Prasad RaghavendraGeorgia Institute of Technology,

Atlanta, GA

Complexity of

Constraint Satisfaction Problems Exact and Approximate

akaWhere does the number 0.878 come from?

Constraint Satisfaction Problem

A constraint satisfaction problem Λ : Λ = (A finite domain [q]={1,2,..q},

Set of Predicates/Relations {P1,P2,.. Pr})Example:

MaxCut = ({0,1}, {P(a,b) = a NOTEQUAL b})3-SAT = ({0,1},

{P1(a,b,c) = a ѵ b ѵ c, P2(a,b,c) = ¬a ѵ b ѵ c …. P8(a,b,c) = ¬a ѵ ¬ b ѵ ¬ c P9(a,b) = a ѵ b, P10(a,b) = ¬a ѵ b … P12(a,b) = ¬a ѵ ¬ b )

P1

Instance of Λ –CSP I:• Set of variables: {x1 , x2, .. Xn}• Set of constraints: Predicates from Λ applied to

subsets of variables.

x1 x2 x3 x9 xnP31 P13

Max-Λ-CSP: “Given an instance of Λ-CSP, find an assignment to the variables, that satisfies the maximum number of constraints.”

Remarks:1) Use fraction of constraints instead of number of constraints (objective value always between

0 and 1).2) The constraints can be weighted, then one maximizes the total weight. Easily seen to be

equivalent.3) Predicates can be replaced by bounded real valued payoffs (Generalized CSPs)

Exact-Λ-CSP: “Given an instance of Λ-CSP, is it possible to satisfy all the constraints?”

Exact-Λ-CSP

Complexity of Exact-Λ-CSP Theorem [Schaefer]Among boolean CSPs Λ , Exact-Λ-CSP is in P for the following cases:• 2-SAT ¬a ѵ b • Horn-SAT b a ѵ c ѵ d• Dual Horn-SAT a ^ b ^ c d• Linear Equations Mod 2• CSPs where all 0s or all 1s is a solution.

The problem is NP-hard for all other CSPs.

Any Pattern?

The Pattern

P1x1 x2 x3 x9 xnP31 P13

0 1 1 0 0 1 1 1 0 1

1 1 1 1 1 1 1 1 1 1

1 0 1 0 0 1 0 1 0 1

Given an instance I of Linear Equations Mod 2

For every 3 solutions {X1, X2, X3} to instance I,

X1 =

X3 =

X2 =

is also a solution to instance I (Here XOR is applied for each variable separately)

0 1 1 01 1 1 1 1 1

321 XXX

321 XXX

PolymorphismsA function F : [q]R -> [q] for some constant R is a “polymorphism” for a CSP Λ if,

P1x1 x2 x3 x9 xnP31 P13

0 1 1 0 0 1 1 1 0 1

1 1 1 1 1 1 1 1 1 1

1 0 1 0 0 1 0 1 0 1

For every instance I of CSP Λ,

For every set of R solutions {X1, X2, … XR} to instance I,

X1 =

XR =

X2 =

F(X1, X2, … XR) is also a solution to instance I (Here F is applied for each variable separately)

0 1 1 01 1 1 1 1 1F(X1, X2, … XR) =

Polymorphisms and Complexity of Exact CSP

Examples: The dictator functions F(x) = xi are polymorphisms for all CSPs Λ

(Algebraic Dichotomy Conjecture) Exact CSP Λ is in P if and only if there are “non-trivial” polymorphisms , i.e., polymorphisms that are very different from dictators. (precisely defined in [Bulatov-Jeavons-Krohkin])

Tractable Boolean CSPsLinear Equations Mod 2 XOR2-SAT MajorityHorn-SAT ANDDual Horn-SAT OR

Conjecture is proven for domain sizes 2 [Schraefer] and 3 [Bulatov]

P1x1

x2

x3

x9

xn

P31 P130 1 1 0 0 1 10 1

1 1 1 1 1 1 1 1 1

1 0 0 1 0 1 0 1

Max-Λ-CSP

Max-Λ-CSP

Approximability Threshold:“αΛ is the largest constant for which there is an αΛ approximation for Max- Λ-CSP”

Approximability: An algorithm A is an α-approximation for a Max-Λ-CSP if for every instance I,

A(I) ≥ α OPT(I)∙

Approximability Curve: αΛ(c)“Given an instance I of CSP Λ, with value at least c what is the largest constant αΛ(c) for which there is an αΛ(c) approximation for Max- Λ-CSP?”

