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

2. © S.Safra Network Power Say you have a network, with links between some components Each link requires power supply, hence, you need to supply power to a set of nodes that cover all links Obviously, you’d like to connect the smallest number of nodes

3. © S.Safra VERTEX-COVER Instance: an undirected graph G=(V,E). 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. Problem: find a set C  V of minimal size s.t. for any (u,v)  E, either u  C or v  C. Instance: an undirected graph G=(V,E). 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. Problem: find a set C  V of minimal size s.t. for any (u,v)  E, either u  C or v  C. Example:

4. © S.Safra 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. Proof: Two vertices outside a vertex-cover cannot be connected by an edge.

5. © S.Safra VC Approximation Algorithm C   C   E’  E E’  E while E’   while E’   do let (u,v) be an arbitrary edge of E’ do let (u,v) be an arbitrary edge of E’ C  C  {u,v} C  C  {u,v} remove from E’ every edge incident to either u or v. remove from E’ every edge incident to either u or v. return C. return C.

6. © S.Safra Demo

7. Polynomial Time C   C   E’  E E’  E while E’   do while E’   do let (u,v) be an arbitrary edge of E’ let (u,v) be an arbitrary edge of E’ C  C  {u,v} C  C  {u,v} remove from E’ every edge incident to either u or v remove from E’ every edge incident to either u or v return C return C O(n 2 ) O(1)O(n) O(n 2 )

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

9. © S.Safra How Good an Approximation is it? Observe the set of edges our algorithm chooses  any VC contains 1 in each our VC contains both, hence at most twice as large no common vertices!

10. © S.Safra How well can VC be Approximated? Upper bound A little better (w/hard work) : 2-o(1) A little better (w/hard work) : 2-o(1) Hardness results Previously: 7/6 Previously: 7/6 Thm: NP-hard to approximate to within 10  5-21  1.36 (> 4/3) Thm: NP-hard to approximate to within 10  5-21  1.36 (> 4/3) Conjecture: NP-hard to within 2-   >0 Conjecture: NP-hard to within 2-   >0

11. © S.Safra (m,r)-co-partite Graph G=(M  R, E) Comprise m=|M| cliques of size r=|R|: E  {(, ) | i  M, j 1 ≠ j 2  R} Comprise m=|M| cliques of size r=|R|: E  {(, ) | i  M, j 1 ≠ j 2  R} m

12. © S.Safra m m Gap Independent-Set Instance: an (m,r)-co-partite graph G=(M  R, E) Problem: distinguish between Good: IS(G) = m Good: IS(G) = m Bad: every set I  V s.t. |I|>  m contains an edge Bad: every set I  V s.t. |I|>  m contains an edge Thm: IS( r,  ) is NP-hard as long as r  ( 1 /  ) c for some constant c h-Clique hIS(r, h,  ) h m , r and h constant!

13. © S.Safra Proof Take a PCP in which every local- constraint depends on a constant D number of variables, and where the error-probability does not depend on D Take a PCP in which every local- constraint depends on a constant D number of variables, and where the error-probability does not depend on D Have a clique in G for every local- constraint, and a vertex in it for every assignments (to its D variables) satisfying it Edges correspond to inconsistency Have a clique in G for every local- constraint, and a vertex in it for every assignments (to its D variables) satisfying it Edges correspond to inconsistency

14. © S.Safra Proof (for PCPholics) Apply the Parallel Repetition Lemma to a gap3SAT, with k repetitions Apply the Parallel Repetition Lemma to a gap3SAT, with k repetitions Have one clique for every sequence of k clauses (m = n k ) Have one clique for every sequence of k clauses (m = n k ) In which one vertex for each satisfying assignment to all k clauses (r= 7 k ) In which one vertex for each satisfying assignment to all k clauses (r= 7 k ) No h-clique in V’ implies an assignment satisfying |V’|/h 2 of constraints No h-clique in V’ implies an assignment satisfying |V’|/h 2 of constraints

