Graph Routing Problems: Approximation, Hardness ... - TTIC

GraphRoutingProblems:Approximation,Hardness,and

Graph-TheoreticInsights

JuliaChuzhoyToyotaTechnologicalInstituteat

Chicago

GraphRoutingProblems

maximums-tflow

maximummulticommodity flow

maximumnode-disjointpaths

(NDP)

Node-DisjointPaths(NDP)

Input:GraphG,source-sinkpairs(s1,t1),…,(sk,tk).Goal:Routeasmanypairsaspossiblevianode-disjointpaths

s1t1

s2

t2

s3t3



s1t1

s2

t2

s3t3

Solutionvalue:2

OPT:valueofbestpossiblesolution



s1t1

s2

t2

s3t3

Solutionvalue:2

Edge-disjointPaths(EDP):pathsmustbeedge-disjoint


VLSIdesign OpticalNetworks



GraphMinortheory



GraphDecomposition NetworkFlows

GraphMinortheory

GraphSparsifiers

ExcludedGrid

Theorem

GraphCrossingNumber Fixed

ParameterTractability

…



s1t1

s2

t2

s3t3

terminals



s1t1

s2

t2

s3t3

Assumption: Allterminalsaredistinct

n– numberofgraphverticesk– numberofdemandpairs



s1t1

s2

t2

s3t3

Canwesolveitefficiently?

k=1? ✔

k=2?

NDPwithk=2• NP-hardindirectedgraphs[Fortune,Hopcroft,Wyllie'80]

• Efficientlysolvableinundirectedgraphs [Jung‘70,Shiloach ’80,Thomassen ‘80,Robertson-Seymour’90]

s1

t1

s2

t2G

Largerk?

flatgraph

Largerk?

• Constantk:efficientlysolvable [Robertson,Seymour’90]

– Runningtime:f(k)�n2 [Kawarabayashi,Kobayashi,Reed‘12]

f(k) = 222

.

.

.

k

Largerk?

• Constantk:efficientlysolvable [Robertson,Seymour’90]

– Runningtime:f(k)�n2 [Kawarabayashi,Kobayashi,Reed‘12]• NP-hardwhenkispartofinput[Knuth,Karp’74]

Example

G

S T

s t

• SetofdemandpairsisSxT:cansolveefficiently• DemandpairsareaspecificmatchingbetweenSandT:NP-hard

Maxs-tflow

Example

G

S T

• SetofdemandpairsisSxT:cansolveefficiently• DemandpairsareaspecificmatchingbetweenSandT:NP-hard

DealingwithNP-HardnessAnα-approximationalgorithm:

• efficientalgorithm• alwaysproducessolutionsroutingatleastOPT/αdemandpairs.

DealingwithNP-HardnessAnα-approximationalgorithm:

• efficientalgorithm• alwaysproducessolutionsroutingatleastOPT/αdemandpairs.

optimumsolutionvalue

OnApproximationFactors

• Asimplewaytocomparealgorithms– α=1+ε– α=2– α=O(logn)– α=O(√n)– …


• Asimplewaytocomparealgorithms

• Designalgorithmswithgoodapproximationfactorsα

• Establishbestpossibleapproximationfactorαforagivenproblem

Hardnessofapproximationresults





Goal:powerful,simplealgorithmictechniqueswithprovablebounds





Understandingwhatmakesaproblemdifficult

Bettermodelsforreal-lifeproblems

DealingwithNP-Hardness

Multicommodity Flowrelaxation:sendasmuchflowaspossiblebetweenthesi-ti pairs.

Anα-approximationalgorithm:• efficientalgorithm• alwaysproducessolutionsroutingatleastOPT/αdemandpairs.

Example

s1

t1

t2

s2t3

s3

Example

s1

t1

t2

s2

s3

t3

Totalflowthroughavertex

atmost1

Example

s1

t1

t2

s2t3

• send½flowunitoneachofthe3paths• solutionvalue:3/2

s3

NDPsolutionvalue:1

Multicommodity Flows

• Canbecomputedefficiently

• OPTflow ≥OPT

fractionalsolution

multicommodityflowLP-relaxation

integralsolution

Multicommodity Flows

• Canbecomputedefficiently

• OPTflow ≥OPT

• UsetheflowtofindintegralroutingofatleastOPTflow/αdemandpairs

α-approximationalgorithm

multicommodityflowLP-relaxation

LP-roundingtechnique

ApproximationAlgorithm[Kolliopoulos,Stein‘98]WhilethereisapathPwithf(P)>0:• AddsuchshortestpathPtothesolution• ForeachpathP’sharingverticeswithP,setf(P’)to0

ApproximationAlgorithm[Kolliopoulos,Stein‘98]WhilethereisapathPwithf(P)>0:• AddsuchshortestpathPtothesolution• ForeachpathP’sharingverticeswithP,setf(P’)to0

-approximationO(pn)

CanWeDoBetter?

