The Importance of Being
Biased
Irit Dinur
S. Safra
(some slides borrowed from Dana Moshkovitz)
VERTEX-COVER
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
Instance: an undirected graph G=(V,E).
Problem: find a set CV of minimal size
s.t. for any (u,v)E, either uC or vC.
Example:
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Minimum VC NP-hard
Observation: Let G=(V,E) be an undirected
graph. The complement V\C of a vertexcover C is an independent-set of G.
Proof: Two vertices outside a vertex-cover
cannot be connected by an edge. 
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VC Approximation Algorithm
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C
E’  E
while E’  
 do let (u,v) be an arbitrary edge of E’

C  C  {u,v}

remove from E’ every edge incident
to either u or v.
return C.
Demo
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Polynomial Time
C
O(n2)
 E’  E
 while E’   do
 let (u,v) be an arbitrary edge of E’
 C  C  {u,v}
O(n2)
 remove from E’ every edge incident to
either u or v
 return C

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O(1)
O(n)
Correctness
The set of vertices our algorithm returns is
clearly a vertex-cover, since we iterate until
every edge is covered.
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How Good an Approximation is it?
Observe the set of edges our algorithm chooses
no common vertices!  any VC contains 1 in each
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our VC contains both, hence at most twice as large
How well can VC be
Approximated?
Upper bound

A little better (w/hard work) : 2-o(1)
Hardness results
Previously: 7/6
 Thm: NP-hard to approximate to
within 105-21  1.36 (> 4/3)
 Conjecture: NP-hard to within 2- 
>0
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(m,r)-co-partite Graph G=(MR, E)

Comprise m=|M| cliques of size r=|R|:
E  {(<i,j1>, <i,j2>) | iM, j1≠j2 R}
m
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h-Clique
Gap Independent-Set
Instance: an (m,r)-co-partite
graph G=(MR, E)
Problem: distinguish between
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Good: IS(G) = m
Bad: every set I  V s.t. |I|>
m
m contains an edge
h-Clique
m
Thm: hIS(r,
IS( r,h,)) is NP-hard as long as r  ( h1 / )c
for some constant c
, r and h constant!
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Hardness of Vertex-Cover
Problem: the size of G’s Vertex-Cover is
Good: (1-1/r)  |G|
Bad: (1- /r)  |G|
Resulting in a factor smaller than 1+1/r
We show: A reduction from hIS(G) to a graph H
Good:
1
 IS(H)  3 H  o(1) H
hIS(G)   m  IS(H)  1 H  o(1) H
9
IS(G)  m
Bad:

implying NP-hardness of 4/3 factor for
Vertex-Cover
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Encode I.S.’s Representatives
supposedly encoding IS’s representative
jR
IS
 assignment:
Edges:
two vertices
can’t
both
if inof
the
ISbe 1 in
Replace clique iM bythat
a 1set
vertices,
the
any Apply
encoding
of an IS
0
if
out
1 for each bit of some binary-code
of R
of
G
long-code
m
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Long-Code of R
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One bit (vertex) for every subset of R
Long-Code of R
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One bit (vertex) for every subset of R
to encode an element eR
0
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0
1
1
1
Long-Code to Co-partite’s
what edges do weI.S.
have within a part?
non-intersecting:
F1F2 =
VLC = M  P[R]
m
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ELC = {(F1,F2) | F1  F2  E}
Problem: all F, |F| >½r are IS
In
Between
each part:
parts:
intersecting
assume a co-matching
m
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Weighted Graphs
Assign weights to V - hence G = (V, E, )
Consider a probability distribution :V[0,1]
and let the size of a set of vertices be
(S)  Pr  v  S 
hence
v 
 (v)
v S
IS(G)  max (I)
i.s.IV
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Easily reducible to graphs with no weights
Biased Long-Code
Consider the p-biased product
distribution p:
Def: The probability of a subset F
pR F  p F  (1  p) R\F
and for a family of subsets 
pR    Pr F    pR F
R
F p
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F
p <½-  F‘s of size >½r Vanish
discriminating against large subsets
solves the >½ problem,
however…
m
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Problem: consistent large subsets
almost all subsets
have a representative in
those subsets
m
Si
Sj
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what if any pair of
cliques i & j have a pair
of large subsets Si & Sj
that are all-wise
consistent
The (m’,r’)-co-partite Graph GB
Fix a large lT and l=r·2lT
V 
B   ,
m' B
l
 
