The Importance of Being Biased Irit Dinur S. Safra (some slides borrowed from Dana Moshkovitz) Irit...

The Importance of Being The Importance of Being BiasedBiased

The Importance of Being The Importance of Being BiasedBiased

Irit DinurIrit Dinur

S. SafraS. Safra

(some slides borrowed from Dana Moshkovitz)(some slides borrowed from Dana Moshkovitz)

Irit DinurIrit Dinur

S. SafraS. Safra

(some slides borrowed from Dana Moshkovitz)(some slides borrowed from Dana Moshkovitz)

©©S.SafraS.Safra

VERTEX-COVERVERTEX-COVER

InstanceInstance:: an undirected graph an undirected graph G=(V,E)G=(V,E).. ProblemProblem:: find a set find a set CCVV of of minimal sizeminimal size

s.t. for any s.t. for any (u,v)(u,v)EE, either , either uuCC or or vvCC..

InstanceInstance:: an undirected graph an undirected graph G=(V,E)G=(V,E).. ProblemProblem:: find a set find a set CCVV of of minimal sizeminimal size

s.t. for any s.t. for any (u,v)(u,v)EE, either , either uuCC or or vvCC..

Example:Example:

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Minimum VC NP-hardMinimum VC NP-hard

ObservationObservation:: Let Let G=(V,E)G=(V,E) be an undirected be an undirected graph. The complement graph. The complement V\CV\C of a vertex- of a vertex-cover cover CC is an independent-set of is an independent-set of GG..

ProofProof:: Two vertices outside a vertex-cover Two vertices outside a vertex-cover cannot be connected by an edge. cannot be connected by an edge.

©©S.SafraS.Safra

VC Approximation AlgorithmVC Approximation Algorithm

C C E’ E’ E E whilewhile E’ E’

dodo let let (u,v)(u,v) be an arbitrary edge of be an arbitrary edge of E’E’ C C C C {u,v} {u,v} remove from remove from E’E’ every edge every edge

incident to either incident to either uu or or vv.. return return CC..

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DemoDemo

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Polynomial TimePolynomial Time

C C E’ E’ E E whilewhile E’ E’ dodo

let let (u,v)(u,v) be an arbitrary edge of be an arbitrary edge of E’E’ C C C C {u,v} {u,v} remove from remove from E’E’ every edge incident every edge incident

to either to either uu or or vv return return CC

O(n2)

O(1)

O(n)

O(n2)

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CorrectnessCorrectness

The set of vertices our algorithm returns is The set of vertices our algorithm returns is clearly a vertex-cover, since we iterate clearly a vertex-cover, since we iterate until every edge is covered.until every edge is covered.

©©S.SafraS.Safra

How Good an Approximation is How Good an Approximation is it?it?

Observe the set of edges our algorithm choosesObserve the set of edges our algorithm chooses

any VC contains 1 in each any VC contains 1 in each

our VC contains both, hence at most twice as large our VC contains both, hence at most twice as large

no common vertices!

no common vertices!

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How well can VC be How well can VC be Approximated?Approximated?

Upper boundUpper bound A little better (w/hard work) : A little better (w/hard work) : 2-o(1)2-o(1)

Hardness resultsHardness results PreviouslyPreviously:: 7/67/6 ThmThm:: NP-hard to approximate to NP-hard to approximate to

within within 10105-21 5-21 1.36 (> 4/3)1.36 (> 4/3) ConjectureConjecture:: NP-hard to within NP-hard to within 2- 2- >0>0

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((m,rm,r)-co-partite Graph )-co-partite Graph G=(MG=(MR, E)R, E)

Comprise Comprise m=|M|m=|M| cliques of size cliques of size r=|r=|R|R|:: E E {(<i,j {(<i,j11>, <i,j>, <i,j22>) | i>) | iM, jM, j11≠≠jj2 2 R}R}

mm

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mm

Gap Independent-SetGap Independent-Set

InstanceInstance:: an an (m,r)(m,r)-co-partite -co-partite graph graph G=(MG=(MR, E)R, E)

ProblemProblem:: distinguish distinguish betweenbetween GoodGood: : IS(G) = mIS(G) = m BadBad: every set : every set I I V V s.t. s.t. |I||I|

> > mm contains an edge contains an edge

ThmThm:: IS( r, IS( r, ) ) is NP-hard as long as is NP-hard as long as r r ( 1 / ( 1 / ))cc for some constantfor some constant c c

h-Clique

h-Clique

hIS(r, h, hIS(r, h, )) h

mm

, r and h constant!, r and h constant!