• Notifweusethemaximummulticommodityflowapproach!

BadExamples1 s2 sk…

tk t1t2…

s3

t3


tk t1t2…

s3

t3

OPTflow=k/3OPT=1gap:

�(k) = �(pn)


tk t1t2…

s3

t3

OPTflow=k/3OPT=1gap:

�(k) = �(pn)

Integralitygapoftheflowrelaxation

CanWeDoBetter?

• Notifweusethemaximummulticommodityflowapproach!

• -hardnessofapproximationforany[Andrews,Zhang‘05],[Andrews,C,Guruswami,Khanna,Talwar,Zhang’10]

�(log1/2�� n)✏

ApproximationStatusofNDP

• -approximationalgorithmUntilrecently:– evenonplanargraphs– evenongridgraphshs


O(pn)

�(log1/2�� n) ✏

OnlyNP-hardnessknownforplanargraphsandgrids

s1

t1

s2

t2

s3

t3

NDPinGrids




O(pn)

�(log1/2�� n) ✏

[C,Kim’15]-approximationO(n1/4)

OnlyNP-hardnessknownforplanargraphsandgrids




O(pn)

�(log1/2�� n) ✏

[C,Kim,Li’16]-approximationO(n9/19)





O(pn)

�(log1/2�� n) ✏



[C,Kim,Nimavat ’16]-hardnessofapproximation

forsubgraphs ofgrids2⌦(

plogn)




O(pn)

�(log1/2�� n) ✏



[C,Kim,Nimavat ’16]-hardnessofapproximation

forsubgraphs ofgrids2⌦(

plogn)

WorkinProgress:AlmostpolynomialhardnessforNDPingridgraphs[C,Kim,Nimavat ‘17]

ApproximationStatusofEDP

• -approximationalgorithm[Chekuri,Khanna,Shepherd’06]

• -hardnessofapproximationevenforsubgraphs ofwallgraphs[C,Kim,Nimavat ’16]

O(pn)

2⌦(

plogn)

ApproximationStatusofEDP

• -approximationalgorithm[Chekuri,Khanna,Shepherd’06]

• -hardnessofapproximationevenforsubgraphs ofwallgraphs[C,Kim,Nimavat ’16]

• Workinprogress:almostpolynomialhardnessforEDPonwallgraphs[C,Kim,Nimavat ‘17]

O(pn)

2⌦(

plogn)

SummarysoFar

EDP and NDP do not have reasonableapproximation algorithms, even on planar graphs

Whatifweallowsomecongestion?

EDP/NDPwithCongestion

An-approximationalgorithmwithcongestioncroutes. demandpairswithcongestionatmostc.

uptocpathscanshareanedgeoravertex

↵OPT/�

EDP/NDPwithCongestion

An-approximationalgorithmwithcongestioncroutes. demandpairswithcongestionatmostc.

optimumnumberofpairswithnocongestionallowed

↵OPT/�

EDPwithCongestion• CongestionO(logn/loglogn):constantapproximation[Raghavan,Thompson’87]

• Congestionc: -approximation[Azar,Regev ’01],[Baveja,Srinivasan ’00],[Kolliopoulos,Stein‘04]

• Congestionpoly(loglogn): polylog(n)-approx[Andrews‘10]

• Congestion2: -approximation [Kawarabayashi,Kobayashi’11]

• Congestion14:polylog(k)-approximation[C,‘11]• Congestion2:polylog(k)-approximation[C,Li’12]• polylog(k)-approximationforNDP withcongestion

2 [Chekuri,Ene ’12],[Chekuri,C‘16]

O(n1/c)

O(n3/7)







O(n1/c)

O(n3/7)

Alltheseresultsarebasedonthemulticommodity

flowrelaxation

“Tight”duetoknownhardnessresults







O(n1/c)

O(n3/7) Structuralresultsaboutgraphs

newresultsingraphtheory!







O(n1/c)

O(n3/7)

Edge-DisjointPathswithConstantCongestion

EDPonExpanders

Inastrongenoughexpander,ifthesetofdemandpairsisnottoolarge,canroutealmostallofthemonNode-DisjointPaths!

A B

E0

|E0| � min{|A|, |B|}2

MainIdea:ExploitAlgorithmsforExpanders!

Butourgraphisnothinglikeanexpander

Findexpander-likestructureinthegraphanduseitforrouting!

