optimizing cycles and bases · short course on computational topology january 5, 2011 jeﬀ...

OptimizingCyclesandBasesAMSShortCourseonComputationalTopologyJanuary5,2011

JeffEricksonComputerScienceDepartment

UniversityofIllinoisatUrbana‐[email protected]

Optimalbasesandcycles

GivenatopologicalspaceX,findanyofthefollowing:

Optimalbases:‣ Shortestsetofloopsthatgenerateπ1(X)

‣ ShortestsetofcyclesthatgenerateH1(X)

‣ Minimum‐weightsetofk‐cyclesthatgenerateHk(X)

Optimalcycles:‣ Shortestpath/cyclehomotopictoagivenpath/cycleinX

‣ Shortest1‐cyclehomologoustoagiven1‐cycleinX

‣ Minimum‐weightk‐cyclehomologoustoagivenk‐cycleinX

2

Motivation:localization

3

[Dey,Sun,Cohen‐Steiner2008]

Motivation:surfaceparameterization

4

[Tong,Alliez,Cohen‐Steiner,Desbrun2006]

Anygenus‐ggraphadmitsastochasticembeddingintoaplanargraphwithexpecteddistortionO(logg).

⇒ Lotsofgoodapproximationalgorithmsforgenus‐ggraphs.

Motivation:planarembedding

5

[BorradaileIndykLeeSidiropoulos’07–‘10]

Motivation:minimalsurfaces

6

[Sullivan1990;Dunfield,Hirani2010]

Theminimum‐areasurfaceboundedbyaknotKistheminimum‐cost2‐chainhomologoustoanysurfaceboundedbyK

Preliminaries

7

Combinatorialspaces

‣ Inputspacesarefiniteabstractsimplicial,cubical,orotherCWcomplexes

‣ Cellshavenon‐negativeweights/lengths—“geometry”

‣Weonlyconsiderpaths/cyclesinthe1‐skeleton‣ Otherwise,wedon’tknowhowtocomputeshortestpathsefficiently!

‣ Weonlyconsiderk‐chainsinthek‐skeleton(cellularhomology)

‣Manyalgorithmsinthistalkarerestrictedto2‐manifolds

‣ Homotopyproblemsareundecidableotherwise.

8

Combinatorialsurfaces

‣ Abunchofpolygonsgluedtogetherintoasurface

‣GraphGembeddedona2‐manifoldΣsoeveryfaceisadisk

9

[Riemann1857;Heawood1890;Poincaré1895;Heffter1898;DehnHeegaard1907;Kerékjártó1923;Radó1937;Edmonds1960;

Youngs1963;GrossTucker1987;MoharThomassen2001]


‣ EdgesofGhavenon‐negativeweights

‣Weconsideronlywalkson(orsubgraphsof)thegraphG

10

‣ Theinternalgeometryoffacesisirrelevant/undefined

Inputparameters

Letn=|V|andassumegenusg=O(n1–ε)

Euler’sformula|V|–|E|+|F|=2–2g⟹|E|=O(n)and|F|=O(n)

11

[Euler1726,L'Huilier1811]

Graphduality

12

AnysurfacegraphGhasanaturaldualgraphG*:‣ verticesofG*=facesofG

‣ edgesofG*=edgesofG

‣ facesofG*=verticesofG

u*

v*

f* g*f g

u

v

Tree‐cotreedecomposition

ApartitionoftheedgesofGintothreedisjointsubsets:‣ AspanningtreeT

‣ AspanningcotreeC—C*isaspanningtreeofG*

‣ LeftoveredgesL:=E\(C∪T)—Euler’sformulaimplies|L|=2g

13

[vonStaudt1847;Eppstein2003]

Optimalhomotopybases

14

‣GivenacombinatorialsurfaceΣandabasepointx∈Σ,find2gloopsofminimumtotallengththatgenerateπ1(Σ,x).