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Hardness of Vertex-CoverHardness of Vertex-Cover

ProblemProblem:: the size of the size of GG’s Vertex-Cover is’s Vertex-Cover isGoodGood: : (1-1/r) (1-1/r) |G| |G|BadBad: : (1- (1- /r) /r) |G| |G|

Resulting in a factor smaller than Resulting in a factor smaller than 1+1/r1+1/rWe showWe show:: A reduction from A reduction from hIS(G)hIS(G) to a graph to a graph

HHGoodGood:: BadBad::

implying NP-hardness of implying NP-hardness of 4/34/3 factor for factor for Vertex-CoverVertex-Cover

1hI S(G) m I S(H) H o(1) H9 1hI S(G) m I S(H) H o(1) H9 1I S(G) m I S(H) H o(1) H3 1I S(G) m I S(H) H o(1) H3

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mm

Encode Encode I.S.I.S.’s ’s RepresentativesRepresentatives

Replace clique Replace clique iiMM by a set of vertices, by a set of vertices,11 for each bit of some binary-code of for each bit of some binary-code of RRApply the Apply the

long-codelong-codeApply the Apply the long-codelong-code

supposedly encoding supposedly encoding ISIS’s representative ’s representative jjRR

supposedly encoding supposedly encoding ISIS’s representative ’s representative jjRR IS assignment:

1 if in the IS 0 if out

IS assignment:1 if in the IS

0 if out

Edges: two vertices that can’t both be 1 in any encoding of

an IS of G

Edges: two vertices that can’t both be 1 in any encoding of

an IS of G

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Long-Code of Long-Code of RR

One bit (vertex) for every subset of One bit (vertex) for every subset of RR

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Long-Code of RLong-Code of R

One bit (vertex) for every subset of One bit (vertex) for every subset of RR

to encode an element to encode an element eeRR

00 00 11 11 11

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VVLCLC = M = M P[R] P[R]VVLCLC = M = M P[R] P[R]

Long-Code to Co-partite’s Long-Code to Co-partite’s I.S.I.S.

EELCLC = {(F = {(F11,F,F22) | F) | F1 1 FF22 E}E}EELCLC = {(F = {(F11,F,F22) | F) | F1 1 FF22 E}E}

mm

what edges do we have within a part?what edges do we have within a part?

non-intersecting: F1non-intersecting: F1F2 F2 ==non-intersecting: F1non-intersecting: F1F2 F2 ==

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Between parts: assume a co-matchingBetween parts: assume a co-matchingIn each part: intersectingIn each part: intersecting

ProblemProblem: all : all FF, , |F||F| >½r >½r areare ISIS

mm

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Weighted GraphsWeighted Graphs

Assign Assign weights to to VV - hence - hence G = (V, E, G = (V, E, ))

Consider a probability distribution Consider a probability distribution :V:V[0,1][0,1]and let the size of a set of vertices beand let the size of a set of vertices be

hencehence

Easily reducible to graphs with no weightsEasily reducible to graphs with no weights

Assign Assign weights to to VV - hence - hence G = (V, E, G = (V, E, ))

Consider a probability distribution Consider a probability distribution :V:V[0,1][0,1]and let the size of a set of vertices beand let the size of a set of vertices be

hencehence

Easily reducible to graphs with no weightsEasily reducible to graphs with no weights

i.s.I VI S(G) max (I )

i.s.I VI S(G) max (I )

v

v S

(S) Pr v S (v)

v

v S

(S) Pr v S (v)

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Consider the p-biased product distribution p:

DefDef: : The probability of a subset The probability of a subset FF

and for a family of subsets and for a family of subsets

Consider the p-biased product distribution p:

DefDef: : The probability of a subset The probability of a subset FF

and for a family of subsets and for a family of subsets

Biased Long-CodeBiased Long-Code

F R\ FRp F p (1 p) F R\ FRp F p (1 p)

RF p

R Rp p

F

Pr F F

RF p

R Rp p

F

Pr F F

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discriminating against large subsetsdiscriminating against large subsets

p <½-p <½- FF‘s of size ‘s of size >½r >½r Vanish Vanish

solves the the >½>½ problem, however…problem, however…

solves the the >½>½ problem, however…problem, however…

mm

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mm

ProblemProblem: consistent large : consistent large subsetssubsets

Si Si

Sj Sj

what if any pair of cliques i & j have a

pair of large subsets Si & Sj that are all-wise

consistent

what if any pair of cliques i & j have a

pair of large subsets Si & Sj that are all-wise

consistent

almost all subsetshave a representative

in those subsets

almost all subsetshave a representative

in those subsets

©©S.SafraS.Safra

Fix a large Fix a large llT T and and l=r·2ll=r·2lTTFix a large Fix a large llT T and and l=r·2ll=r·2lTT

m’m’m’m’mm

}l)T(a|}F,T{]l[:a{'R T1 }l)T(a|}F,T{]l[:a{'R T1

The (The (m’,r’m’,r’)-co-partite Graph )-co-partite Graph GGBB

B'm,l

VB

B'm,

l

VB

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m’m’m’m’mm



}l)T(a|}F,T{B:a{'R T1 }l)T(a|}F,T{B:a{'R T1

B'm,l

VB

B'm,

l

VB

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VerticesVertices::Fix a large Fix a large llT T and and l=r·2ll=r·2lTT letlet BB=V=V(l)(l),, m’ =| m’ =|BB|| For everyFor every BBBB

EdgesEdges:: Let Let B’ = VB’ = V(l-1)(l-1):: B B11=B’=B’{v{v11}, B}, B22=B’=B’{v{v22} } (a (a11, a, a22) ) EEBB for for aa11RRB1B1, , aa22RRB2B2 if if aa11||B’B’ a a22||B’B’

oror (v(v11, v, v22))EE and and aa11(v(v11) = a) = a22(v(v22) = T) = T

1B TR a:B T,F | a (T) l 1B TR a:B T,F | a (T) l

PropProp: : IS(G) = m IS(G) = m IS(G IS(GBB) > m’ (1-2) > m’ (1-2––

(l(lTT))))

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Now Apply Long-Code to Now Apply Long-Code to GGBB

The final graphThe final graph H = ( H = (BB P[ R P[ RB B ], E], EBBLCLC, , ))

VerticesVertices:: one one B B BB and a subset and a subset F F P[RP[RBB]]

EdgesEdges:: EEBBLCLC (F (F11, F, F22) ) for for FF11 P[RP[RBB11]],, FF22P[RP[RBB22] ] ifif

FF11 F F22 E EBB

WeightsWeights:: (F) = (F) = pp(F) / |(F) / |BB||

Prop (Completeness)Prop (Completeness)::IS(H) IS(H) p · IS(G p · IS(GBB) / m’ ) / m’

Thm (Soundness)Thm (Soundness): : For p≤(3-5)/2,hIS(G) < hIS(G) < m m IS(H) < P IS(H) < P + + ’ ’ [for[for p p 1/3 1/3::

PP=p=p22]]

Proof: given an IS in GB, I, consider the

corresponding set of singletons in H; take monotone extension

Proof: given an IS in GB, I, consider the

corresponding set of singletons in H; take monotone extension

©©S.SafraS.Safra

m’m’m’m’mm



}l)T(a|}F,T{B:a{'R T1 }l)T(a|}F,T{B:a{'R T1

B'm,l

VB

B'm,

l

VB

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Soundness for GSoundness for GBB

LemmaLemma:: an IS of size an IS of size m’m’ in in GGBB implies IS of implies IS of size size ½½mm in G in G

ProofProof: For an IS : For an IS I’I’ of of GGBB

Fix a Fix a B’B’ in in VVl-1l-1 for which (such must exist) for which (such must exist)