G

Well-Linkedness[Robertson,Seymour],[Chekuri,Khanna,Shepherd],[Raecke]

terminals

G


SetTofterminalsiswell-linkedinG,iff foranypartition(A,B)ofV(G), A B

|E(A,B)| � min{|A ⇥ T |, |B ⇥ T |}

terminals

G


SetTofterminalsiswell-linkedinG,iff foranypartition(A,B)ofV(G), A B

|E(A,B)| � min{|A ⇥ T |, |B ⇥ T |}

edge/node-disjoint

EDP:Well-LinkedInstances

• Terminals:verticesparticipatinginthedemandpairs

• Aninstanceiswell-linkediff thesetofterminalsiswell-linkedinG.

Theorem [Chekuri,Khanna Shepherd‘04]:anα-approximationalgorithmonwell-linkedinstancesgivesanO(αlog2k)-approximationonanyinstance.

EDP:Well-LinkedInstances

• Terminals:verticesparticipatinginthedemandpairs

• Aninstanceiswell-linkediff thesetofterminalsiswell-linkedinG.

Theorem [Chekuri,Khanna Shepherd‘04]:anα-approximationalgorithmonwell-linkedinstancesgivesanO(αlog2k)-approximationonanyinstance.

Onlytrueifthealgorithmroundstheflowrelaxation

G

MainIdea[Chekuri,Khanna,Shepherd],[Rao,Zhou]

X

EmbedanexpanderovertheterminalsintoG!

terminalsofG

G


X

EmbedanexpanderovertheterminalsintoG!

AnedgeofGmaybelongtoatmost2clusters/paths

GX

1.EmbedanexpanderovertheterminalsintoG


2.Findaroutingonnode-disjointpathsintheexpander

3.Translateitintocongestion-2routinginG


G

EmbeddinganExpanderintoG

Routingonvertex-disjointpathsinXgivesagoodroutinginG!

Xs ts

t

G

MainIdea

X

1.EmbedanexpanderovertheterminalsintoG


2.Findaroutingonnode-disjointpathsintheexpander

3.Translateitintocongestion-2routinginG

Cut-MatchingGame[Khandekar,Rao,Vazirani ’06]

CutPlayer:wantstobuildanexpanderMatchingPlayer:wantstodelayitsconstruction

B3A3B2A2B1A1

Cut-MatchingGame[Khandekar,Rao,Vazirani ’06]

CutPlayer:wantstobuildanexpanderMatchingPlayer:wantstodelayitsconstruction

Thereisastrategyforcutplayer,s.t.afterO(log2n)iterations,wegetanexpander!

EmbeddingExpanderintoGraphG

EmbeddingExpanderintoGraphG

AfterO(log2k)iterations,wegetanexpanderembeddedintoG.

Problem:congestionΩ(log2k)

Path-of-SetsSystem

C2 C3 … CL

…w

C1

APath-of-SetsSystem

• L disjointconnectedclusters• TwodisjointsetsAi,Bi ofwverticesineachclusterCi• Ai Bi iswell-linkedinCi• Foralli,setPi ofwdisjointpathsconnectingBi toAi+1• Allpathsaredisjointfromeachotherandinternallydisjointfromclusters

[

A1 B1

widthwlengthL

A2 B2 A3 B3 AL BL

FromWell-Linkedness toPath-of-SetsC2 C3 … CL

…w

C1

A1 B1 A2 B2 A3 B3 AL BL

Theorem[C,’11],[C,Li’12],[Chekuri,C’13]:SupposeGhasasetofkwell-linkedvertices.Thenwecanefficientlyconstructapath-of-setssysteminGwithparametersLandw,if: w · L48 < O(k)


…w

C1


Theorem[C,’11],[C,Li’12],[Chekuri,C’13]:SupposeGhasasetofkwell-linkedvertices.Thenwecanefficientlyconstructapath-of-setssysteminGwithparametersLandw,if:

Extras:• CanconnectwterminalstoA1 bydisjointpaths• Canmakesuretheyformdemandpairs!

We’lluse:L=O(log2k)w=k/polylog k

w · L48 < O(k)


…w

C1


Thepathsaredisjointfromeachotherandthe

PoS system

Theterminalsformdemandpairs

GiventhePoS,canembedanexpander!

C2 C3 … CL

…w

C1

EmbeddingtheExpander

Ci

Ai Bi iswell-linkedinsideCi

[

Ai Bi

C2 C3 … CL

…w

C1


X

C2 C3 … CL

…w

C1


XExpandervertex thepathcontainingtheterminal

C2 C3 … CLC1


XExpandervertex thepathcontainingtheterminal

C2 C3 … CLC1


Expanderedges? cut-matchinggame!