Optimalhomotopybasis

15

Aneasyhomotopybasis

‣ Let(T,L,C)beanytree‐cotreedecompositionofG‣ ComputedinO(n)timeviadepth‐orbreadth‐firstsearch

‣ Foranyedgeuv∈L,letℓ(uv):=T[x,u]⋅uv⋅T[v,x]‣ T[s,t]=uniquepathinTfromstot

‣ ComputedinO(1)timeperedge

‣ ℓ(L):={ℓ(e)|e∈L}isabasisforπ1(Σ,x)

‣ Totalconstructiontime:O(n+k)=O(ng)

16

[Eppstein2003]

Agreedyhomotopybasis

‣ Let(T,L,C)bethegreedytree‐cotreedecompositionofG‣ T:=shortestpathtreerootedatx

‣ C*:=maximumspanningtreeofG*wherew(e*):=|ℓ(e)|

‣ ℓ(L):={ℓ(e)|e∈L}istheshortestbasisforπ1(Σ,x)[Erickson,Whittlesey2005;ColindeVerdière2010]

‣ TextbookalgorithmscomputeTandC*inO(nlogn)time[Dijkstra1959;Borůvka1926;Jarník1930=Prim1957;Kruskal1956]

‣ Ifg=O(n1–ε),bothTandC*canbecomputedinO(n)time[Henzinger,Klein,Rao,Subramanian1997;Borůvka1926;Mareš2004]

17

[Erickson,Whittlesey2005]

Summary

‣ Shortestbasisofπ1(Σ,x)canbecomputedinO(n+k)=O(gn)time.

‣ Ifnobasepointspecified,tryeverybasepoint:O(n2)time.

18

Canonicalhomotopybasis

‣ Ahomotopybasiswithanyfixedincidencepattern(dependingonlyonthegenus)

‣ Theshortesthomotopybasisisnotnecessarilycanonical!

19

12

12 3

4

34

Canonicalhomotopybasis

‣ AcanonicalhomotopybasiswithanygivenbasepointcanbecomputedinO(gn)time.[Dey,Schipper1995;Lazarus,Pocchiola,Vegter,Verroust2001]

‣Givenahomotopybasiswithcomplexityk,theshortesthomotopicbasiscanbecomputedinO(g4nk4)time.[ColindeVerdière,Lazarus2005;ColindeVerdière,Lazarus2006]

Open:Cantheshortestcanonicalhomotopybasisbecomputedinpolynomialtime,oristhatproblemNP‐hard?

20

Optimalhomologybases

21

Optimalhomologybasis

‣GivenacombinatorialsurfaceΣ,find2gcyclesofminimumtotallengththatgenerateH1(Σ).

22

Aneasyhomologybasis

‣ Let(T,L,C)beanytree‐cotreedecompositionofG‣ ComputedinO(n)timeviadepth‐orbreadth‐firstsearch

‣ Foranyedgee∈L,letγ(uv)betheuniquecycleinT∪e‣ ComputedinO(1)timeperedge

‣ {γ(e)|e∈L}isabasisforΗ1(Σ)

‣ Totalconstructiontime:O(n+k)=O(ng)

‣ Theshortesthomologybasismaynothavethisform.

23

[Eppstein2003]

Shortesthomologybasis

‣ FixanarbitrarycoefficientfieldR.

‣ EachcycleintheshortesthomologybasisoverRisalsoaloopinthegreedyhomotopybasisatsomebasepoint.[Dey,Sun,Wang2010]

‣ AllO(gn)candidatecycles,alongwithvectorsencodingtheirhomologyclasses,canbecomputedinO(gn2)time.

24


‣ BecauseH1(Σ;R)isavectorspace,findingtheshortestbasisamongthecandidatecyclesisamatroidoptimizationproblem.

