Ryan O'Donnell (CMU, IAS)Yi Wu (CMU, IBM)Yuan Zhou (CMU)

Solving linear equations• Given a set of linear equations over reals, is there a

solution satisfying all the equations?– Easy : Gaussian elimination.

Noisy version• Given a set of linear equations for which there is a

solution satisfying 99% of the equations, – can we find a solution that satisfies at least 1% of

the equations?

• I.e. 99% vs 1% approximation algorithm for linear equations over reals?

Hardness of Max-3Lin(q)• Theorem. [Håstad '01] Given a set of linear

equations modulo q, it is NP-hard to distinguish between– there is a solution satisfying (1 - ε)-fraction of the

equations – no solution satisfies more than (1/q + ε)-fraction of

the equations

• Equations are sparse, and are of the form xi + xj - xk = c (mod q)

• (1 - ε) vs (1/q + ε) approx. for Max-3Lin(q) is NP-Hard

• A 3-query PCP of completeness (1 - ε), soundness (1/q + ε)

Sparser equations: Max-2Lin(q)

• Theorem. [KKMO '07] Assuming Unique Games Conjecture, for any ε, δ > 0, there exists q > 0, such that (1 - ε) vs δ approx. for Max-2Lin(q) is NP-Hard

Max-3Lin Max-2Lin

over [q]

(1 - ε) vs (1/q + ε)

NP-hardness[Håstad '01]

(1 - ε) vs δ UG-hardness[KKMO '07]

overintegers/reals ? ?

Equations over integers: Max-3Lin(Z)• Approximate Max-3Lin/Max2Lin over large domains?

• Intuitively, it should be harder, because when domain size increases,– soundness becomes smaller in both [Håstad '01]

and [KKMO '07]

• Obstacle of getting hardness– "Long code" becomes too long (even infinitely

long)

Hardness of Max-3Lin(Z)• Theorem. [Guruswami-Raghavendra '07] For all ε, δ

> 0, it is NP-Hard to (1 - ε) vs δ approximate Max-3Lin(Z) – 3-query PCP over integers– Implies the hardness for Max-3Lin(R)

• Proof follows [Håstad '01], but much more involved– derandomized Long Code testing

– Fourier analysis with respect to an exponential distribution on Z+

Max-3Lin Max-2Lin

over [q]

(1 - ε) vs (1/q + ε)

NP-hardness[Håstad '01]

(1 - ε) vs δ UG-hardness[KKMO '07]

over integers/real

s

(1 - ε) vs δ NP-hardness

[GR '07]?

Unique Games over Integers?• Can we use the techniques in [Guruswami-

Raghavendra '07] prove a (1 - ε) vs δ UG-hardness for Max-2Lin(Z)?

– Seems difficult

– Open question from Raghavendra's thesis [Raghavendra '09] :

Our results

• Relatively easy to modify the KKMO proof to get

– Theorem. For all ε, δ > 0, it is UG-Hard to (1 - ε) vs δ approximate Max-2Lin(Z) • Also applies to Max-2Lin over reals and large

domains

– Simpler proof (and better parameters) of Max-3Lin(Z) hardness

Dictatorship Test

• Theorem. For all ε, δ > 0, it is UG-Hard to (1 - ε) vs δ approximate Max-2Lin(Z)

• By [KKMO '07], only need to design a (1 - ε) vs δ 2-query dictatorship test over integers.

Dictatorship Test (cont'd)

• f: [q]d -> Z is called a dictator if f(x1, x2, ..., xd) = xi (for some i)

• Dictatorship test over [q]: a distribution over equations

f(x) - f(y) = c (mod q)– Completeness: for dictators, Pr[equation holds]

≥ 1 - ε– Soundness: for functions far from dictators,

Pr[equation holds] < δ

(1 - ε) vs δ hardness of Max-2Lin(q)

Dictatorship Test over Integers• A distribution over equations f(x) - f(y) = c

– Completeness: for dictators, Pr[f(x) - f(y) =c] ≥ 1 - ε

– Soundness: for functions far from dictators, Pr[f(x) - f(y) = c mod q] < δ

• It is UG-Hard to distinguish between– a Max-2Lin(Z) instance is (1 - ε)-satisfiable– the instance is not δ-satisfiable even when the the

equations are modulo q

Recap of KKMO Dictatorship Test

Back to KKMO Dictatorship Test•Dictatorship test over [q]: a distribution over equations f(x) - f(y) = c (mod q)

•Completeness: for dictators, Pr[equation holds] ≥ 1 - ε•Soundness: for functions far from dictators, Pr[equation holds] < δ

•KKMO Test

•Pick x ∈ [q]d by random•Get y by rerandomizing each coordinate of x w.p. ε•Test f(x) - f(y) = 0 (mod q)