C2 C3 … CLC1


Expanderedges? cut-matchinggame!

node-disjointpaths

C2 C3 … CLC1


C2 C3 … CLC1


AfterO(log2k)iterations,weobtainanexpanderembeddedintoGwithcongestion2.…

AlgorithmforEDPwC inWell-LinkedInstances

FindaPath-of-SetsSystem

Findvertex-disjointroutingintheexpander

TransformintoroutinginG

EmbedanexpanderintoG

StructuralResult

IfGcontainsalargewell-linkedsetofvertices,thenitcontainsalargePath-of-SetsSystem

Excludedgridtheorem

Large-treewidthgraph

decompositions

Treewidthsparsifiers

Vertexflowsparsifiers

ExcludedGridTheorem[Robertson,Seymour]


Simplegraphs


ComplicatedgraphsSimplegraphs



Treewidth kè DP-basedalgorithmswithrunning

time2O(k)poly(n).

Treewidth:measureshowcomplexthegraphis.



Treewidth:measureshowcomplexthegraphis.

Originaldefinition:Treewidth isthesmallest“width”ofatree-likestructurethatcorrectly“simulates”thegraph.

(Almost)Equivalentdefinition:Treewidth isthecardinalityofthelargestwell-linkedsetofverticesinthegraph.

Treewidth

Trees Low-TreewidthGraphs

High-TreewidthGraphs

ExcludedGridTheorem[Robertson,Seymour‘86]

Ifthetreewidth ofGislarge,thenGcontainsalargegridasaminor.

CanembedalargegridintoGwithno

congestion


Ifthetreewidth ofGislarge,thenitcontainsalargegridminor,so:• Gcontainsmanydisjointcycles• Gcontainsmanydisjointcyclesoflength0modm

• Gcontainsaconvenientroutingstructure• ThesizeofthevertexcoverinGislarge• …

Applications

• Fixedparametertractability• Erdos-Posa typeresults• Graphminortheory– AlgorithmforNDPwherekissmall

• Algorithmsforgraphcrossingnumber• …


Ifthetreewidth ofGisk,thenGcontainsagridofsizef(k)xf(k)asaminor.

• [Robertson,Seymour‘94]:–

• Conjecture[Robertson,Seymour‘94]: Thisistight.

f(k) = O⇣p

k/ log k⌘


Ifthetreewidth ofGisk,thenGcontainsagridofsizef(k)xf(k)asaminor.

Howlargeisf(k)?

ExcludedGridTheorem• [Robertson,Seymour,Thomas‘89]:• [Diestel,Gorbunov,Jensen,Thomassen ‘99] – simplerproof• [Kawarabayashi,Kobayashi‘12],[Leaf,Seymour‘12]:

• [Chekuri,C‘13]:• [C,‘16]:

f(k) = �⇣log1/5 k

⌘

f(k) = �

slog k

log log k

!

f(k) = �⇣k1/98

⌘

f(k) = ⌦⇣k1/19

⌘

C2 C3 … CL

…w

C1

MainIdea widthwlengthL

Thm: IfGcontainsapath-of-setssystemofwidthandlengthΘ(g2),thenthereisa(gxg)-gridminorinG.

[Leaf,Seymour‘12][Chekuri,C’13]

C2 C3 … CL

…w

C1

MainIdea widthwlengthL

Thm: IfGcontainsapath-of-setssystemofwidthandlengthΘ(g2),thenthereisa(gxg)-gridminorinG.

Ghaslargetreewidth

largewell-linkedsetofvertices

largePath-of-Setssystem

ExcludedGridTheorem

NodeDisjointPaths

[Robertson-Seymour‘90]

[C,Chekuri ‘13]

ExcludedGridTheorem

NodeDisjointPaths

[Robertson-Seymour‘90]

[C,Chekuri ‘13]

HistoricalNote• WorkonroutinggaveslightlyweakerstructurethanPath-of-SetsSystem,calledTree-of-SetsSystem

• WelatermodifiedittogetthePath-of-SetssystemfortheExcludedGridtheorem.

• Thisinturnhelpedimproveresultsforroutingproblems.

ApproximationAlgorithms

HardnessofApproximation

GraphTheory

FixedParameterTractability

RoutingProblems

GraphTheory

OpenProblems

• GettingtightboundsfortheExcludedGridTheorem.

• SimpleralgorithmsforNDPwithconstantk• Congestionminimization:– O(logn/loglogn)-approximationalgorithm– Ω(loglogn)-hardnessofapproximation– Integralitygapofthemulticommodity LPrelaxationopen

OpenProblems

• GettingtightboundsfortheExcludedGridTheorem.

• SimpleralgorithmsforNDPwithconstantk• Congestionminimization:– O(logn/loglogn)-approximationalgorithm– Ω(loglogn)-hardnessofapproximation– Integralitygapofthemulticommodity LPrelaxationopen

Thankyou!

Planargraphs?

Graph Routing Problems: Approximation, Hardness ... - TTIC

Documents

Transcript of Graph Routing Problems: Approximation, Hardness ... - TTIC