lower bounds based on seth
TRANSCRIPT
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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?
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SAT
Input:Formula!withmclausesovernboolean
variables.
Ques;on:Doesthereexistanassignmenttothe
variablesthatsaCsfiesallclauses?
Note:Inputcanhavesizesuperpolynomialinn!
FastestalgorithmforSAT:2npoly(m)
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d-SAT
Hereallclauseshavesize≤d
Inputsize≤nd
Fastestalgorithmfor2-SAT: n+m
Fastestalgorithmfor3-SAT: 1.31n
Fastestalgorithmfor4-SAT: 1.47n
…
Fastestalgorithmford-SAT: 2↑(1− %⁄' )* FastestalgorithmforSAT: 2↑*
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StrongETH
Let +↓' =inf{% :d-SAThasa2↑%* algorithm}
Know:0≤s
d≤ s↓∞ ≤1
ETH:s3>0 SETH: s↓∞ =1
Let +↓∞ = lim┬'→∞ +↓'
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ShowingLowerBoundsunderSETH
YourProblem
Toofastalgorithm?
d-SAT
1.99999↑*
Thenumberof9’sMUST
beindependentofd
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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?
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SATàk-DominaCngSet
Variables
SAT-formula
kgroups,eachon
n/kvariables.
Onevertexforeachofthe2n/k
assignmentstothevariables
inthegroup.
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x y
x yx y x y
x y
Variables
SAT-formula
kgroups,eachon
n/kvariables.
SelecCngonevertexfromeach
cloudcorrspondstoselecCng
anassignmenttothevariables.
Cliques
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x y
x yx y x y
x y
Variables
SAT-formula
kgroups,eachon
n/kvariables.
Onevertexperclauseintheformula
EdgeiftheparCalassignment
saCsfiestheclause
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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↑* )
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DominaCngSet,wrappingup
AO(n2.99
)algorithmfor3-DominaCngSet,or
aO(n3.99
)algorithmfor4-DominaCngSet,ora
aO(n4.99
)algorithmfor5-DominaCngSet,ora…
…wouldviolateSETH.
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Treewidth
• Wehaveseen:2tnO(1)
,3tnO(1)
,etc.algorithmsandno
2o(t)nO(1)
algorithmsassumingETH.
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IndependentSet/Treewidth
Input:GraphG,integerk,tree-decomposiConof
Gofwidth≤t.
Ques;on:DoesGhaveanindependentsetofsizeatleastk?
DP:O(2tn)Cmealgorithm
Canwedoitin1.99tpoly(n)Cme?
Next:Ifyes,thenSETHfails!
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IndependentSet/Treewidth
Willreducen-variabled-SATtoIndependentSet
ingraphsoftreewidtht,wheret≤ n+d.
Soa1.99tpoly(N)algorithmforIndependentSet
givesa1.99n+d
poly(n)≤O(1.999n)Cmealgorithm
ford-SAT.
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IndependentSetsonanEvenPath
t f t f t f t f t f
True
False
Inindependentset:
NotinsoluCon:
first
True
then
False
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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
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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?
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Dealingwithtrueàfalse
a
b
c
d
Clause
gadgets
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Everyvariableflipstrueàfalseatmostonce!
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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
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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
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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
– …
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3tlowerboundforDominaCngSet?
Needtoreducek-SATformulasonn-variablesto
DominaCngSetingraphsoftreewidtht,where
3↑4 ≈2↑*
Sot≈*⁄log 3 ≈0.58*
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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
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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
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d-SATvsHiongSet
AcnalgorithmforHiongSetmakesac
2n
algorithmford-SAT.
Since1.412n<1.9999
n,a1.41
nalgorithmfor
HiongSetviolatestheSETH.
Havea2n
algorithmanda1.41nlowerbound.
Next:2n
lowerbound
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HiongSet
Foranyfixedϵ>0,willreducek-SATwithnvariablestoHiongSetwithuniversewithat
most(1+5)nelements.)nelements.
Soa1.99nalgorithmforHiongSetgivesa
1.99n(1+5)≤1.999nCmealgorithmfork-SAT
)≤1.999nCmealgorithmfork-SAT
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Somedeepmath
Forevery5>0thereexistsanaturalnumberg
suchthat,fort= ⌊g(1+ϵ)⌋↓odd wehave:(4¦⌈4⁄2 ⌉ )≥ 2↑;
Whyisthisrelevant?
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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
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Analyzingagroup
Groupofgvariables
Groupoftelements
2gassignmentstovariables
(4¦⌈4⁄2 ⌉ )subsetsofelementsofsize
exactly⌈4⁄2 ⌉.
InjecCon
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ForcingsoluCon⌈4⁄2 ⌉verCcesinagroup?
Addallsubsetsofthegroupofsize⌈4⁄2 ⌉tothefamilyF.
Anysetthatpickslessthan⌈4⁄2 ⌉elementsthegroupmissesaguard.
Anysetthatpicksatleast⌈4⁄2 ⌉elementsfrom
eachgrouphitsalltheguards
Letscallthesesetsguards
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Analyzingagroup
Groupofgvariables
Groupoftelements
assignmentstovariables
subsetsofelementsofsizeexactly⌈4⁄2 ⌉.
InjecCon
Whatabouttheelementsubsetsofsize⌈4⁄2 ⌉thatdonotcorrespondtoassignments?
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Setsofsize⌊4⁄2 ⌋Addingasetofsize⌊4⁄2 ⌋tothefamilyF
ensuresthatthe«groupcomplement»setisnot
picked.
Allothersetsofsize⌈4⁄2 ⌉inthegroupmaysCll
bepickedinsoluCon.
Forbidsetsofsize⌈4⁄2 ⌉thatdonotcorrespondtoassignments.
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d-SAT≤HiongSet
Variables:g g
g gg
t
t tt t
Elements:
potenCal
soluCons
assignments
Want:SoluCons↔SaCsfyingassignments
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ForbiddingparCalassignments
Pickanydgroupsofvariables,andconsider
someassignmenttothesevariables.
Ifthisassignmentfalsifies!wewanttoforbidthecorrespondingsetintheHiongSetinstance
frombeingselected.
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ForbiddingparCalassignments
Variables
…
…
Badassignment
SetaddedtoFtoforbidthebadassignment
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ForbiddingparCalassignments
Foreachbadassignmenttoatmostdgroups,
forbiditbyaddinga«badassignmentguard»
ThisaddsO(nd2gd)=O(n
d)setstoF.
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SaCsfyingAssignments↔HiongSets
AsaCsfyingassignmenthasnobad
sub-assignmentsàcorrespondstoahiongset.
Ahiongsetcorrespondstoanassignment.
IfthisassignmentfalsifiedaclauseC,the
assignmentwouldbebadforthe≤dgroupsC
livesin,andmissabadassignmentguard.
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HiongSetwrapup
Canreducenvariabled-SATton(1+ϵ)element
HiongSet.
SoacnalgorithmforHiongSetyieldsa(c+5)n)n
algorithmford-SAT.
A1.99nalgorithmforHiongSetwouldviolate
SETH.
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Conclusions
SETHcanbeusedtogiveveryCghtrunningCme
bounds.
SETHrecentlyhasbeenusedtogivelower
boundsforpolynomialCmesolvableproblems,
andforrunningCmeofapproximaCon
algorithms.
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ImportantOpenProblems
Canweshowa2nlowerboundforSetCover
assumingSETH?
Canweshowa1.00001lowerboundfor3-SAT
assumingSETH?