Let Let I = { v | (<B’,v>, a) I = { v | (<B’,v>, a) I’ and a(v) = T } I’ and a(v) = T }II is an IS of is an IS of GG of size of size ½½mm

VB ,v B

la(B,a) I ' ,a(v) T 2rPr

VB ,v B

la(B,a) I ' ,a(v) T 2rPr

v G( B' ,v ,a) I ' ,a(v) T 2rPr

v G( B' ,v ,a) I ' ,a(v) T 2rPr

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IS IS of size of size PP even in even in BadBad CaseCase

Partition Partition VV into into VV11 and and VV22

For every block For every block BB, let, let aa11 assign assign TT to to VV11 and and FF to to VV22

aa22 assign assign TT to to VV22 and and FF to to VV11

and letand letBB = { F = { F {a {a11, a, a22} }} }

These These BB‘s‘s form an form an IS IS of weight of weight pp22 in in HH

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Erdös-Ko-RadoErdös-Ko-Rado

DefDef:: A family of subsets A family of subsets P[R] P[R] is is tt-intersecting if for every-intersecting if for everyFF11, F, F22 ,, |F |F11 F F22| | t t

ThmThm[Wilson,Frankl,Ahlswede-Khachatrian]:[Wilson,Frankl,Ahlswede-Khachatrian]:For a For a tt-intersecting -intersecting ,,

wherewhere

CorollaryCorollary: : pp(() > P) > P is not is not 22-intersecting-intersecting

p p i,ti

( ) max (A ) p p i,ti

( ) max (A )

i,tA F | F 1,...,2i t i t i,tA F | F 1,...,2i t i t

p p i,2i

( ) max (A ) p p i,2i

( ) max (A ) PP = =

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Soundness ProofSoundness Proof

Important ObservationImportant Observation::Assume Assume I I is a is a maximalmaximal ISIS in in HH II’s intersection with any block’s intersection with any block

I[B] I[B] I I P[ R P[ RB B ]]is is monotonemonotone and and intersectingintersecting

It follows:It follows: qq(I[B])(I[B]) is a non-decreasing function of is a non-decreasing function of qq

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We proveWe prove: If : If H H has an has an IS I IS I s.t. s.t. (I) > P(I) > P + 500+ 500 thenthen hIS(G) > hIS(G) > mm

LetLet BB[I][I] == { B | { B | pp(I[B]) > P(I[B]) > P + 250+ 250 } }

PropProp:: | |BB[I]| > 250[I]| > 250 | |BB||

ObservationObservation:: VB' B B' V \ B'

l 1

V

B' B B' V \ B'l 1

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(Naïve) Plan:(Naïve) Plan: Find, for every Find, for every B B BB [I] [I],, a a distinguisheddistinguished

block-assignment block-assignment aaBB LetLet

VVB’B’ ={ v | B’ ={ v | B’{v} {v} BB [I] [I] andand a aB’B’{v}{v}

(v)=T}(v)=T}

There must be There must be B’ B’ VV(l-1)(l-1) s.t. s.t. |V|VB’B’| > 124| > 124mm

Now, show that Now, show that VVB’ B’ contains no contains no hh-clique-clique

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Long-Code’s JuntaLong-Code’s Junta

DefDef:: A family of subsets A family of subsets P[R] P[R] is is CC--decideddecided if membership of if membership of FF in in is is decided according to decided according to FFCC

P[R]P[R] is is CC-decided to within -decided to within if if there exists a there exists a CC-decided -decided ’’ so that so that

(( ’) ’)

We refer to We refer to CC as the as the ((q, q, )-core)-core of of

Are I[B]’s juntas?Are I[B]’s juntas?