ALGORITHMS[Charikar-Makarychev-Makarychev 06]

[Goemans-Williamson][Charikar-Wirth]

[Lewin-Livnat-Zwick][Charikar-Makarychev-Makarychev 07]

[Hast] [Charikar-Makarychev-Makarychev 07]

[Frieze-Jerrum][Karloff-Zwick]

[Zwick SODA 98][Zwick STOC 98]

[Zwick 99][Halperin-Zwick 01]

[Goemans-Williamson 01][Goemans 01]

[Feige-Goemans][Matuura-Matsui]

[Trevisan-Sudan-Sorkin-Williamson]

Approximability of Max-CSP

MAX CUT

MAX 2-SAT

MAX 3-SAT

MAX 4-SAT

MAX DI CUT

MAX k-CUT

Unique GamesMAX k-CSP

MAX Horn SAT

MAX 3 DI-CUTMAX E2 LIN3

MAX 3-MAJ

MAX 3-CSPMAX 3-AND

0 1

NP HARD

NP Hardness Results[Hastad]

[Samorodnitsky-Trevisan]

Unique GamesA Special Case

E2LIN mod pGiven a set of linear equations of the form:

Xi – Xj = cij mod pFind a solution that satisfies the maximum number of equations.

x-y = 11 (mod 17)x-z = 13 (mod 17)

…….

z-w = 15(mod 17)

Unique Games Conjecture [Khot 02]

An Equivalent Version [Khot-Kindler-Mossel-O’Donnell]

For every ε> 0, the following problem is NP-hard for large enough prime p

Given a E2LIN mod p system, distinguish between:• There is an assignment satisfying 1-ε fraction of the equations.• No assignment satisfies more than ε fraction

of equations.

Assuming UGCUGC Hardness

Results[Khot-Kindler-Mossel-O’donnell]

[Austrin 06][Austrin 07]

[Khot-Odonnell][Odonnell-Wu]

[Samorodnitsky-Trevisan]

NP HARDUGC HARD

0 1

MAX CUT

MAX 2-SAT

MAX 3-SAT

MAX 4-SAT

MAX DI CUT

MAX k-CUT

Unique GamesMAX k-CSP

MAX Horn SAT

MAX 3 DI-CUTMAX E2 LIN3

MAX 3-MAJ

MAX 3-CSPMAX 3-AND

Any Pattern?

Approximate Polymorphisms

A function F : [q]R -> [q] for some constant R is an “α-approximate polymorphism” for a CSP Λ if,

P1x1 x2 x3 x9 xnP31 P13

0 1 1 0 0 1 1 1 0 1

1 1 1 1 1 1 1 1 1 1

1 0 1 0 0 1 0 1 0 1

For every instance I of CSP Λ, and c > 0,

For every set of R assignments {X1, X2, … XR} to instance I, that satisfy c fraction of constraints

X1 =

XR =

X2 =

F(X1, X2, … XR) satisfies at least α c fraction of constraints (Here F is applied for each variable separately)

0 1 1 01 1 1 1 1 1F(X1, X2, … XR) =

“(α,c) –approximate polymorphisms”

Fix the value of c in the CSP instance.

Distributional Function

Definition: A distributional function is a map F: [q]R {Probability distribution over [q]}}Alternately, F: [q]R such that F1(x) + F2(x) + .. Fq (x) = 1 and Fi(x) ≥ 0

Definition: A DDF Ψ is a probability distribution over distributional functions F Є Ψ over [q]R

F: [q]R {Probability distribution over [q]}}

q

Approximate PolymorphismsA DDF Ψ for some constant R is an “α-approximate polymorphism” for a CSP Λ if,

For every instance I of CSP Λ, and c > 0,For every set of R assignments {X1, X2, … XR} to instance I, that satisfy c fraction of constraints

•Sample a distributional function F Є Ψ•Apply F to each bit separately.

The expected value of the solution returned is at least α c

P1x1 x2 x3 x9 xnP31 P13

0 1 1 0 0 1 1 1 0 1

1 1 1 1 1 1 1 1 1 1

1 0 1 0 0 1 0 1 0 1

0 1 1 01 1 1 1 1 1P1 P2 P3

InfluencesDictator functions are trivially

1-approximate polymorphisms for every CSP.Non-trivial not like a dictator.