R' { a : [l]  {T, F } | a 1 (T )  lT }
m
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m’
The (m’,r’)-co-partite Graph GB
Fix a large lT and
l=r·2l
V  T
B   ,
m' B
l
 
R' { a : B  {T, F } | a 1 (T )  lT }
m
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m’
The (m’,r’)-co-partite Graph GB
Vertices:
Fix a large lT and l=r·2lT
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let B=V(l), m’ =|B|
For every BB
Prop: IS(G) = m 
IS(GB) > m’ (1-2–(lT))

RB  a : B  T,F | a 1 (T)  lT

Edges:
Let B’ = V(l-1): B1=B’{v1}, B2=B’{v2}
(a1, a2) EB for a1RB1, a2RB2 if
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a1|B’  a2|B’
or
(v1, v2)E and a1(v1) = a2(v2) = T
Now Apply Long-Code to GB
The final graph H = (B  P[ RB ], EBLC, )
Vertices: one  B B and a subset F P[RB]
Edges: EBLC  (F1, F2) for FProof:
P[R
] if
1 P[Rgiven
B1], F2an
ISB2in
F1  F2  EGB , I, consider the
B
corresponding set of
Weights: (F) = p(F) / |B|
singletons in H; take
monotone extension
Prop (Completeness):
IS(H)  p · IS(GB) / m’
Thm (Soundness): For p≤(3-5)/2,
hIS(G) < m  IS(H) < P + ’
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[for p 1/3: P=p2]
IS of size P even in Bad Case
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Partition V into V1 and V2
For every block B, let
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a1 assign T to V1 and F to V2
a2 assign T to V2 and F to V1
and let
B = { F {a1, a2} }
These B‘s form an IS of weight p2 in H
Erdös-Ko-Rado


Def: A family of subsets
  P[R] is t-intersecting if for every
F1, F2  , |F1  F2|  t
p (P) = max p (Ai,2 )
i
Thm[Wilson,Frankl,Ahlswede-Khachatrian]:
For a t-intersecting ,
where
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p ( )  max p (Ai,t )
i


Ai,t  F | F  1,...,2i  t  i  t
Corollary: p() > P   is not 2-intersecting
Soundness Proof
Important Observation:
Assume I is a maximal IS in H
I’s intersection with any block
I[B]  I  P[ RB ]
is monotone and intersecting
It follows:
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q(I[B]) is a non-decreasing function of q
Soundness Proof
We prove: If H has an IS I s.t. (I) > P + 500 then
hIS(G) > m
Let
B[I] = { B | p(I[B]) > P + 250 }
Prop:
|B[I]| > 250 |B|
Observation:
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 V 
B'  

l

1


B  B'  V \ B'
Soundness Proof
(Naïve) Plan:
Find, for every B  B [I], a
distinguished block-assignment aB
 Let
VB’ ={ v | B’{v}  B [I] and
aB’{v}(v)=T}
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There must be B’  V(l-1) s.t.
|VB’| > 124m
Are I[B]’s juntas?
Long-Code’s Junta
Def: A family of subsets   P[R] is Cdecided if membership of F in  is
decided according to FC
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  P[R] is C-decided to within  if
there exists a C-decided ’ so that
( ’)  
We refer to C as the (q, )-core of 
Influence and Sensitivity

The influence of an element e R on a family
 P[R], according to q is
influenceeq ()  PrR F  {e}    F \ {e}  
Fq

The average-sensitivity of  is the sum of
element’s influences:
asqR ( )   influence eq ( )
eR
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Friedgut’s Lemma
Thm[Friedgut]: A Family of subsets
  P[R] of average-sensitivity
k = asq() is C-decided to within , where |C|
 2kO(k/)
Namely,
 has a (q, )-core C  R of size
|C|  2O(k/)
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Thm [Margulis-Russo]:
For monotone 
asq ( ) 
d q ( )
dq
Hence
Lemma:
For monotone 
 > 0,  q[p, p+] s.t. asq()  1/
Proof: Otherwise p+() > 1
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Now Comes the Hard Part
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Hence I[B] has low, 1/, average-sensitivity
with regards to q
Which, for any , implies a small (q, )-core CB
Let the core-family
CF B
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

