Subhash Khot IAS Elchanan Mossel UC Berkeley
description
Transcript of Subhash Khot IAS Elchanan Mossel UC Berkeley
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Subhash KhotIAS
Elchanan MosselUC Berkeley
Guy KindlerDIMACS
Ryan O’DonnellIAS
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Our main theorem
Unique Games Conjecture
+
Majority Is Stablest Conjecture
It is NP-hard to approximate MAX-CUT to within any factor better than αGW = .878…
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Conjectures? What?
Usual modus operandi in Mathematics:
Prove theorem, give talk.
Non-usual modus operandi in Mathematics:
Fail to prove two theorems, give talk.
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Why this is still interesting
Part 1: The status of the conjectures
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Unique Games conjecture
[Khot ’02]: A certain graph-coloring problem is NP-hard.
A simple way to think about it:
MAX-2LIN(m)
Input: Some two-variable linear equations mod m=10⁶, over n variables. You are promised that there is an assignment satisfying 99% of them.
Goal: Find an assignment satisfying 1% of them.
Status of UGC: ???. It would be a pity if it were false.
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Majority Is Stablest conjecture
Roughly speaking: among all boolean functions in which each coordinate has “small influence,” the Majority function is least susceptible to noise in the input.
Status of MISC: Almost certainly true, we claim.
• Preponderance of published evidence supports it
• Preponderance of expert opinion supports it
• We have some partial results
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“Beating Goemans-Williamson – i.e., approximating MAX-CUT to a factor .879 –
is formally harder* than the problem of
satisfying 1% of a given set of 99%-satisfiable two-variable linear equations mod 10⁶.”
So, Uri Zwick et al,
please work on this problem,
rather than this problem.
How we want you tointerpret our result
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Why this is still interesting
Part 2: More justification
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More justification• Natural, simple problem; no progress made on it in
years.
• Seemed as though there ought to be plenty of room for improving the GW algorithm.
• αGW is a funny number.
• Insight into the Unique Games Conjecture.
• Fourier methods and results independently interesting.
• Motivates algorithmic progress on other 2-variable CSPs: MAX-2SAT, MAX-2LIN(m), …
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Plan for the talk:
1. Describe the Unique Games Conjecture.
2. State the Majority Is Stablest Conjecture.
3. Sketch proof of main theorem.
4. Evidence for Majority Is Stablest Conjecture.
5. Conclusions and open problems
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Plan for the talk:
1. Describe the Unique Games Conjecture.
2. State the Majority Is Stablest Conjecture.
3. Sketch proof of main theorem.
4. Evidence for Majority Is Stablest Conjecture.
5. Conclusions and open problems
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Unique Games Conjecture
“Unique Label Cover” with m colors:
n
Labels
[m]
πu
v
πuv :
Bijections
Input
πuv
πuv
πuv
πuv
πuv
πuv
πuv
πuv
πuv
πuv
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Unique Games Conjecture
“Unique Label Cover” with m colors:
n
Labels
[m]
πu
v
πuv :
Bijections
Solution
πuv
πuv
πuv
πuv
πuv
πuv
πuv
πuv
πuv
πuv
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Unique Games Conjecture
Unique Games Conjecture [Khot ’02]:
“For every constant ε > 0,
there exists a constant m = m(ε)
such that
it is NP-hard to distinguish between
(1−ε)-satisfiable and ε-satisfiable
instances of Unique Label Cover with m labels.”
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Unique Games Conjecture
• A strengthening of the PCP Theorem of AS+ALMSS+Raz
• Implies hardness of MAX-2LIN(m).
• Implies MIN-2SAT-Deletion hard to approximate to within any constant factor [Khot ’02, Håstad], Vertex-Cover hard to approximate to within any factor smaller than 2 [Khot-Regev ’03]
• These results need an appropriate “Inner Verifier” – correctness follows from deep theorem in Fourier analysis. [Bourgain ’02; Friedgut ’98]
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Plan for the talk:
1. Describe the Unique Games Conjecture.
2. State the Majority Is Stablest Conjecture.
3. Sketch proof of main theorem.
4. Discuss the Majority Is Stablest Conjecture.
5. Conclusions and open problems
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Majority Is Stablest Conjecture
• Introduced formally by us in the present work.