Definition: Influence of the ith co-ordinate on a function F:[q]R under a product distribution μR is defined as:

Infiμ (F) = E [ Variance [F] ]

q

Random Fixing of All Other Coordinates from μR-1

over changing the ith coordinate as per μ

Definition: A function is τ-quasirandom if for all product distributions μR and all i, Infi

μ (F) ≤ τ

(For the ith dictator function : Infiμ (F) is as large as variance of F)

Example 1

• MaxCut101010101010101010101001 01010101001010101001010 1 101010100101010101010010

----------------------------------- ??????????

-----------------------------------– Majority function is a polymorphism for Exact-2-SAT– Majority function is -approximate

polymorphism for Max-Cut.)1,1(

1

1015

3

71

1

1

-1

-1

-1-1

-1

Example 2

Submodular CSPs like MinCutP is submodular if

P(X) + P(Y) >= P(X and Y) + P(X or Y)

So if X, Y in {0,1}n have value = c, then, With probability ½ use X and Y With probability ½ use X or Y The expected cost of combined solution is <= c “½ AND + ½ OR” is a 1-approximate polymorphism.

For a CSP Λ, αΛ = largest constant such that there are

αΛ–approximate non-trivial polymorphisms

Complexity of Approximability

αΛ–approximate τ –quasirandom polymorphisms for every τ > 0

(Analogue of Algebraic Dichotomy Conjecture):“For every Max-CSP Λ, αΛ is the threshold of

approximability of Max-CSP Λ”.

Algorithm: can approximate to factor αΛ

Hardness: cannot approximate to better than αΛ factor

True for all known approximation

thresholds

Define: αΛ (c) -Restricted to instances with value ≥ c.

Theorem [R 08]For every ε >0, every Max-CSP Λ, can be approximated

within a ratio αΛ - ε in time exp(exp(poly(1/ ε, |Λ|))· poly(n).

For every Max-CSP Λ and c , ε >0, it is NP-hard to approximate the problem on instances with value c-ε to a factor better than αΛ (c)

Hardness and Algorithm

For every Max-CSP Λ, it is NP-hard to approximate the problem better than αΛ

Unique Games Conjecture [Khot 02]

• Generalization of the reduction of [Khot-Kindler-Mossel-O’Donnell]• A slightly more general version of hardness is equivalent to Unique Games Conjecture.

• Algorithm based on semidefinite programming – LC relaxation.• Settles the approximability of every CSP under Unique Games Conjecture.

More Generally,Theorem Under UGC, this semidefinite program (LC)

yields the “optimal” approximation for:Every Generalized Constraint Satisfaction Problem: bounded real valued functions instead of predicates. (minimization problems also - Metric Labelling)

[R 08]

On instances with value c,• Under UGC, it is hard to approximate instances with

value c to a factor better than αΛ (c+ ε)+ ε• The LC relaxation approximates instances with value c

to αΛ (c- ε)- ε

Remarks

Theorem [R 08]

For every Max-CSP Λ, the value of αΛ can be computed within an error ε in time

exp(exp(poly(1/ε,|Λ|)))

• Approximation threshold αΛ is not very explicit.

•Including all local constraints on up to 2O(loglogn^0.5) variables to LC Semidefinite program does not improve approximation ratios for any CSP, Ordering CSP or Labelling problem.

[R-Steurer 09][Khot-Saket]•LC Semidefinite program can be solved in near linear time in number of constraints. [Steurer 09]

Rest of the Talk

• Algorithm:– Intuitive Idea– Invariance Principle– Description of LC semidefinite program– Random Projections

• Hardness: – Dictatorship Tests– Polymorphisms vs Dictatorship Tests

• Future Work

AlgorithmTheorem [R 08]

For every ε >0, every Max-CSP Λ, can be approximated within a ratio αΛ - ε in time

exp(exp(poly(1/ ε, |Λ|))· poly(n).

Design an αΛ -approximation algorithm using the existence of αΛ - approximate polymorphism.