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Influence and SensitivityInfluence and Sensitivity

The influence of an element The influence of an element e e RR on a on a family family P[R] P[R], according to , according to qq is is

The average-sensitivity of The average-sensitivity of is the sum of is the sum of element’s element’s influenceinfluences:s:

Rq

eq

F( ) Pr F {e} F \ {e}

influence

Rq

eq

F( ) Pr F {e} F \ {e}

influence

R eq q

e R

as ( ) ( )

influenceR eq q

e R

as ( ) ( )

influence

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Friedgut’s LemmaFriedgut’s Lemma

ThmThm[Friedgut]:[Friedgut]: A Family of subsets A Family of subsets P[R] P[R] of average-sensitivityof average-sensitivityk = ask = asqq(()) is is CC-decided to within -decided to within , where , where ||C|C| 22O(k/O(k/))

Namely,Namely, has a (has a (q,q, )-)-core core C C R R of size of size |C| |C| 2 2O(k/O(k/))

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ThmThm [Margulis-Russo]: [Margulis-Russo]:

For monotoneFor monotone

HenceHenceLemmaLemma::For monotoneFor monotone > 0 > 0, , q q[p, p+[p, p+]] s.t. s.t. asasqq(() ) 1/ 1/

ProofProof:: Otherwise Otherwise p+p+(() > 1) > 1

qq

d ( )as ( )

dq

q

q

d ( )as ( )

dq

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Now Comes the Hard PartNow Comes the Hard Part

Hence Hence I[B]I[B] has low, has low, 1/1/, average-sensitivity , average-sensitivity with regards to with regards to qq

Which, for any Which, for any , implies a small (, implies a small (q,q, )-core )-core CCBB

Let the Let the core-familycore-family

Thus Thus CF[B]CF[B] is of size is of size > P> P

hence there existhence there exist a aB B andand F Fьь, F, F## CF[B] CF[B] s.t.s.t. FFььFF## ={a ={aBB}}

aaBB is the is the distinguished block-assignmentdistinguished block-assignment of of BB

R \ CB Bq

B F'3CF B F P C | Pr F F' I B 4

R \ CB Bq

B F'3CF B F P C | Pr F F' I B 4

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Now Comes the Harder PartNow Comes the Harder Part

Assuming Assuming CCBB is preserved with is preserved with respect to respect to B’B’ifif I[B] I[B] were exactly the extensions of were exactly the extensions of CF[B]CF[B]

Let’s show that if there is an Let’s show that if there is an hh-clique -clique QQ in in VVB’B’, , II would not have been an would not have been an ISIS

Apply Sunflower construction, Apply Sunflower construction, Pigeon-Hole-Principle, to find two Pigeon-Hole-Principle, to find two blocks with ‘same’ blocks with ‘same’ FFьь, F, F##

©©S.SafraS.Safra

Sunflower Lemma [Erdös-Rado]Sunflower Lemma [Erdös-Rado]

Every family Every family of subsets of a domain of subsets of a domain UU of of large enough size has a subfamily large enough size has a subfamily ’’ s.t. each element of s.t. each element of UU either either Belongs to Belongs to nono subset subset F F’’ Belongs to Belongs to 11 subset subset F F’’ Belongs to Belongs to allall subset subset F F’’

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mm

For some For some q q [p, p+ [p, p+]]For some For some q q [p, p+ [p, p+]]

GG,, G GBB andand H H

m’m’ m’m’

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m’m’m’m’mm

Assume Assume VVB’B’ contains an contains an hh-clique -clique QQAssume Assume VVB’B’ contains an contains an hh-clique -clique QQ

VVB’B’

B’

RB’RB’

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m’m’m’m’

Apply Sunflower lemma and PHPApply Sunflower lemma and PHPApply Sunflower lemma and PHPApply Sunflower lemma and PHP

VVB’B’

partial-viewson B’


To obtain a kernel To obtain a kernel KK and two blocks and two blocks

BB11 and and BB22 of of QQ whose restriction to whose restriction to

partial-views of partial-views of B’B’ is same on is same on KK and anddisjoint outside disjoint outside KK

To obtain a kernel To obtain a kernel KK and two blocks and two blocks

BB11 and and BB22 of of QQ whose restriction to whose restriction to

partial-views of partial-views of B’B’ is same on is same on KK and anddisjoint outside disjoint outside KK

