Fine-GrainedComplexityandAlgorithmDesignBootCamp

LowerBoundsBasedonSETH

DánielMarx(slidesbyDanielLokshtanov)

SimonsIns;tute,Berkeley,CA

September4,2015

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Tightlowerbounds

HaveseenthatETHcangiveCghtlowerbounds

HowCght?ETH«ignores»constantsinexponent

HowtodisCnguish1.85nfrom1.0001

n?

SAT

Input:Formula!withmclausesovernboolean

variables.

Ques;on:Doesthereexistanassignmenttothe

variablesthatsaCsfiesallclauses?

Note:Inputcanhavesizesuperpolynomialinn!

FastestalgorithmforSAT:2npoly(m)

d-SAT

Hereallclauseshavesize≤d

Inputsize≤nd

Fastestalgorithmfor2-SAT: n+m

Fastestalgorithmfor3-SAT: 1.31n

Fastestalgorithmfor4-SAT: 1.47n

…

Fastestalgorithmford-SAT: 2↑(1− %⁄' )* FastestalgorithmforSAT: 2↑* 

StrongETH

Let +↓' =inf{% :d-SAThasa2↑%* algorithm}

Know:0≤s

d≤ s↓∞ ≤1

ETH:s3>0 SETH: s↓∞ =1

Let +↓∞ = lim┬'→∞  +↓'  

ShowingLowerBoundsunderSETH

YourProblem

Toofastalgorithm?

d-SAT

1.99999↑* 

Thenumberof9’sMUST

beindependentofd

DominaCngSet

Input:nverCces,integerkQues;on:IsthereasetSofatmostkverCces

suchthatN[S]=V(G)?

Naive:nk+1

Smarter:nk+o(1)

AssumingETH:nof(k)no(k)

nk/10

?

nk-1?

SATàk-DominaCngSet

Variables

SAT-formula

kgroups,eachon

n/kvariables.

Onevertexforeachofthe2n/k

assignmentstothevariables

inthegroup.

x y

x yx y x y

x y

Variables

SAT-formula

kgroups,eachon

n/kvariables.

SelecCngonevertexfromeach

cloudcorrspondstoselecCng

anassignmenttothevariables.

Cliques

x y

x yx y x y

x y

Variables

SAT-formula

kgroups,eachon

n/kvariables.

Onevertexperclauseintheformula

EdgeiftheparCalassignment

saCsfiestheclause

SATàk-DominaCngSetanalysis

Toofastalgorithmfork-DominaCngSet:nk-0.01

Foranyfixedk(likek=3)

Theoutputgraphhas

k2n/k+m≤2k⋅2n/kverCces

Ifm≥2n/kthen2nisatmostmk,

whichispolynomial!

Som≤2n/k

( 22⋅2↑*/2 )↑2−0.01 

≤ (22)↑2 ⋅ 2↑*2−0.01/2  

=3( 1.999↑* )

DominaCngSet,wrappingup

AO(n2.99

)algorithmfor3-DominaCngSet,or

aO(n3.99

)algorithmfor4-DominaCngSet,ora

aO(n4.99

)algorithmfor5-DominaCngSet,ora…

…wouldviolateSETH.

Treewidth

•  Wehaveseen:2tnO(1)

,3tnO(1)

,etc.algorithmsandno

2o(t)nO(1)

algorithmsassumingETH.

IndependentSet/Treewidth

Input:GraphG,integerk,tree-decomposiConof

Gofwidth≤t.

Ques;on:DoesGhaveanindependentsetofsizeatleastk?

DP:O(2tn)Cmealgorithm

Canwedoitin1.99tpoly(n)Cme?

Next:Ifyes,thenSETHfails!

IndependentSet/Treewidth

Willreducen-variabled-SATtoIndependentSet

ingraphsoftreewidtht,wheret≤ n+d.

Soa1.99tpoly(N)algorithmforIndependentSet

givesa1.99n+d

poly(n)≤O(1.999n)Cmealgorithm

ford-SAT.

IndependentSetsonanEvenPath

t f t f t f t f t f

True

False

Inindependentset:

NotinsoluCon:

first

True

then

False

d-SAT≤IndependentSetproofbyexample

!=(a∨b∨c)∧(a∨c∨d)∧(b∨c∨d)=(a∨b∨c)∧(a∨c∨d)∧(b∨c∨d)

t f t f t f

t f t f t f

t f t f t f

t f t f t f

a

b

c

d

a c

b

a d

c

b d

c

IndependentSets↔Assignments

!=(a∨b∨c)∧(a∨c∨d)∧(b∨c∨d)=(a∨b∨c)∧(a∨c∨d)∧(b∨c∨d)

t f t f t f

t f t f t f

t f t f t f

t f t f t f

a

b

c

d

a c

b

a d

c

b d

c

True

True

False

False

Butwhataboutthe

firsttruethenfalseindependentsets?

Dealingwithtrueàfalse

a

b

c

d

Clause

gadgets

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

Everyvariableflipstrueàfalseatmostonce!

TreewidthBoundbypicture

t f t f t f

t f t f t f

t f t f t f

t f t f t f

a

b

c

d

a c

b

a d

c

b d

c

…

…

…

…

n

d

Formalproof-exercise

IndependentSet/Treewidth

wrapup

Reducedn-variabled-SATtoIndependentSetin

graphsoftreewidtht,wheret≤ n+d.

A1.99tpoly(N)algorithmforIndependentSet

givesa1.99n+d

poly(n)≤O(1.999n)Cmealgorithm

ford-SAT.