‣ Solvedbystandardgreedyalgorithm[Kruskal1956]

‣GreedyalgorithmrunsinO(gnlogn+g3n)time‣ Sorthomologyvectors;useGaussianeliminationtotestindependence


25

B←Øforeachcandidatecycleγinincreasinglengthorder

ifγislinearlyindependentfromBaddγtoB



‣ ForanycombinatorialsurfaceΣandanyfieldR,theshortestbasisforH1(Σ;R)canbecomputedinO(gnlogn+g3n)time.

26

[Chen,Friedman2010;Dey,Sun,Wang2010]


‣ ForanycombinatorialsurfaceΣandanyfieldR,theshortestbasisforH1(Σ;R)canbecomputedinO(gnlogn+g3n)time.

‣ ForanysimplicialcomplexΣandanyfieldR,theshortestbasisforH1(Σ;R)canbecomputedinO(n4)time.

‣ Usepersistenthomologyinsteadoftree/cotreedecompositionsandexplicitelimination.

26

[Chen,Friedman2010;Dey,Sun,Wang2010]

Higher‐dimensionalhomology

‣ Foranyfixedp≥2,forarbitrarycomplexesΣ,computingtheminimum‐volumebasisforHp(Σ;ZZ2)isNP‐hard.‣ volumeofp‐cycle=totalweightofallp‐cells

27

[Chen,Friedman2007;Chen,Friedman2010]

Higher‐dimensionalhomology

‣ Foranyfixedp≥2,forarbitrarycomplexesΣ,computingtheminimum‐volumebasisforHp(Σ;ZZ2)isNP‐hard.‣ volumeofp‐cycle=totalweightofallp‐cells

‣ Foranyp≥2,forarbitrarycomplexesΣ,theminimum‐radiusbasisforHp(Σ;R)canbecomputedinpolynomialtimeforanyfieldR.‣ radiusofp‐cycle=radiusofsmallestballcontainingthecycle

‣ usesexactlythesamegreedyapproach

27

[Chen,Friedman2007;Chen,Friedman2010]

WhataboutZZ?

Openproblem:CantheoptimalbasisforH1(Σ,ZZ)becomputedinpolynomialtime,orisitNP‐hard?

‣ Thecharacterizationofcyclesrequiresthecoefficientringtobeafield!

‣ Thegreedyalgorithmrequiresthecoefficientringtobeafield!

‣ H1(Σ,ZZ)isnotavectorspace,soamaximalsetoflinearlyindependentvectorsisnotnecessarilyabasis,evenforsurfaces.

28


Shortesthomotopicpaths/cycles

29

Shortesthomotopicpaths

30

‣ Computetheshortestpathπ'homotopictoagivenpathπinacombinatorialsurfaceΣ.

‣ Thepathπ'istheprojectionoftheshortestpathintheuniversalcoverΣ~betweentheendpointsofanyliftπofπ.

‣ Thischaracterizationdoesnotimmediatelygiveusanalgorithm;theuniversalcoverΣ~isinfinite!

‣ InsteadweneedtoconstructsomefiniterelevantsubsetofΣ~.

Polygonswithholes

(1) Triangulatethepolygon,andlabeleachdiagonal

31

[HershbergerSnoeyink1994]

A

B

C

D

E

F GH

I

J

K

L

M

NO

P

Q

R

S

TU

V

W

X

YZ12

3

4

Polygonswithholes

(2) Computethecrossingsequenceofπ

32


UTSRR21Z4334YXWVTSRQPJIHHHGFCBBAABDEKLMNOPPJ

A

B

C

D

E

F GH

I

J

K

L

M

NO

P

Q

R

S

TU

V

W

X

YZ12

3

4

Polygonswithholes

(3) Reducethecrossingsequenceofπ

33


UTSRR21Z4334YXWVTSRQPJIHHHGFCBBAABDEKLMNOPPJUTS21Z44YXWVTSRQPJIHGFCBDEKLMNOJUTS21ZYXWVTSRQPJIHGFCBDEKLMNOJ

A

B

C

D

E

F GH

I

J

K

L

M

NO

P

Q

R

S

TU

V

W

X

YZ12

3

4

Polygonswithholes

(4) Computethesleeveofthereducedcrossingsequence

34


Thesleeveistherelevantsubsetoftheuniversalcover!