Intuitive Idea

P1x1 x2 x3 x9 xnP31 P13

0 1 1 0 0 1 1 1 0 1

1 1 1 1 1 1 1 1 1 1

1 0 1 0 0 1 0 1 0 1

0 1 1 01 1 1 1 1 1

Input: An instance I of the Max-CSP ΛWe Know: For every τ > 0 αΛ–approximate τ –quasirandom polymorphisms.

F : [q]R [q]

R Optimal Solutions

Foolish Algorithm: Take R Optimal Solutions X1 , X2 ,.. , XR, Apply F(X1,..XR) to get a solution that is αΛ–approximate optimal.

A Plausible Algorithm:• Relax the constraint that solutions are integral {0,1} - allow real values. Using semidefinite programming, one can generate “real valued optimal solutions”.• Feed in these real valued optimal solutions.• A τ –quasirandom polymorphism cannot distinguish between integral solutions and these real valued solutions.

0.5 1.2 -1 0.3 0.3 1.2 1.1 1.1 0.1 1

1.1 1 -1 1 1 1 1 1 0.8

1.2 0.3 1 0 0 1.1 0 1 0.1

Multi-linear Expansion

Every function F : {-1,1}R can be written as a multilinear polynomial using Fourier Expansion.

Example: AND(x(1) ,x(2)) = (1-x(1)) )(1-x(2))/4 OR(x(1),x(2)) = 1 – (1-x(1))(1-x(2))

More generally, the polynomial PF corresponding to F

Any function F:[q]R ! can be written as a multilinear polynomial after suitably arithmetizing the inputs.

][

ˆ)(RS Si

iSF xFxP

where ])([ˆ}1,1{

Si

ix

S xxFEFR

If Z1, Z2 , Z3 and Y1 , Y2 , Y3 are two sets of random variables with matching first two moments.

Z11, Z2

1 , Z31 Y1

1 , Y21, Y3

1 Z1

2, Z22 , Z3

2 Y12 , Y2

2 , Y32

Z13, Z2

3 , Z33 Y1

3 , Y23 , Y3

3

…… ,………….

Z1R, Z2

R , Z3R Y1

R , Y2R , Y3

R

<---->

Invariance Principle for Low Degree Low Influence Polynomials[Rotar] [Mossel-O’Donnell-Oleszkiewich], [Mossel 2008]

If P(X1 ,… XR) is a constant degree polynomial,• Infi (P) ≤ τ for all i (All influences are small),• Z1 , Z2 ,.. ZR are iid random variables,then the distribution of P(Z1 ,Z2 , .. ZR) only depends on the first and second

moments of random variables Zi. (P cannot distinguish between two sets of random variables with the same first two moments)

Central Limit Theorem:``Sum of large number of {-1,1} random variables

has similar distribution asSum of large number of Gaussian random variables.”

For the polynomial P(X) = , the distribution of P(X) is similar for X being iid {-1,1} or iid Gaussian vector.

1pR

P Ri=1 X i

P(Z1) P(Z2) P(Z3) P(Y1) P(Y2) P(Y3)

Noise 0 1 1 0 0 1 1 1 0 1

1 1 1 1 1 1 1 1 1 1

1 0 1 0 0 1 0 1 0 1

P1x1 x2 x3 x9 xnP31 P13

Proof: Let X1 ,X2 ,.. XR be solutions to instance I with value c.Perturb each coordinate of each solution Xi probability ε

Yi = Xi with ε noiseThe expected value of each perturbed solution is > c – O(ε)The expected value of solution F(Y1 , .. , YR) is at least αΛ (c-O(ε))

Lemma:Let H(X ) = EY [F(Y)] where Y = X with ε noise Yi = Xi with 1- ε probability, random bit with prob ε.Then H is a αΛ - O(ε) polymorphism.

Advantage: H is a αΛ - O(ε) polymorphism essentially a low-degree function due to averaging.

Let F be an αΛ -approximate polymorphism

Semidefinite Program for CSPs

Variables :For each variable Xa

Vectors {V(a,0) , V(a,1)}

For each clause P = (xa ν xb ν xc),Scalar variables

μ(P,000) , μ(P,001) , μ(P,010) , μ(P,100) , μ(P,011) , μ(P,110) , μ(P,101) , μ(P,111)

))()()(( 145532532321 xxxxxxxxxxxx

Xa = 1 V(a,0) = 0 V(a,1) = 1

Xa = 0 V(a,0) = 1 V(a,1) = 0

If Xa = 0, Xb = 1, Xc = 1

μ(P,000) = 0 μ(P,011) = 1μ(P,001) = 0 μ(P,110) = 0μ(P,010) = 0 μ(P,101) = 0μ(P,100) = 0 μ(P,111) = 0

Objective Function :

PClauses sassignment

PP3}1,0{

),()(

Constraints : For each clause P,

0 ≤μ(P,α) ≤ 1

For each clause P (xa ν xb ν xc), For each pair Xa , Xb in P,

consitency between vector and LP variables.

