The Importance of Being Biased Irit Dinur S
The Importance of Being Biased Irit Dinur S. Safra (some slides borrowed from Dana Moshkovitz)
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: ©S. Safra
Minimum VC NP-hard Observation: Let G=(V, E) be an undirected graph. The complement VC of a vertex-cover C is an independent-set of G. Proof: Two vertices outside a vertex-cover cannot be connected by an edge. ©S. Safra
VC Approximation Algorithm ©S. Safra 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 ©S. Safra
Polynomial Time C O(n 2) E’ E while E’ do let (u, v) be an arbitrary edge of E’ O(1) C C {u, v} O(n 2) remove from E’ every edge incident to O(n) either u or v return C ©S. Safra
Correctness The set of vertices our algorithm returns is clearly a vertex-cover, since we iterate until every edge is covered. ©S. Safra
How Good an Approximation is it? Observe the set of edges our algorithm chooses no common vertices! any VC contains 1 in each ©S. Safra 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 ©S. Safra
(m, r)-co-partite Graph G=(M R, E) Comprise m=|M| cliques of size r=|R|: E {(<i, j 1>, <i, j 2>) | i M, j 1≠j 2 R} m ©S. Safra
h-Clique 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 m contains an edge h-Clique m , h, )) is NP-hard as long as r ( h 1 / )c Thm: h. IS(r IS( r, for some constant c , r and h constant! ©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 h. IS(G) to a graph H Good: Bad: ©S. Safra implying NP-hardness of 4/3 factor for Vertex-Cover
Encode I. S. ’s Representatives supposedly encoding IS’s representative j R assignment: IS Edges: two vertices can’t both if inof the ISbe 1 in Replace clique i M bythat a 1 set 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 ©S. Safra
Long-Code of R ©S. Safra One bit (vertex) for every subset of R
Long-Code of R One bit (vertex) for every subset of R to encode an element e R 0 ©S. Safra 0 1 1 1
Long-Code to Co-what partite’s edges do we. I. S. have within a part? non-intersecting: F 1 F 2 = VLC = M P[R] m ©S. Safra ELC = {(F 1, F 2) | F 1 F 2 E}
Problem: all F, |F| >½r are IS In Between each part: parts: intersecting assume a co-matching m ©S. Safra
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 ©S. Safra Easily reducible to graphs with no weights
Biased Long-Code Consider the p-biased product distribution p : Def: The probability of a subset F and for a family of subsets ©S. Safra
p <½- F‘s of size >½r Vanish discriminating against large subsets solves the >½ problem, however… m ©S. Safra
Problem: consistent large subsets almost all subsets have a representative in those subsets m Si Sj ©S. Safra 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 l. T and l=r· 2 l. T m ©S. Safra m’
The (m’, r’)-co-partite Graph GB Fix a large l. T and l=r· 2 l. T m ©S. Safra m’
The (m’, r’)-co-partite Graph GB Vertices: Fix a large l. T and l=r· 2 l. T let B=V(l), m’ =|B| For every B B Prop: IS(G) = m IS(GB) > m’ (1 -2– (l. T)) Edges: Let B’ = V(l-1): B 1=B’ {v 1}, B 2=B’ {v 2} (a 1, a 2) EB for a 1 RB 1, a 2 RB 2 if ©S. Safra a 1|B’ a 2|B’ or (v 1, v 2) E and a 1(v 1) = a 2(v 2) = 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 (F 1, F 2) for. Proof: F 1 P[given RB 1], Fan P[Rin 2 IS B 2]Gif B, F 1 F 2 EBI, consider the corresponding set of Weights: (F) = p(F) / |singletons B| in H; take monotone extension Prop (Completeness): IS(H) p · IS(GB) / m’ Thm (Soundness): For p≤(3 - 5)/2, h. IS(G) < m IS(H) < P + ’ [for p 1/3: P =p 2] ©S. Safra
The (m’, r’)-co-partite Graph GB Fix a large l. T and l=r· 2 l. T m ©S. Safra m’
Soundness for GB Lemma: an IS of size m’ in GB implies IS of size ½ m in G Proof: For an IS I’ of GB Fix a B’ in Vl-1 for which (such must exist) Let I = { v | (<B’, v>, a) I’ and a(v) = T } I is an IS of G of size ½ m ©S. Safra
IS of size P even in Bad Case Partition V into V 1 and V 2 For every block B, let a 1 assign T to V 1 and F to V 2 a 2 assign T to V 2 and F to V 1 ©S. Safra and let B = { F { a 1 , a 2 } } These B‘s form an IS of weight p 2 in H
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 P = Thm[Wilson, Frankl, Ahlswede-Khachatrian]: For a t-intersecting , where ©S. Safra 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: ©S. Safra 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 h. IS(G) > m Let B[I] = { B | p(I[B]) > P + 250 } Prop: |B[I]| > 250 |B| Observation: ©S. Safra
Soundness Proof (Naïve) Plan: Find, for every B B [I], a distinguished block-assignment a. B Let VB’ ={ v | B’ {v} B [I] and a. B’ {v}(v)=T} There ©S. Safra must be B’ V(l-1) s. t. |VB’| > 124 m Now, show that VB’ contains no h-clique
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 ©S. Safra 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 ©S. Safra 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:
Friedgut’s Lemma Thm[Friedgut]: A Family of subsets P[R] of average-sensitivity k = asq( ) is C-decided to within , where |C| 2 O(k/ ) Namely, has a (q, )-core C R of size |C| 2 O(k/ ) ©S. Safra
Thm [Margulis-Russo]: For monotone Hence Lemma: For monotone > 0, q [p, p+ ] s. t. asq( ) 1/ Proof: Otherwise p+ ( ) > 1 ©S. Safra
Now Comes the Hard Part ©S. Safra 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 a. B and Fь, F# CF[B] s. t. Fь F# ={a. B} a. B is the distinguished block-assignment of B
Now Comes the Harder Part ©S. Safra 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#
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 ©S. Safra Belongs to no subset F ’ Belongs to 1 subset F ’ Belongs to all subset F ’
For some q [p, p+ ] m ©S. Safra G, GB and H m’
VB’ Assume VB’ contains an h-clique Q m ©S. Safra B’ RB’ m’
Apply Sunflower lemma and PHP VB’ partial-views on B’ m’ 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 ©S. Safra
Yet Harder ©S. Safra Given an h-Clique Q in VB’: Let e. CB be the set of partial-views of B of non -negligible (>2–O(|C|)) influence Redefine VB’ ={ v | B’ {v} B [I] and a. B’ {v}(v)=T and e. CB’ {v} preserved on B’} Prop: VB’ still large! Apply Sunflower construction on e. C’s, Pigeon. Hole-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 ©S. Safra m’
Non-negligible Partial-Views m ©S. Safra 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 a. B’ {v}(v)=T and e. CB preserved on B’} ©S. Safra
Almost There ©S. Safra Assume an h-clique Q of VB’ Consider the projection of e. CB on B’ for all B Q Apply the Sunflower lemma to obtain Q’ (a set of blocks whose e. C’s form a Sunflower) These e. C’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, Fь, F#
An Edge between I[B 1] and I[B 2] Extend Fь within I[B 1] and F# within I[B 2] so as not to agree on any a’ in RB’ Not on C’s “spouses” ©S. Safra 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
Open Problems ©S. Safra 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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