = 1/k. Then,byproperty 2
{ }
oforthogonal separators, Pr(v Si∗(t) i(t) = i∗) α v¯i∗ 2. Weget
∈ | ≤ k k
Pr i(t) =i∗ andv Si∗(t) A(t) (t0)= ∗ αkv¯i∗k2 Pr v A(t) (t0)= ∗ .
(cid:0) ∈ ∩ | C C (cid:1) ≤ k · (cid:0) ∈ | C C (cid:1)
8
Wenowplugthisexpression in(9)anduseLemma2.2,
E[XM,i∗ | C(t0) = C∗] ≤ X αkv¯i∗kk2µi∗(v) · XPr(cid:0)v ∈ A(t) | C(t0) = C∗(cid:1)
v∈V t∈M
α v¯i∗ 2µi∗(v) k
k k
≤ X k · (1 2δ)α(1 f(m 1))
v∈V − − −
v¯i∗ 2µi∗(v)
= k k .
X (1 2δ)(1 f(m 1))
v∈V − − −
FthienaSlDly,Pocbosnesrtvreainthta(t21).−Hefn(cme,−E[1X)M≥,i∗1/2(bty0)th=e in∗d]ucti2vρei∗h/y(p1othe2sδis),.aBnydMPavrk∈oVvk’sv¯ii∗nke2qµuai∗li(tvy), ≤ ρi∗ by
| C C ≤ −
2ρi∗ 1
Pr XM,i∗ 4(1+ε)ρi∗ ,
(cid:0) ≥ (cid:1) ≤ 4(1 2δ)(1+ε)ρi∗ ≤ 2
−
sinceδ = ε/4. Thisconcludes theproof.
Asanimmediatecorollary ofLemmas2.2and2.3,wegetthatforallu V,
∈
∞ m
2k 4k
Pr(u A(t)) = lim Pr(u A(t)) . (10)
X ∈ m→∞X ∈ ≤ (1 2δ)α ≤ α
t=0 t=0 −
ProofofTheorem2.1. We now prove Theorem 2.1. We first bound the expected running time. At every
iteration ofthealgorithm t < T,thesetA(t)isnotempty. Hence,using(10),weget
∞ ∞
4k
E[T] E A(t) +1= Pr(v A(t))+1 n +1= 4kn2+1.
≤ hX| |i XX ∈ ≤ · α
t=0 v∈V t=0
Wenowupperboundtheexpectedsizeofthecut. Foreveryedge(u,v) E weestimatetheprobability
∈
that (u,v) is cut. Suppose that (u,v) is cut. Then, u and v belong to distinct sets P (T). Consider the
i
iteration tatwhichuandv areseparated thefirsttime. Apriori,therearetwopossible cases:
1. At iteration t, u and v are active, but only one of the vertices u or v is added to some set P (t+1);
i
theothervertexremainsinthesetA(t+1).
2. At iteration t, u and v are in some set P (t), but only one of the vertices u or v is removed from the
i
setP (t+1).
i
It is easy to see that, in fact, the second case is not possible, since if u and v were never separated before
iteration t, then u and v must have the same depth (i.e., depth (t) = depth (t)) and thus uand v may be
u v
removedfromthebinionlytogether.
Consider the first case, and assume that u P (t +1) and v A(t +1). Here, as in the proof of
i(t)
∈ ∈
Lemma2.3,wedenotetheindexichosenatiterationtbyi(t). Sinceu P (t+1)andv A(t+1),we
i(t)
∈ ∈
haveu S (t)andv / S (t). Write
i(t) i(t)
∈ ∈
Pr(u,v A(t); u S (t); v / S (t)) =
i(t) i(t)
∈ ∈ ∈
= Pr(u,v A(t)) Pr(u S (t); v / S (t))
i(t) i(t)
∈ · ∈ ∈
k
Pr(u S (t); v / S (t) i(t) = i)
i i
= Pr(u,v A(t)) ∈ ∈ | .
∈ ·X k
i=1
9
We replace Pr(u,v A(t)) with Pr(u A(t)) Pr(u,v A(t)), and then use the inequality Pr(u
∈ ∈ ≥ ∈ ∈
S (t); v / S (t)) αD u¯ v¯ 2, which follows from the third property of orthogonal separators. We
i i i i
∈ ≤ k − k
get
k
1
Pr(u,v A(t); u S (t); v / S (t)) Pr(u A(t)) αD u¯ v¯ 2 .
∈ ∈ i(t) ∈ i(t) ≤ ∈ ×(cid:16)k X k i − ik (cid:17)
i=1
Thus,theprobability thatuandv areseparatedatiterationtisupperboundedby Pr(u A(t))+Pr(v
(cid:16)
∈ ∈
A(t)) 1 k αD u¯ v¯ 2 .Theprobabilitythattheedge(u,v)iscut(atsomeiteration)isatmost
(cid:17)×(cid:16)k Pi=1 k i− ik (cid:17)
∞ k
1
Pr(u A(t))+Pr(v A(t)) αD u¯ v¯ 2
(cid:16)X ∈ ∈ (cid:17)×(cid:16)k X k i − ik (cid:17) ≤
t=0 i=1
k k
8k 1
αD u¯ v¯ 2 = 8 D u¯ v¯ 2.
≤ α (cid:16)k X k i − ik (cid:17) X k i − ik
i=1 i=1
To bound the first term on the left hand side we used inequality (10). We get the desired bound on the
expectednumberofcutedges:
k
Pr((u,v)iscut) 8 D u¯ v¯ 2 = 16D SDP,
X ≤ X X k i − ik ·
(u,v)∈E (u,v)∈E i=1
whereSDP istheSDPvalue.
References
A.Amir,J.Ficler,R.Krauthgamer,L.Roditty,andO.SarShalom.MultiplyBalancedk-Partitioning.LATIN
2014.
N. Bansal, U. Feige, R. Krauthgamer, K. Makarychev, V. Nagarajan, J. Naor, and R. Schwartz. Min-max
GraphPartitioning andSmallSetExpansion. FOCS2011.
E. Chlamtac, K. Makarychev, and Y. Makarychev. How to Play Unique Games Using Embeddings. FOCS
2006.
R.Krauthgamer, J.Naor,R.Schwartz.Partitioning graphsintobalanced components. SODA2009.
R.Krauthgamer,J.Naor,R.SchwartzandK.Talwar(2014).Non-UniformGraphPartitioning.SODA2014.
A.LouisandK.Makarychev. Approximation AlgorithmforSparsestk-Partitioning. SODA2014.
H.Ra¨cke.Optimalhierarchical decompositions forcongestion minimization innetworks. STOC2008.
10