F  P CB | PrF'RB\CB F  F'  I B  3
4
q
Thus CF[B] is of size > P
hence there exist aB and Fь, F#  CF[B] s.t.
FьF# ={aB}
aB is the distinguished block-assignment of B
The (m’,r’)-co-partite Graph GB
Fix a large lT and
l=r·2l
V  T
B   ,
m' B
l
 
R' { a : B  {T, F } | a 1 (T )  lT }
m
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m’
Now Comes the Harder Part
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Assuming CB is preserved with respect
to B’
if I[B] were exactly the extensions of
CF[B]
Let’s show that if there is an h-clique Q
in VB’, I would not have been an IS
Apply Sunflower construction, PigeonHole-Principle, to find two blocks with
‘same’ Fь, F#
Sunflower Lemma [Erdös-Rado]

Every family  of subsets of a domain U of
large enough size has a subfamily ’ s.t.
each element of U either
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Belongs to no subset F’
Belongs to 1 subset F’
Belongs to all subset F’
G, GB and H
For some q  [p, p+]
m
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m’
VB’
Assume VB’ contains an h-clique Q
m
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B’
RB’
m’
VB’
Apply Sunflower lemma and PHP
partial-views
on B’
m’
To obtain a kernel K and two blocks
B1 and B2 of Q whose restriction to
partial-views of B’ is same on K and
disjoint outside K
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Yet Harder
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Given an h-Clique Q in VB’:
Let eCB be the set of partial-views of B of
non-negligible (>2–O(|C|)) influence
Redefine VB’ ={ v | B’{v}  B [I] and
aB’{v}(v)=T and
eCB’{v} preserved on B’}
Prop: VB’ still large!
Apply Sunflower construction on eC’s, PigeonHole-Principle on C, Fь, F#, to find two blocks
with ‘same’ Fь, F#
Non-negligible Partial-Views
Extended-Core {a | influencea > 2–O(|C|) }
m
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m’
Non-negligible Partial-Views
m
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B’
m’
Taken Care of Kernel
Fь1 and F#2 disagree on K
partial-views
on B’
m’
Let us redefine
VB’ = { v | B’{v}  B [I] and
aB’{v}(v)=T and
eCB preserved on B’}
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Almost There
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Assume an h-clique Q of VB’
Consider the projection of eCB on B’ for all
BQ
Apply the Sunflower lemma to obtain Q’ (a set
of blocks whose eC’s form a Sunflower)
These eC’s are thus disjoint outside the
Sunflower’s kernel K
Q’ being large enough, by PHP it must contain
two blocks B1 and B2 with ‘same’ C, Fь, F#
An Edge between I[B1] and I[B2]

Extend Fь within I[B1] and F# within I[B2] so
as not to agree on any a’ in RB’

Not on C
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Not on C’s “spouses”
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Fь disagrees with F# except for the distinguished partialview
which is assigned T in both blocks
Make the extension in each block avoid the other’s
spouses; as all spouses have low influence, this changes
little the size of the extension, leaving it bounded away
from ½
Now show outside C and spouses, there exist two
extensions that disagree
An Edge between I[B1] and I[B2]
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As long as q is so that 1-q(1-q)
Open Problems
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Conj: Vertex-Cover is hard to
approximate to within 2-o(1)
Conj: Coloring a 3-Colorable graph with
>O(1) colors is hard
Free Bit Complexity
Max-Cut
Property-Testing
Max-Bisection
Fourier Transform

Consider all ‘linear’ functions, one for each
character S[n]
S (x)  1

Sx
Given a function
f : 1, 1  1, 1
n

Let the Fourier coefficients of f be
f(S) : E f(x)   S (x) 
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x
Simple Observations
f     E f(x) 

Claim:

Let the influence function be
x
fi  x  
fx  f x  i

2
the Fourier coefficients of f be
fi Let
x 
1
1
f(S)  S  x    f(S)  S  x    S  i  

2 S
2 S
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Simple Observations

Claim:

Let the influence function beiS

Let the Fourier coefficients of f be
influencei  f   f(S)
asi  f    f(S) S
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S