15. © S.Safra 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

16. © S.Safra m m Encode I.S.’s Representatives Replace clique i  M by a set of vertices, 1 for each bit of some binary-code of R Apply the long-code supposedly encoding IS’s representative j  R 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

17. © S.Safra Long-Code of R One bit (vertex) for every subset of R One bit (vertex) for every subset of R

18. © S.Safra Long-Code of R One bit (vertex) for every subset of R to encode an element e  R One bit (vertex) for every subset of R to encode an element e  R 0 0 0 0 1 1 1 1 1 1

19. © S.Safra V LC = M  P[R] Long-Code to Co-partite’s I.S. E LC = {(F 1,F 2 ) | F 1  F 2  E} m m what edges do we have within a part? non-intersecting: F1  F2 = 

20. © S.Safra Between parts: assume a co-matching In each part: intersecting Problem: all F, |F| >½r are IS m m

21. © S.Safra Weighted Graphs Assign to V - hence G = (V, E,  ) Consider a probability distribution  :V  [0,1] and let the size of a set of vertices be hence 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 Easily reducible to graphs with no weights Assign to V - hence G = (V, E,  ) Consider a probability distribution  :V  [0,1] and let the size of a set of vertices be hence 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 Easily reducible to graphs with no weights

22. © S.Safra Consider the p-biased product distribution  p : Def: The probability of a subset F and for a family of subsets  Consider the p-biased product distribution  p : Def: The probability of a subset F and for a family of subsets  Biased Long-Code

23. © S.Safra discriminating against large subsets p ½r Vanish the >½ problem, however… solves the >½ problem, however… m m

24. © S.Safra m m Problem: consistent large subsets SiSi SiSi SjSj SjSj what if any pair of cliques i & j have a pair of large subsets S i & S j that are all-wise consistent almost all subsets have a representative in those subsets almost all subsets have a representative in those subsets

25. © S.Safra Fix a large l T and l=r·2l T m’m’ o/w a:B  {F} m m The (m’,r’)-co-partite Graph G B

26. © S.Safra m’m’ m m Fix a large l T and l=r·2l T The (m’,r’)-co-partite Graph G B

27. © S.Safra The (m’,r’)-co-partite Graph G B Vertices: Fix a large l T and l=r·2l T let B=V (l), m’ =|B| let B=V (l), m’ =|B| For every B  B For every B  B Edges: Let B’ = V (l-1) : B 1 =B’  {v 1 }, B 2 =B’  {v 2 } (a 1, a 2 )  E B for a 1  R B1, a 2  R B2 if a 1 | B’  a 2 | B’ or a 1 | B’  a 2 | B’ or (v 1, v 2 )  E and a 1 (v 1 ) = a 2 (v 2 ) = T (v 1, v 2 )  E and a 1 (v 1 ) = a 2 (v 2 ) = T Prop: IS(G) = m  IS(G B ) > m’ (1-2 –  (l T ) )

28. © S.Safra Now Apply Long-Code to G B The final graph H = (B  P[ R B ], E B LC,  ) Vertices: one  B  B and a subset F  P[R B ] Edges: E B LC  (F 1, F 2 ) for F 1  P[R B 1 ], F 2  P[R B 2 ] if F 1  F 2  E B Weights:  (F) =  p (F) / |B| Prop (Completeness): IS(H)  p · IS(G B ) / m’ Thm (Soundness): hIS(G) <  m  IS(H) < P  +  ’ [for p  1/3: P  =p 2 ] Thm (Soundness): For p≤(3-  5)/2, hIS(G) <  m  IS(H) < P  +  ’ [for p  1/3: P  =p 2 ] Proof: given an IS in G B, I, consider the corresponding set of singletons in H; take monotone extension

29. © S.Safra m’m’ m m Fix a large l T and l=r·2l T The (m’,r’)-co-partite Graph G B