Polygonswithholes

(5) Computetheshortestpathinthesleevebetweenendpointsofπ

35


Usestandard“funnel”algorithm[Chazelle1982;Lee,Preparata1984]

Polygonswithholes

‣GivenapolygonalpathπwithksegmentsinapolygonPwithholes,theshortestpathinPhomotopictoπcanbecomputedinO(nlogn+kn)time.‣ BuildingtheinitialtriangulationtakesO(nlogn)time.

‣ EverythingelsetakesO(kn)time.

‣ Asimilaralgorithmfindstheshortestcyclefreelyhomotopictoagivencycle,inthesametimebound.

36



GivenapathπwithkedgesinacombinatorialsurfaceΣ,theshortestpathhomotopictoπcanbefoundinO(gn(logn+k))time.

(1) BuildatighthexagonaldecompositionofΣ—O(gnlogn)time

(2)Reducethepathπ—O(gnk)time

(3)BuildrelevantregionofΣ~—O(gnk)time

(4)Findtheshortestpathintherelevantregion—O(gnk)time

37

[ColindeVerdière,Erickson2006]

Tighthexagonaldecomposition

4gcycles,eachasshortaspossibleinitsfreehomotopyclass,thatdecomposeΣinto“right‐angledhexagons”

38


Universalcover

39


40

M.C.Escher,CircleLimitIV:HeavenandHell(1960)

Universalcover

Relevantregion

41


Relevantregion

42


Summary

‣GivenapathπwithkedgesinacombinatorialsurfaceΣ,theshortestpathhomotopictoπcanbefoundinO(gn(logn+k))time.

‣ AsimilaralgorithmcomputestheshortestcyclefreelyhomotopictoagivencycleinO(gnklognk)time.

43


Shortesthomologouscycles

44

Whatcoefficients?

45

Whatcoefficients?

45

Optimalhomologybases:‣ PolynomialtimeforH1withfieldcoefficients(likeZZ2orIR)

‣ NP‐hardforhigher‐dimensionalhomologywithZZ2coefficients

‣ OpenforH1withZZcoefficients,evenforsurfaces

Whatcoefficients?

45

Optimalhomologybases:‣ PolynomialtimeforH1withfieldcoefficients(likeZZ2orIR)

‣ NP‐hardforhigher‐dimensionalhomologywithZZ2coefficients

‣ OpenforH1withZZcoefficients,evenforsurfaces

Optimalhomologouscycles:‣ NP‐hardforH1withZZ2coefficients,evenforsurfaces

‣ NP‐hardforH1withZZcoefficients

‣ PolynomialtimeforanydimensionwithIRcoefficients

‣ PolynomialtimeformanifoldsofanydimensionwithZZcoefficients

ShortestZZ2‐homologouscycles

46

ZZ2‐Homology

47

‣ 1‐chain=subgraphof1‐skeleton

‣ 1‐cycle=evensubgraphof1‐skeleton

‣ 1‐boundary=boundaryoftheunionof2‐cells

‣ Twoevensubgraphsarehomologousifftheirsymmetricdifferenceisaboundarysubgraph

ShortestZZ2‐homologous“cycles”

48

‣Givena{cycle,closedwalk,evensubgraph}γinacombinatorialsurface,findthe{cycle,closedwalk,evensubgraph}ofminimumlengthintheZZ2‐homologyclassofγ.

‣Unfortunately,everyvariantisNP‐hard![Chambers,ColindeVerdière,Erickson,Lazarus,Whittlesey2006;Chambers,Erickson,Nayyeri2009]

‣ Butsolvablein2O(g)nlogntime.[Erickson,Nayerri2011]

NP‐hard

49

Hamiltoniancycleingridgraphs

[Chambers,ColindeVerdière,Erickson,Lazarus,Whittlesey2006]

ZZ2‐homologycoverΣQ

TheuniqueconnectedcoveringspaceofΣwhosegroupofdecktransformationsisΗ1(Σ;ZZ2).