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Yet HarderYet Harder

Given an Given an hh-Clique -Clique QQ in in VVB’B’:: Let Let eCeCBB be the set of partial-views of be the set of partial-views of BB of non- of non-

negligible (negligible (>2>2–O(|C|)–O(|C|)) ) influenceinfluence Redefine Redefine VVB’B’ ={ v | B’ ={ v | B’{v} {v} BB [I] [I] andand

a aB’B’{v}{v}(v)=T (v)=T andand eCeCB’B’{v} {v} preservedpreserved on on

B’}B’} PropProp:: V VB’B’ still large!still large! Apply Sunflower construction on Apply Sunflower construction on eCeC’s, Pigeon-’s, Pigeon-

Hole-Principle on Hole-Principle on C,C, FFьь, F, F##, to find two blocks , to find two blocks with ‘same’ with ‘same’ FFьь, F, F##

©©S.SafraS.Safra

m’m’m’m’mm

Extended-Core Extended-Core {a | {a | influenceinfluenceaa > 2 > 2–O(|C|)–O(|C|) } }Extended-Core Extended-Core {a | {a | influenceinfluenceaa > 2 > 2–O(|C|)–O(|C|) } }

Non-negligible Partial-Non-negligible Partial-ViewsViews

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m’m’m’m’

Taken Care of KernelTaken Care of Kernel



FFьь11 and FF##

22 disagree on KFFьь11 and FF##

22 disagree on K

Let us redefineLet us redefine VVB’B’ = { v | B’ = { v | B’{v} {v} BB [I] [I] andand a aB’B’{v}{v}(v)=T (v)=T andand eCeCB B preservedpreserved on on B’}B’}

Let us redefineLet us redefine VVB’B’ = { v | B’ = { v | B’{v} {v} BB [I] [I] andand a aB’B’{v}{v}(v)=T (v)=T andand eCeCB B preservedpreserved on on B’}B’}

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Almost ThereAlmost There

Assume an Assume an hh-clique -clique QQ of of VVB’B’

Consider the projection of Consider the projection of eCeCBB on on B’B’ for all for all BBQQ

Apply the Sunflower lemma to obtainApply the Sunflower lemma to obtain Q’ Q’ (a (a set of blocks whoseset of blocks whose eC eC’s form a Sunflower)’s form a Sunflower)

These These eCeC’s are thus disjoint outside the ’s are thus disjoint outside the Sunflower’s kernel Sunflower’s kernel KK

Q’ Q’ being large enough, by PHP it must being large enough, by PHP it must contain two blockscontain two blocks B B11 andand B B22 with ‘same’ with ‘same’ C, FC, Fьь, F, F##

©©S.SafraS.Safra

An Edge between An Edge between I[B1]I[B1] and and I[B2]I[B2]

Extend Extend FFьь withinwithin I[B1] I[B1] and and FF## within within I[B2] I[B2] so as not to agree on anyso as not to agree on any a’ a’ inin R RB’B’ Not onNot on C C

FFьь disagrees with disagrees with FF# # except for the distinguished except for the distinguished partial-viewpartial-view

which is assigned which is assigned TT in both blocks in both blocks Not on Not on CC’s “’s “spousesspouses””

Make the extension in each block avoid the other’s Make the extension in each block avoid the other’s spouses; as all spouses have low spouses; as all spouses have low influenceinfluence, this changes , this changes little the size of the extension, leaving it bounded little the size of the extension, leaving it bounded away from ½away from ½

Now show outside Now show outside CC and spouses, there exist and spouses, there exist two extensions that disagreetwo extensions that disagree

©©S.SafraS.Safra

Open ProblemsOpen Problems

ConjConj:: Vertex-Cover is hard to Vertex-Cover is hard to approximate to within approximate to within 2-o(1)2-o(1)

ConjConj:: Coloring a Coloring a 33-Colorable graph -Colorable graph with with >O(1)>O(1) colors is hard colors is hard

Free Bit ComplexityFree Bit Complexity Max-CutMax-Cut Property-TestingProperty-Testing Max-BisectionMax-Bisection

The Importance of Being Biased Irit Dinur S. Safra (some slides borrowed from Dana Moshkovitz) Irit...

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