V(a,0) V∙ (b,0) = μ(P,000) + μ(P,001) V(a,0) V∙ (b,1) = μ(P,010) + μ(P,011) V(a,1) V∙ (b,0) = μ(P,100) + μ(P,101) V(a,1) V∙ (b,1) = μ(P,100) + μ(P,101)

1),(

P

Semidefinite Relaxation for CSPSDP solution for =:

SDP objective:

for every constraint ɸ in I- local distributions µφ over

assignments to the variables of ɸ

Example of local distr.: ɸ = 3XOR(x3, x4, x7)

x3 x4 x7 ¹Á0 0 0 0.10 0 1 0.010 1 0 0 …1 1 1 0.6for every variable xi in =

- vectors vi,1 , … , vi,q

constraints

(also for first moments)

Explanation of constraints:first and second moments of distributions are consistent and form PSD matrix

maximize

Simplest SDP that yields, Local Distributions over Integral Assignments with

first two moments matching inner product of SDP vectors.

Gaussian ProjectionsSample g = a random Gaussian vector inGenerate real valued solution Z = Z1 , Z2 ,… …… Zn-1 Zn by random projection along gwhere Zi = vi ¢ g

n

Lemma: For every constraint Á in the instance I, the first two moments of random variables {Zi | i 2 Á} agree with local distribution ¹Á. Formally, for all i ,j 2 Á,

Exi,xj » ¹Á [xi xj] = E[Zi Zj] Exi » ¹Á

[xi] = E[Zi]

Algorithm:•Sample R independent Gaussian vectors {g1,g2,.. gR} in•Generate the corresponding Gaussian projections.•Apply the polynomial PH corresponding to H. •Outputs are not distributions, but close to distributions over {0,1}. Round them to the nearest distribution, and sample from them.

Algorithm

Setup: • {v1, v2 , v3 ,… vn } - SDP

vectors. • H be a low degree

αΛ - O(ε) approximate polymorphism.

n

0.5 1.2 -1 0.3 0.3 1.2 1.1 1.1 0.1 1

1.1 1 -1 1 1 1 1 1 0.8

1.2 0.3 1 0 0 1.1 0 1 0.1

P1x1 x2 x3 x9 xnP31 P13

P(Z1) P(Z2) P(Z3)Q (Z1) Q (Z2) Q (Z3)0 1 0 0 0 0 0

Each row satisfies in expectation constraints. Output satisfies in expectation.

Analysis of AlgorithmConsider a predicate φ of the instance I. Let µφ be local distribution for φ

)]([ xEx

φx1 x2 x3 x4

0 1 1 01 1 1 10 1 1 0

0 1 1 01 1 1 1

Xi from µφ

0.5 1.2 0.1 01 1.1 1.2 10 1.3 1.3 0

0 3.2 1.3 01 4.3 0.9 1

Zi from Gaussian projection

)]([ xEx

By

invarianceOutput satisfiesin expectation.

)]([ xEx

Summing up over all constraints,Output value ≥ αΛ * SDP value

≥ αΛ *OPT

Hardness

Dictatorship TestGiven a function F : {-1,1}R {-1,1}•Toss random coins•Make a few queries to F •Output either ACCEPT or REJECT

F is a dictator functionF(x1 ,… xR) = xi

F is far from every dictator function

(No influential coordinate)

Pr[ACCEPT ] = Completeness

Pr[ACCEPT ] =Soundness

UG Hardness

Rule of Thumb: [Khot-Kindler-Mossel-O’Donnell]A dictatorship test where • Completeness = c and Soundness = αc•the verifier’s tests are predicates from a CSP Λ

It is UG-hard to approximate CSP Λ to a factor better than α

Polymorphisms and Dictatorship TestsBy definition of αΛ ,

For every α > αΛ there is no α-approximate polymorphism. there is some instance I and R solutions of value c such that

every low influence function F yields solution of value < αc.