• A related conjecture was made in [G. Kalai ’02], a paper about “Social Choice” theory from economics.
• Folkloric inklings of it have existed for a while. [Ben-Or-Linial ‘90, Benjamini-Kalai-Schramm ’98, Mossel-O. ’02, Bourgain ’02]
To state it, a few definitions are needed.
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Influences on boolean functions
Let f : {−1,1}ⁿ {−1,1} be a boolean function.
We view {−1,1}ⁿ as a probability space, uniform distribution.
Def: Let i [n]. Pick x at random and let y be x with the ith bit flipped. The influence of i on f is
Inf i ( f ) = Pr [ f(x) ≠ f(y) ].
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Influence examples
• Let f be the Dictator function, f (x) = x1.
Inf 1 ( f ) = 1, Inf i ( f ) = 0 for all i
≠ 1.
• Let f be the Parity function on n bits.Inf i ( f ) = 1 for all i.
• Let f be the Majority function on n bits.
Inf i ( f ) = + o(1) for all i. √n √2/π
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Noise sensitivityLet −1 < ρ < 1. Given a string x, “applying ρ-noise”
means:Pick y at random by choosing each coord. independently
and w.p. s.t. E[xi yi] = ρ. (Hence E[ x, y ] = ρn.)
For ρ > 0, this means for each bit of x,leave it alone w.p. ρ, replace it with a random bit w.p. 1−ρ.
[For ρ < 0, first set x = −x, ρ = −ρ, then do the above.]
Def: The noise sensitivity of f at ρ is
NSρ ( f ) = Pr [ f(x) ≠ f(y) ].
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Noise sensitivity examples
• Let f be the Dictator function.NSρ ( f ) = ½ − ½ ρ.
• Let f be the Parity function on n bits. NSρ ( f ) = ½ − ½ ρⁿ.
• Let f be the Majority function on n bits.NSρ ( f ) = (arccos ρ)/π ± o(1). [Central Lim.
Th.]
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NSρ ( Dict ) = ½ − ½ ρ
½
NS 1
10ρ
−1
NSρ( Maj ) = (arccos ρ)/π
−.69 =: ρ*
87.8%
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Majority Is Stablest Conjecture
Conjecture:
“Fix 0 < ρ < 1.
Let f : {−1,1}ⁿ {−1,1} be any boolean function* satisfying
f is balanced: E[ f ] = 0;
f has small influences: Inf i ( f ) < δ for all i.
Then
NSρ ( f ) ≥ (arccos ρ)/π − oδ(1).”
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Plan for the talk:
1. Describe the Unique Games Conjecture.
2. State the Majority Is Stablest Conjecture.
3. Sketch proof of main theorem.
4. Evidence for Majority Is Stablest Conjecture.
5. Conclusions and open problems
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Sketch of the main theorem• The main theorem gives a (poly-time) reduction from
Unique Label Cover to Gap-MAX-CUT.
• The reduction is parameterized by −1 < ρ < 1.
• (1−ε)-satisfiable ULC instances MAP TO: weighted graphs with cuts of weight
½ − ½ ρ − oε(1)
• ε-satisfiable ULC instances MAP TO: weighted graphs with no cuts more than
(arccos ρ)/π + oε(1)
• We choose our favorite ρ, viz. ρ*, and then MAX-CUT hardness is ratio of second quantity to first quantity.
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Sketch of the main theorem
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Sketch of the main theorem
{−1,1}m
(1,1,1)(1,1,−1) (−1,−1,−1)
· · ·
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Sketch of the main theorem
πuv
πuv
fv
fv
fv
fv
fv
fv
fv
fv
fv
fv
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Plan for the talk:
1. Describe the Unique Games Conjecture.
2. State the Majority Is Stablest Conjecture.
3. Sketch proof of main theorem.
4. Evidence for Majority Is Stablest Conjecture.
5. Conclusions and open problems.
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Noise stabilityIn working on the Majority Is Stablest Conjecture it
is more convenient to work with a linear fcn. of noise sensitivity.
Def: The stability of f at ρ is
Sρ ( f ) = 1 − 2 NS ρ ( f ).
Note: Sρ ( f ) = 1 − 2 Pr [ f(x) ≠ f(y) ] = 1 − 2 E[ ½ − ½ f(x) f(y) ]= E[ f(x) f(y) ].