30. © S.Safra Soundness for G B Lemma: an IS of size  m’ in G B implies IS of size ½  m in G Proof: For an IS I’ of G B Fix a B’ in V l-1 for which (such must exist) Let I = { v | (, a)  I’ and a(v) = T } I is an IS of G of size ½  m

31. © S.Safra IS of size P  even in Bad Case Partition V into V 1 and V 2 Partition V into V 1 and V 2 For every block B, let For every block B, let a 1 assign T to V 1 and F to V 2 a 1 assign T to V 1 and F to V 2 a 2 assign T to V 2 and F to V 1 a 2 assign T to V 2 and F to V 1 and let  B = { F  {a 1, a 2 } } and let  B = { F  {a 1, a 2 } } These  B ‘s form an IS of weight p 2 in H These  B ‘s form an IS of weight p 2 in H

32. © S.Safra Erdös-Ko-Rado Def: A family of subsets   P[R] is t-intersecting if for every F 1, F 2  , |F 1  F 2 |  t Def: A family of subsets   P[R] is t-intersecting if for every F 1, F 2  , |F 1  F 2 |  t Thm[Wilson,Frankl,Ahlswede-Khachatrian]: For a t-intersecting , where Thm[Wilson,Frankl,Ahlswede-Khachatrian]: For a t-intersecting , where Corollary:  p (  ) > P    is not 2-intersecting Corollary:  p (  ) > P    is not 2-intersecting P  = P  =

33. © S.Safra Soundness Proof Important Observation: Assume I is a maximal IS in H I’s intersection with any block I[B]  I  P[ R B ] is monotone and intersecting It follows:  q (I[B]) is a non-decreasing function of q  q (I[B]) is a non-decreasing function of q

34. © S.Safra 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:

35. © S.Safra Soundness Proof (Naïve) Plan: Find, for every B  B [I], a distinguished block-assignment a B Find, for every B  B [I], a distinguished block-assignment a B Let V B’ ={ v | B’  {v}  B [I] and a B’  {v} (v)=T} Let V B’ ={ v | B’  {v}  B [I] and a B’  {v} (v)=T} There must be B’  V (l-1) s.t. |V B’ | > 124  m There must be B’  V (l-1) s.t. |V B’ | > 124  m Now, show that V B’ contains no h-clique Now, show that V B’ contains no h-clique

36. © S.Safra 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  (   ’)     P[R] is C-decided to within  if there exists a C-decided  ’ so that  (   ’)   We refer to C as the (q,  )-core of  We refer to C as the (q,  )-core of  Are I[B]’s juntas?

37. © S.Safra Influence and Sensitivity The influence of an element e  R on a family  P[R], according to  q is 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 influence s: The average-sensitivity of  is the sum of element’s influence s:

38. © S.Safra Friedgut’s Lemma Thm[Friedgut]: A Family of subsets   P[R] of average-sensitivity k = as q (  ) is C-decided to within , where |C|  2 O(k/  ) Namely,  has a (q,  )- core C  R of size |C|  2 O(k/  )

39. © S.Safra Thm [Margulis-Russo]: For monotone  Hence Lemma: For monotone   > 0,  q  [p, p+  ] s.t. as q (  )  1/  Proof: Otherwise  p+  (  ) > 1

40. © S.Safra Now Comes the Hard Part Hence I[B] has low, 1/ , average-sensitivity with regards to  q Hence I[B] has low, 1/ , average-sensitivity with regards to  q Which, for any , implies a small (q,  )-core C B Which, for any , implies a small (q,  )-core C B Let the core-family Let the core-family Thus CF[B] is of size > P  Thus CF[B] is of size > P  hence there exist a B and F ь, F #  CF[B] s.t. F ь  F # ={a B } hence there exist a B and F ь, F #  CF[B] s.t. F ь  F # ={a B } a B is the distinguished block-assignment of B a B is the distinguished block-assignment of B