‣ CutΣalonganysystemofloopsℓ1,ℓ2,...,ℓ2gintoadiskD.

‣ Make22gcopies(D,h)ofD,oneforeachhomologyclassh∈Η1(Σ;ZZ2)=(ZZ2)2g.

‣ Foreachclasshandindexi,glue(D,h)to(D,h⊕2i)alongℓi.

‣ Σ�hasn=22gnvertices(v,h)andgenusg=22g(g–1)+1.

50

[Erickson,Nayerri2011]

ZZ2‐homologycoverΣQ

51

00

10

01

11

1 2

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

00

11

01

10

1

1

1

12

2

2

2


TheuniqueconnectedcoveringspaceofΣwhosegroupofdecktransformationsisΗ1(Σ;ZZ2).

ShortestZZ2‐homologousclosedwalks

‣ EveryclosedwalkωinΣthroughavertexvistheprojectionofawalkωfrom(v,0)to(v,[ω])inΣ�.

‣ TheshortestclosedwalkinΣinclasshistheprojectionoftheshortestpathinΣ�fromanynode(v,o)tocorrespondingnode(v,h)

‣ nindependentshortestpathcomputations:O(nn)=O(22gn2)time.[Henzinger,Klein,Rao,Subramanian1997]

‣ parametricshortestpathdatastructures:O(ggnlogn)=2O(g)nlogntime.[Cabello,Chambers,Erickson2011]

52


Moregeneralspaces

‣ Inmoregeneralsimplicialcomplexes,shortesthomologousevensubgraphsareNP‐hardtoapproximatetoanyconstantfactor,evenwhenβ1=1.[ChenFreedman2010]

53

ShortestIR‐andZZ‐homologous“cycles”

54

IR‐chains

55

‣ FixafinitecellcomplexXofanydimensionandanon‐negativeintegerp≥1.

‣ Letm=#p‐cellsandn=#(p+1)‐cells.

‣ Realp‐chainscanbeidentifiedbyrealvectorsc=(c1,c2,...,cm)∈IRm

‣ Theweightofap‐chainis||c||1=Σi|ci|

Minimum‐weightIR‐homologouschains

56

‣ Problem:Givenarealp‐chaincinsomecellcomplexX,computetheminimum‐weightp‐chainxthatisIR‐homologoustoc.

‣ Thisproblemcanbesolvedinpolynomialtimebyexpressingitasalinearprogrammingproblem.

[Sullivan1990;Dey,Hirani,Krishnamoorthy2010]

IR‐homologylinearprogram

57

‣c∈IRm—inputp‐chain

‣x=x+–x–∈IRm—outputp‐chain

‣y∈IRn—(p+1)‐chain

‣ [∂]∈IRm×n—boundarymatrix

minimize Σi(xi++xi

–)subjectto x+–x–=c+[∂]y

x+,x–≥0


Minimum‐weightZZ‐homologouschains

58

‣ SwitchingfromIR‐homologytoZZ‐homologymakestheproblemNP‐hard.[Hirani,Dunfield2010]

‣ Itcanbeformulatedasanintegerprogrammingproblem:

minimize Σi(xi++xi

–)subjectto x+–x–=c+[∂]y

x+,x–≥0x+,x–∈ZZmy∈ZZm


Totalunimodularity

‣ Amatrixistotallyunimodularifeverysquareminorhasdeterminant–1,0,or1.

‣ IfAistotallyunimodular,thenforanyintegervectorb,everyvertexofthepolyhedron{x|Ax≤b}hasintegercoordinates.

‣ Iftheconstraintmatrixforalinearprogramistotallyunimodular,itssolutionisanintegervector.