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S ρ ( Dict ) = ρ
− 1
S 1
10ρ
−1
S ρ ( Maj ) = (2/π) arcsin ρ
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Evidence for Maj. Is Stablest
Note that Majority is “Uniformly Stable” – for fixed ρ, as n ∞, S ρ ( Majorityn ) is bounded
away from 0.
On the other hand, Parity is “Asymptotically Sensitive” – for fixed ρ, as n ∞, S ρ ( Parityn ) = ρⁿ 0.
The family of all boolean halfspaces – functions of the form sign(a1 x1 + · · · + an xn) –
are Uniformly Stable ([BKS ’98, Peres ’98]),
and in fact more is true…
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Evidence for Maj. Is Stablest
[BKS ’98] shows that the set of boolean halfspaces “asymptotically span” the Uniformly Stable functions:
• Uniformly Stable families of functions have Ω(1) correlation with the family of boolean halfspaces
• (monotone) function families are Asymptotically Sensitive iff they are asymptotically orthogonal to the set of boolean halfspaces
Theorem [us]: The Majority Is Stablest Conjecture is true when restricted to the set of boolean halfspaces.
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g(x) = Σ cS · Π
xi
Fourier detourAny g : {−1,1}ⁿ R can be expressed as a
multilinear polynomial:
Def: For 0 ≤ k ≤ n, the weight of g at level k is
w(k) = Σ ĝ(S)²
g(x) = Σ ĝ(S) · Π xi
S [n] i S
|S| = k
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Fourier facts• if g : {−1,1}ⁿ [−1,1],
Σk w(k) ≤ 1 (equality if
{−1,1})
• if g is balanced (E[g] = 0), then w(0) = 0
• Inf i ( g ) = Σ ĝ(S)²
• Sρ ( g ) = Σk w(k) · ρk
“The more g’s weight is at lower levels, the stabler g is.”
S i
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Maj. Is Stablest evidence
Conjecture [Kalai ’02]: The “symmetric” boolean-valued function [“symmetric” implies small influences] with most weight on levels 1… k is Majority.
Thm [Bourgain ’02]: Boolean-valued functions with small influences have at least as much weight beyond level k as Majority (asymptotically).
Thm [us]: Bounded functions with small influences have no more weight at level 1 than 2/π, precisely the weight of Majority at level 1.
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Corollary of our level-1 result
Weakened version of Majority Is Stablest Conjecture:
Thm: If f : {−1,1}ⁿ [−1,1] has small influences and ρ < 0,
NSρ ( f ) ≥ ½ − ρ/π − (½ − 1/π)ρ³ − o(1).
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½
NS 1
10ρ
−1 −½ =: ρ*
¾ + 1/2π
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Corollary of our level-1 result
Weakened version of Majority Is Stablest Conjecture:
Thm: If f : {−1,1}ⁿ [−1,1] has small influences and ρ < 0,
NSρ ( f ) ≥ ½ − ρ/π − (½ − 1/π)ρ³ − o(1).
Cor: The Unique Games Conjecture implies it is NP-hard to approximate MAX-CUT to any factor larger than
¾ + 1/2π = .909… < 16/17 = .941…
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Plan for the talk:
1. Describe the Unique Games Conjecture.
2. State the Majority Is Stablest Conjecture.
3. Sketch proof of main theorem.
4. Evidence for Majority Is Stablest Conjecture.
5. Conclusions and open problems.
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Conclusions and open problems
“Beating Goemans-Williamson is harder than cracking Unique Label Cover or MAX-2LIN(m).”
Open problems:
• Prove Majority Is Stablest Conjecture.
• What balanced m-ary function f : [m]ⁿ [m] is stablest?
A conjecture: Plurality.
Thm [us]: Noise stability of Plurality is
m(ρ-1)/(ρ+1) + o(1).
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Conclusions and open problems
Connections between stability conjectures and Unique Games Conjecture:
• Proving that m-ary stability is om(1) is probably
enough to show that UGC implies hardness of (hence, essentially, equivalence with) MAX-2LIN(m).
• Proving a sharp bound for the m-ary stability problem would give strong results for the UGC w.r.t. how big m needs to be as a function of ε.