Thus,no1.99talgorithmfor

IndependentSetassumingSETH

AssumingSETH,thefollowingalgorithmsare

opCmal:

–  2t⋅poly(n)forIndependentSet–  3t⋅poly(n)forDominaCngSet

–  ct⋅poly(n)forc-Coloring–  3t⋅poly(n)forOddCycleTransversal–  2t⋅poly(n)forParCConIntoTriangles–  2t⋅poly(n)forMaxCut

–  2t⋅poly(n)for#PerfectMatching

– …

3tlowerboundforDominaCngSet?

Needtoreducek-SATformulasonn-variablesto

DominaCngSetingraphsoftreewidtht,where

3↑4 ≈2↑* 

Sot≈*⁄log 3  ≈0.58*

HiongSet/n

Input:FamilyF={S1,…,S

m}ofsetsoveruniverse

U={v1,…,v

n},integerk.

Ques;on:DoesthereexistasetX⊆Uofsizeat

mostksuchthatforeverySi∈F,S

i∩ X≠∅?

Naivealgorithmrunsin O(2↑n nm)Cme.

Next: 1.41↑n poly(n,m)impliesthatSETHfails

d-SAT≤HiongSet

a

a 

!=(a∨b∨c)∧(a∨c∨d)∧(b∨c∨d)=(a∨b∨c)∧(a∨c∨d)∧(b∨c∨d)

b

b 

c

c 

d

d Budget=4

d-SATvsHiongSet

AcnalgorithmforHiongSetmakesac

2n

algorithmford-SAT.

Since1.412n<1.9999

n,a1.41

nalgorithmfor

HiongSetviolatestheSETH.

Havea2n

algorithmanda1.41nlowerbound.

Next:2n

lowerbound

HiongSet

Foranyfixedϵ>0,willreducek-SATwithnvariablestoHiongSetwithuniversewithat

most(1+5)nelements.)nelements.

Soa1.99nalgorithmforHiongSetgivesa

1.99n(1+5)≤1.999nCmealgorithmfork-SAT

)≤1.999nCmealgorithmfork-SAT

Somedeepmath

Forevery5>0thereexistsanaturalnumberg

suchthat,fort= ⌊g(1+ϵ)⌋↓odd wehave:(4¦⌈4⁄2 ⌉ )≥ 2↑; 

Whyisthisrelevant?

d-SAT≤HiongSet

Groupthevariablesintogroupsofsizeg,andset

t=⌊;(1+ϵ)⌋↓odd .Variables:

g gg g

g

t

t tt t

Elements≤(1+ϵ)⋅variables

Elements:

SoluConbudget⌈4⁄2 ⌉fromeachgroup

Willforce≥⌈4⁄2 ⌉fromeachgroup

àExactly⌈4⁄2 ⌉fromeachgroup

Analyzingagroup

Groupofgvariables

Groupoftelements

2gassignmentstovariables

(4¦⌈4⁄2 ⌉ )subsetsofelementsofsize

exactly⌈4⁄2 ⌉.

InjecCon

ForcingsoluCon⌈4⁄2 ⌉verCcesinagroup?

Addallsubsetsofthegroupofsize⌈4⁄2 ⌉tothefamilyF.

Anysetthatpickslessthan⌈4⁄2 ⌉elementsthegroupmissesaguard.

Anysetthatpicksatleast⌈4⁄2 ⌉elementsfrom

eachgrouphitsalltheguards

Letscallthesesetsguards

Analyzingagroup

Groupofgvariables

Groupoftelements

assignmentstovariables

subsetsofelementsofsizeexactly⌈4⁄2 ⌉.

InjecCon

Whatabouttheelementsubsetsofsize⌈4⁄2 ⌉thatdonotcorrespondtoassignments?

Setsofsize⌊4⁄2 ⌋Addingasetofsize⌊4⁄2 ⌋tothefamilyF

ensuresthatthe«groupcomplement»setisnot

picked.

Allothersetsofsize⌈4⁄2 ⌉inthegroupmaysCll

bepickedinsoluCon.

Forbidsetsofsize⌈4⁄2 ⌉thatdonotcorrespondtoassignments.

d-SAT≤HiongSet

Variables:g g

g gg

t

t tt t

Elements:

potenCal

soluCons

assignments

Want:SoluCons↔SaCsfyingassignments

ForbiddingparCalassignments

Pickanydgroupsofvariables,andconsider

someassignmenttothesevariables.

Ifthisassignmentfalsifies!wewanttoforbidthecorrespondingsetintheHiongSetinstance

frombeingselected.

ForbiddingparCalassignments

Variables

…

…

Badassignment

SetaddedtoFtoforbidthebadassignment

ForbiddingparCalassignments

Foreachbadassignmenttoatmostdgroups,

forbiditbyaddinga«badassignmentguard»

ThisaddsO(nd2gd)=O(n

d)setstoF.

SaCsfyingAssignments↔HiongSets

AsaCsfyingassignmenthasnobad

sub-assignmentsàcorrespondstoahiongset.

Ahiongsetcorrespondstoanassignment.

IfthisassignmentfalsifiedaclauseC,the

assignmentwouldbebadforthe≤dgroupsC

livesin,andmissabadassignmentguard.

HiongSetwrapup

Canreducenvariabled-SATton(1+ϵ)element

HiongSet.

SoacnalgorithmforHiongSetyieldsa(c+5)n)n

algorithmford-SAT.

A1.99nalgorithmforHiongSetwouldviolate

SETH.

Conclusions

SETHcanbeusedtogiveveryCghtrunningCme

bounds.

SETHrecentlyhasbeenusedtogivelower

boundsforpolynomialCmesolvableproblems,

andforrunningCmeofapproximaCon

algorithms.

ImportantOpenProblems

Canweshowa2nlowerboundforSetCover

assumingSETH?

Canweshowa1.00001lowerboundfor3-SAT

assumingSETH?