Subhash KhotIAS

Elchanan MosselUC Berkeley

Guy KindlerDIMACS

Ryan O’DonnellIAS

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…

Conjectures? What?

Usual modus operandi in Mathematics:

Prove theorem, give talk.

Non-usual modus operandi in Mathematics:

Fail to prove two theorems, give talk.

Why this is still interesting

Part 1: The status of the conjectures

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.

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

“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

Why this is still interesting

Part 2: More justification

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), …

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

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

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

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

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.”

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]

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

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.

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) ].

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

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) ].

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.]

NSρ ( Dict ) = ½ − ½ ρ

½

NS 1

10ρ

−1

NSρ( Maj ) = (arccos ρ)/π

−.69 =: ρ*

87.8%

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).”

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

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.

Sketch of the main theorem

Sketch of the main theorem

{−1,1}m

(1,1,1)(1,1,−1) (−1,−1,−1)

· · ·

Sketch of the main theorem

πuv

πuv

fv

fv

fv

fv

fv

fv

fv

fv

fv

fv

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.

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) ].

S ρ ( Dict ) = ρ

− 1

S 1

10ρ

−1

S ρ ( Maj ) = (2/π) arcsin ρ

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…

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.

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

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

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.

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).

½

NS 1

10ρ

−1 −½ =: ρ*

¾ + 1/2π

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…

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.

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).

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.