59

[Veinott,Dantzig1968]

Totalunimodularity

‣ Theboundarymatrix[∂p+1]ofacellcomplexXistotallyunimodularifandonlyifHp(Y,Z)istorsion‐freeforallpuresubcomplexesZ⊂Y⊆XwithdimY=p+1anddimZ=p.

‣ Thisconditioncanbecheckedinpolynomialtime.[Seymour1980]

‣ Satisfiedbyanyorientable(p+1)‐manifoldwithboundary.

‣ IfXmeetsthiscondition,thenoptimalZZ‐homologouschainsinXcanbefoundinpolynomialtimebylinearprogramming.

60

[Dey,Hirani,Krishnamoorthy2010]

IReal1‐cycles

‣ Real1‐chain=functionφ:E→IR‣ Assignsadirectionandnon‐negativevaluetoeachedge

‣ Boundary∂φ(v):=Σu∈Vφ(u→v)–Σw∈Vφ(v→w)

‣ Real1‐cycle=functionφ:E→IRsuchthat∂φ=0

‣Graphtheory/algorithms/optimizationpeoplecallthisacirculation.

61

Minimum‐costcirculations

‣DirectedgraphG=(V,E)whereeveryedgee∈Ehasanon‐negativecapacityc(e)andacost$(e)

‣ AfeasiblecirculationinGisafunctionφ:E→IRsatisfying...‣ capacityconstraint:0≤φ(e)<c(e)foreveryedgee∈E

‣ conservationconstraint:Σu∈Vφ(u→v)=Σw∈Vφ(v→w)foreveryvertexv∈V

‣ Thecostofacirculationφis$(φ)=Σe∈E$(e)⋅φ(e)

‣ Thefeasiblecirculationofminimumcostcanbecomputedinpolynomialtime[Tardos1985;Goldberg,Tarjan1989]

62

Orientablemanifolds

‣ IfXisanorientable(p+1)‐manifold,thedualoftheIR‐homologylinearprogramdescribesaminimum‐costcirculationproblem.‣ G=dual1‐skeletonofX

‣ nverticesdualto(p+1)‐cells

‣ mdirectededgesdualtoorientedp‐cells

‣ everyedgehascapacity1

‣ costofeachedgeisthecoefficientofcforcorrespondingp‐cell

‣ SooptimalZZ‐homologouschainsinorientablemanifoldscanbecomputedinpolynomialtimeviacirculationalgorithms.

‣ MuchfasterthangenericLPalgorithms,bothintheoryandinpractice.

63

[Sullivan1990]

Maximumflowsinsurfacegraphs

64

Maximumflows

65

‣DirectedgraphG=(V,E),capacityfunctionc:E→IR+,twoverticess,t

‣ Afeasible(s,t)‐flowinGisafunctionφ:E→IRsatisfying...‣ capacityconstraint:0≤φ(e)<c(e)foreveryedgee∈E

‣ conservationconstraint:Σu∈Vφ(u→v)=Σw∈Vφ(v→w)foreveryvertexv∈V\{s,t}

‣ Flowφhasvalue|φ|=Σw∈Vφ(s→w)–Σu∈Vφ(u→s)

‣ Acirculationisjustaflowwithvalue0

‣ Aflowisjustarelative1‐cycle

‣Wewantafeasible(s,t)‐flowwithmaximumvalue.[FordFulkerson1955]

Maximumflows

66

[HarrisRoss’55,FordFulkerson‘56]

/54

Maximumflows

67

Howmuchwatercanweinjectatsandextractatt?

s

t

[“Blush”prototyperadiatordesignbyThorunnArnadottir]

Maximumflowalgorithms

‣ IngraphswithnverticesandO(n)edges:

‣O(n2logn)time[Sleator,Tarjan1983;GoldbergTarjan1988;Goldberg,Grigoriadis,Tarjan1991;Hochbaum2008]