P1x1 x2 x3 x9 xnP31 P13

0 1 1 0 0 1 1 1 0 1

1 1 1 1 1 1 1 1 1 1

1 0 1 0 0 1 0 1 0 1

Dictatorship Test:

Function : F:{0,1}R {0,1}•Pick a random constraint P from the instance I

• Let X1 , X2 ,.. Xk denote the R dimensional vectors in {0,1}R corresponding to variables in P

•Query the function values F(X1) F(X2) … F(Xk)

•Accept if P(F(X1) , F(X2) … , F(Xk) ) is TRUE.

vv v

Completeness = cSoundness = αc

FUTURE WORK

Disproving Unique Games Conjecture

On UG instances with value 1-ε, Alphabet size = p, Instance Size = n, Spectral Gap λ

Algorithm On (1-Є) satisfiable instances

[Khot 02]

[Trevisan]

[Gupta-Talwar] 1 – O(ε logn)

[Charikar-Makarychev-Makarychev]

[Chlamtac-Makarychev-Makarychev]

[Arora-Khot-Kolla-Steurer-Tulsiani-Vishnoi]

)2/( p)loglog(1 pnO

)log(1 3 nO

))/1log((1 5/12 pO

1log1

Disproving Unique Games Conjecture

Theorem: [Arora-Khot-Kolla-Steurer-Tulsiani-Vishnoi] Unique Games is easy if constraint graph is a sufficiently good expander

Spectral Gap (λ) > 10* Completeness Gap (10 ε)

Theorem: [Arora-Barak-Steurer 09] Unique Games with completeness 1- ε can be solved in time 2^{n^poly(ε)}

Above algorithms use the basic SDP cannot solve the known integrality gap

instances.

Theorem: [Arora-Russell-Steurer 09][R-Steurer 09] Unique Games is easy if there is sufficiently good local expansion

Constraint Satisfaction Problems

Metric Labelling Problems

Ordering Constraint Satisfaction Problems

Kernel Clustering Problems

Grothendieck Problem

Unique Games Conjecture

Vertex Cover

Sparsest Cut

MISSING:•Steiner Tree•Assymetric Travelling Salesman•Metric Travelling Salesman

More UG Hardness Results?

Reverse ReductionsCSPs, Vertex Cover, Sparsest Cut, Metric Labelling, kernel

clustering

Unique Games

Failed Approach: Use parallel repetition on MaxCut to reduce it to Unique Games. [Raz 08] [Barak-Hardt-Haviv-Rao-Regev-Steurer] [Kindler-O’Donnell-Rao]

Theorem: [R-Steurer 09]A variant of Sparsest Cut problem reduces to Unique Games.

Power of Linear and Semidefinite Programs

Theorem [Charikar-Makarychev-Makarychev]Even nα rounds of certain Linear programming hierarchies like

Sherali-Adams, Lovasz-Schriver do not disprove UGC.

Possibility: Adding a simple constraint on every 5 variables in the LC SDP

relaxation yields a better approximation for MaxCut and disproves Unique Games Conjecture!

Theorem [R-Steurer 09][Khot-Saket 09]Adding all local constraints on up to 2O(loglogn^{1/4}) variables does not

disprove UGC.

logn

Integrality Gaps via UG Hardness Results

Rule of Thumb: [Khot-Vishnoi 05]Integrality Gaps compose with Hardness Reductions

Unique Games Instance(Integrality Gap)

Instance of MaxCut

(Integrality Gap)

Hardness

Reduction

UGC based hardness results yield integrality gaps – limitations of semidefinite or linear programs.

Back to Exact CSPsIntroduced ε noise to make the polymorphism low degree.

will not get fully satisfiable assignments.

Definition: A function F is noise stable if ε perturbation of inputs changes F with probability g(ε) such that g(ε)0 as ε0

Theorem: If there exists noise stable low influence polymorphisms for a CSP Λ then it is tractable (using semidefinite programming).

• Above condition holds for all boolean CSPs except linear equations.• Semidefinite programs cannot solve linear equations over {0,1}.• SDPs can solve all bounded width CSPs trivially.

OPEN PROBLEM: Can one show that if CSP Λ does not contain the affine type, then it has noise stable low influence polymorphisms?

Thank You