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

2. VERTEX-COVER 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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3. Minimum VC NP-hard
Observation: Let G=(V,E) be an undirected graph. The complement V\C of a vertex-cover C is an independent-set of G.
Proof: Two vertices outside a vertex-cover cannot be connected by an edge. 
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4. VC Approximation Algorithm
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.
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5. Demo
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6. Polynomial Time 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
O(n2)
O(1)
O(n2)
O(n)
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7. 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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8. How Good an Approximation is it?
Observe the set of edges our algorithm chooses
no common vertices!
 any VC contains 1 in each
our VC contains both, hence at most twice as large
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9. 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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10. (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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11. Gap Independent-Set Instance: an (m,r)-co-partite graph G=(MR, E)
h-Clique
hIS(r, h, ) h
Gap Independent-Set
Instance: an (m,r)-co-partite graph G=(MR, E)
Problem: distinguish between
Good: IS(G) = m
Bad: every set I  V s.t. |I|> m contains an edge m m
Thm: IS( r,  ) is NP-hard as long as r  ( 1 / )c for some constant c
, r and h constant!
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12. 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: Bad: implying NP-hardness of 4/3 factor for Vertex-Cover
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13. Encode I.S.’s Representatives
Edges: two vertices that can’t both be 1 in any encoding of an IS of G
IS  assignment:
1 if in the IS
0 if out
supposedly encoding IS’s representative jR
Apply the long-code
Replace clique iM by a set of vertices, 1 for each bit of some binary-code of R m
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14. Long-Code of R
One bit (vertex) for every subset of R
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15. Long-Code of R
One bit (vertex) for every subset of R to encode an element eR 1 1 1
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16. Long-Code to Co-partite’s I.S.
what edges do we have within a part?
non-intersecting: F1F2 =
VLC = M  P[R] m
ELC = {(F1,F2) | F1  F2  E}
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17. Problem: all F, |F| >½r are IS
In each part: intersecting
Between parts: assume a co-matching m
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18. 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 hence
Easily reducible to graphs with no weights
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19. Biased Long-Code Consider the p-biased product distribution p:
Def: The probability of a subset F and for a family of subsets 
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20. p <½-  F‘s of size >½r Vanish
discriminating against large subsets
solves the >½ problem, however… m
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21. Problem: consistent large subsets
almost all subsets
have a representative in those subsets
what if any pair of cliques i & j have a pair of large subsets Si & Sj that are all-wise consistent m Si Sj
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22. The (m’,r’)-co-partite Graph GB
Fix a large lT and l=r·2lT m’ m
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o/w a:B{F}

23. The (m’,r’)-co-partite Graph GB
Fix a large lT and l=r·2lT m’ m
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24. The (m’,r’)-co-partite Graph GB
Vertices: Fix a large lT and l=r·2lT
let B=V(l), m’ =|B|
For every BB
Edges: Let B’ = V(l-1): B1=B’{v1}, B2=B’{v2} (a1, a2) EB for a1RB1, a2RB2 if
a1|B’  a2|B’ or
(v1, v2)E and a1(v1) = a2(v2) = T
Prop: IS(G) = m  IS(GB) > m’ (1-2–(lT))
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25. 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 F1 P[RB1], F2P[RB2] if F1  F2  EB
Weights: (F) = p(F) / |B|
Prop (Completeness): IS(H)  p · IS(GB) / m’
Thm (Soundness): For p≤(3-5)/2, hIS(G) < m  IS(H) < P + ’ [for p 1/3: P=p2]
Proof: given an IS in GB, I, consider the corresponding set of singletons in H; take monotone extension
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26. IS of size P even in Bad Case
Partition V into V1 and V2
For every block B, let
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
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27. Erdös-Ko-Rado
Def: A family of subsets   P[R] is t-intersecting if for every F1, F2  , |F1  F2|  t
Thm[Wilson,Frankl,Ahlswede-Khachatrian]: For a t-intersecting , where
Corollary: p() > P   is not 2-intersecting
P =
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28. 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:
q(I[B]) is a non-decreasing function of q
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29. 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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30. 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}
There must be B’  V(l-1) s.t. |VB’| > 124m
Now, show that VB’ contains no h-clique
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31. Are I[B]’s juntas?
Long-Code’s Junta
Def: A family of subsets   P[R] is C-decided if membership of F in  is decided according to FC
  P[R] is C-decided to within  if there exists a C-decided ’ so that ( ’)  
We refer to C as the (q, )-core of 
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32. Influence and Sensitivity
The influence of an element e R on a family  P[R], according to q is
The average-sensitivity of  is the sum of element’s influences:
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33. 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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34. Thm [Margulis-Russo]:
For monotone 
Hence Lemma: For monotone   > 0,  q[p, p+] s.t. asq()  1/
Proof: Otherwise p+() > 1
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35. Now Comes the Hard Part
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
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
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36. The (m’,r’)-co-partite Graph GB
Fix a large lT and l=r·2lT m’ m
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37. Now Comes the Harder Part
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, Pigeon-Hole-Principle, to find two blocks with ‘same’ Fь, F#
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38. 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
Belongs to no subset F’
Belongs to 1 subset F’
Belongs to all subset F’
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39. G, GB and H
For some q  [p, p+] m’ m
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40. VB’
Assume VB’ contains an h-clique Q m’ B’ m
RB’
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41. VB’ partial-views on B’ To obtain a kernel K and two blocks
Apply Sunflower lemma and PHP m’
partial-views
on B’
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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42. Yet Harder Prop: VB’ still large! 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, Pigeon-Hole-Principle on C, Fь, F#, to find two blocks with ‘same’ Fь, F#
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43. Non-negligible Partial-Views
Extended-Core {a | influencea > 2–O(|C|) } m’ m
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44. Non-negligible Partial-Viewsm’ B’ m
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45. Taken Care of Kernel
Fь1 and F#2 disagree on K m’
partial-views
on B’
Let us redefine VB’ = { v | B’{v}  B [I] and aB’{v}(v)=T and eCB preserved on B’}
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46. Almost There 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#
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47. 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
Fь disagrees with F# except for the distinguished partial-view
which is assigned T in both blocks
Not on C’s “spouses”
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
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48. An Edge between I[B1] and I[B2]
As long as q is so that 1-q(1-q)
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49. Open Problems
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
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50. Fourier Transform
Consider all ‘linear’ functions, one for each character S[n]
Given a function
Let the Fourier coefficients of f be
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51. Simple Observations Claim: Let the influence function be
Let the Fourier coefficients of f be
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52. Simple Observations Claim: Let the influence function be
Let the Fourier coefficients of f be
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