‣O(n3/2lognlogU)timeforintegercapacitieslessthanU[Goldberg,Rao1998]

‣ Inplanargraphs:

‣Undirected:O(nloglogn)time[Wulff‐Nilsen2010;Italiano,Sankowski2010]

‣ OnlyrecentlyimprovedfromO(nlogn)time[Reif1983;HassinJohnson1985;Frederickson1987]

‣Directed:O(nlogn)time[Borradaile,Klein2009;Erickson2010]

68

Cocycles

‣ SupposeGisembeddedonanorientablesurfaceΣ

‣ AcocycleisasubgraphλofGdualtoadirectedcycleλ*inG*

‣Defineφ(λ)=Σe∈γφ(e)andc(λ)=Σe∈γc(e)

69

[Chambers,Erickson,Nayyeri2009]

Homologousflows

‣ An(s,t)‐flowφisjustareal1‐chainwithboundary|φ|(t–s)

‣ Easylemma:Flowsφandψarehomologousiffφ(λ)=ψ(λ)foreverycocycleλ.

‣ SetofhomologyclassesofflowsistherelativehomologygroupH1(Σ,{s,t})=IR2g+1

70


Feasiblehomologyclass

‣ Ahomologyclassofflowsisfeasibleifitcontainsafeasibleflow

‣ Lemma:Thehomologyclass[φ]isfeasibleifandonlyifφ(λ)≤c(λ)foreverycocycleλ.

‣ Feasibilityof[φ]canbetestedinO(g2nlog2n)time‣ ShortestpathalgorithminthedualresidualnetworkG*

φ

‣ Findseitherafeasibleflowhomologoustoφoracocyclethatφoverflows.

71

[Venkatesan’83(planar);Klein,Mozes,Weimann2009(planar);Chambers,Erickson,Nayyeri2009]

Linearprogramming,again

‣ Representflowhomologyclasses[φ]asvectorsinIR2g+1

‣φ(λ)≤c(λ)isalinearconstraintonthehomologyvector[φ]

‣ Flowvalue|φ|isalinearfunctionofthehomologyvector[φ]

‣ Sofindingthefeasibleflowhomologyclassofmaximumvalueisalinearprogrammingproblem

72



Findingthefeasiblehomologyclassofmaximumvalueisalinearprogrammingproblem

‣Goodnews:Only2g+1variables!‣ ThefeasiblehomologypolytopeisjusttheprojectionofthefeasibleflowpolytopeintoIR2g+1

‣ Badnews:nO(g)non‐redundantconstraints—toobigtosolveexplicitly!

73



Wehaveamembership/separationoraclethatrunsinO(g2nlog2n)time,sowecansolvetheflowhomologyLPimplicitly:

‣O(g8nlog4nlog2C)timeforintegercapacitiesthatsumtoC,viacentral‐cutellipsoidmethod[ShorNemirovskyYudin‘72;Khachiyan‘79;GrötschelLovászSchrijver‘81,‘93]

‣ gO(g)n3/2timeviamultidimensionalparametricsearch[Cohen‘91;CohenMegiddo‘93;NortonPlotkinTardos‘92;AgarwalSharirToledo‘93]

74


Flowsinsurfacegraphs

‣Maximumflowsinsurface‐embeddedgraphscanbecomputedinO(g8nlog4nlog2C)timeforintegercapacities,oringO(g)n3/2timeforarbitrarycapacities.

‣ Fasterthansparse‐graphalgorithmsbyO(n1/2)wheng=O(1).

‣ Similaralgorithmscomputeminimum‐costcirculationsinthesametimebounds.

‣Dualalgorithmscomputeminimum‐weightIR‐orZZ‐homologouscirculationsincombinatorialsurfacesinthesametimebounds.

75

Thankyou!

76

Thankyou!Thankyou!Thankyou!Thankyou!Thankyou!Thankyou!

optimizing cycles and bases · short course on computational topology january 5, 2011 jeﬀ...

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