41. © S.Safra Now Comes the Harder Part Assuming C B is preserved with respect to B’ if I[B] were exactly the extensions of CF[B] Assuming C B 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 V B’, I would not have been an IS Let’s show that if there is an h-clique Q in V B’, I would not have been an IS Apply Sunflower construction, Pigeon- Hole-Principle, to find two blocks with ‘same’ F ь, F # Apply Sunflower construction, Pigeon- Hole-Principle, to find two blocks with ‘same’ F ь, F #

42. © S.Safra 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 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 no subset F  ’ Belongs to 1 subset F  ’ Belongs to 1 subset F  ’ Belongs to all subset F  ’ Belongs to all subset F  ’

43. © S.Safra m m For some q  [p, p+  ] G, G B and H m’ m’

44. © S.Safra m’m’ m m Assume V B’ contains an h-clique Q V B’ B’ R B’

45. © S.Safra m’m’ Apply Sunflower lemma and PHP V B’ partial-views on B’ To obtain a kernel K and two blocks B 1 and B 2 of Q whose restriction to partial-views of B’ is same on K and disjoint outside K

46. © S.Safra Yet Harder Given an h-Clique Q in V B’ : Given an h-Clique Q in V B’ : Let eC B be the set of partial-views of B of non-negligible (>2 –O(|C|) ) influence Let eC B be the set of partial-views of B of non-negligible (>2 –O(|C|) ) influence Redefine V B’ ={ v | B’  {v}  B [I] and a B’  {v} (v)=T and eC B’  {v} preserved on B’} Redefine V B’ ={ v | B’  {v}  B [I] and a B’  {v} (v)=T and eC B’  {v} preserved on B’} Prop: V B’ still large! Prop: V B’ still large! Apply Sunflower construction on eC’s, Pigeon- Hole-Principle on C, F ь, F #, to find two blocks with ‘same’ F ь, F # Apply Sunflower construction on eC’s, Pigeon- Hole-Principle on C, F ь, F #, to find two blocks with ‘same’ F ь, F #

47. © S.Safra m’m’ m m Extended-Core {a | influence a > 2 –O(|C|) } Non-negligible Partial-Views

48. © S.Safra m’m’ m m Non-negligible Partial-Views B’

49. © S.Safra m’m’ Taken Care of Kernel partial-views on B’ F ь 1 F # 2 F ь 1 and F # 2 disagree on K Let us redefine V B’ = { v | B’  {v}  B [I] and a B’  {v} (v)=T and eC B preserved on B’}

50. © S.Safra Almost There Assume an h-clique Q of V B’ Assume an h-clique Q of V B’ Consider the projection of eC B on B’ for all B  Q Consider the projection of eC B on B’ for all B  Q Apply the Sunflower lemma to obtain Q’ (a set of blocks whose eC’s form a Sunflower) 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 These eC’s are thus disjoint outside the Sunflower’s kernel K Q’ being large enough, by PHP it must contain two blocks B 1 and B 2 with ‘same’ C, F ь, F # Q’ being large enough, by PHP it must contain two blocks B 1 and B 2 with ‘same’ C, F ь, F #

51. © S.Safra An Edge between I[B1] and I[B2] Extend F ь within I[B1] and F # within I[B2] so as not to agree on any a’ in R B’ Extend F ь within I[B1] and F # within I[B2] so as not to agree on any a’ in R B’ Not on C Not on C F ь disagrees with F # except for the distinguished partial- view F ь disagrees with F # except for the distinguished partial- view which is assigned T in both blocks which is assigned T in both blocks Not on C’s “spouses” 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 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 Now show outside C and spouses, there exist two extensions that disagree

52. © S.Safra Open Problems Conj: Vertex-Cover is hard to approximate to within 2-o(1) Conj: Vertex-Cover is hard to approximate to within 2-o(1) Conj: Coloring a 3-Colorable graph with >O(1) colors is hard Conj: Coloring a 3-Colorable graph with >O(1) colors is hard Free Bit Complexity Free Bit Complexity Max-Cut Max-Cut Property-Testing Property-Testing Max-Bisection Max-Bisection