Back to KKMO Dictatorship Test (cont'd)•KKMO Test

•Pick x ∈ [q]d by random•Get y by rerandomizing each coordinate of x w.p. ε•Test f(x) - f(y) = 0 (mod q)

• Soundness analysis

"Majority Is Stablest" Theorem [MOO '05]– If f is far from dictators and "β-balanced", then

Pr[f passes the test] < βε/2

– f is β-balanced : Pr[f(x) = a mod q] < β for all 0 ≤ a < q

Back to KKMO Dictatorship Test (cont'd)•KKMO Test

•Pick x ∈ [q]d by random•Get y by rerandomizing each coordinate of x w.p. ε•Test f(x) - f(y) = 0 (mod q)

• Soundness analysis– "Folding" trick: to make sure f is β-balanced

– Idea: when query f(x) = f(x1, x2, ..., xn), return

g(x) = f(0, (x2 - x1) mod q, ..., (xn - x1) mod q) + x1

– Dictators not affected in completeness analysis– g(x) is 1/q-balanced

Dictatorship Test for Max-2Lin(Z)• A distribution over equations f(x) - f(y) = c

– Completeness: for dictators, Pr[f(x) - f(y) =c] ≥ 1 - ε– Soundness: for functions far from dictators, Pr[f(x) - f(y) = c mod q] < δ

•

• If we use KKMO test...– Soundness: the same,– Completeness does not hold, because

• when query f(x), get g(x) = (xi - x1) mod q + x1

• when query f(y), get g(y) = (yi - y1) mod q + y1

Max-2Lin(q): Pr[g(x) - g(y) = 0 mod q] ≥ 1 - ε Max-2Lin(Z): Pr[g(x) - g(y) ≠ 0] ≥ Pr["wrap-around" (exactly one of g(x), g(y) ≥ q)] ≈

1/2

Our method

Step IIntroducing the new "active folding"

The new "active folding"

• Completeness:• Soundness:

– Claim. g(x) = f(x1 - c, ..., xn - c) + c is 1/q-balanced

– Proof. Prx,c[f(x1 - c, ..., xn - c) + c = a mod q]

= Ec [Prx[f(x1 - c, ..., xn - c) = a - c mod q] ]

= Ec [Prx[f(x) = a - c mod q] ]

= Ex [Prc[f(x) = a - c mod q] ] ≤ 1/q

•KKMO Test with active folding

•Pick x ∈ [q]d by random•Get y by rerandomizing each coordinate of x w.p. ε

•Pick c, c' ∈ [q] by random, test

f(x1 - c, ..., xn - c) + c = f(y1 - c', ..., yn - c') + c' (mod q)

mod q

Our method

Step II"Partial active folding"

"Partial active folding"

• Completeness:

– f(x1 - c, ..., xn - c) + c = (xi - c) mod q + c

= (xi - c) + c = xi w.p. 1 - 1/q0.5

– f(y1 - c', ..., yn - c') + c' = yi w.p. 1 - 1/q0.5

Pr[f(x1-c, ..., xn-c)+c = f(y1-c', ..., yn-c')+c'] ≥ 1 - ε - 2/q0.5

•KKMO Test with partial active folding for Max-2Lin(Z)

•Pick x ∈ [q]d by random•Get y by rerandomizing each coordinate of x w.p. ε

•Pick c, c' ∈ [q0.5] by random, test

f(x1 - c, ..., xn - c) + c = f(y1 - c', ..., yn - c') + c'

"Partial active folding" (cont'd)

• Completeness:• Soundness:

– Claim. g(x) = f(x1 - c, ..., xn - c) + c is 1/q0.5-balanced

– Proof. Prx,c[f(x1 - c, ..., xn - c) + c = a mod q]

= Ec [Prx[f(x1 - c, ..., xn - c) = a - c mod q] ]

= Ec [Prx[f(x) = a - c mod q] ]

= Ex [Prc[f(x) = a - c mod q] ] ≤ 1/q0.5

•KKMO Test with partial active folding for Max-2Lin(Z)

•Pick x ∈ [q]d by random•Get y by rerandomizing each coordinate of x w.p. ε

•Pick c, c' ∈ [q0.5] by random, test

f(x1 - c, ..., xn - c) + c = f(y1 - c', ..., yn - c') + c'

"Partial active folding" (cont'd)

• Completeness:• Soundness:

– Claim. g(x) = f(x1 - c, ..., xn - c) + c is 1/q0.5-balanced

– By Majority Is Stablest Theorem, when f is far from dictators

Pr[f(x1-c,...,xn-c)+c = f(y1-c',...,yn-c')+c' mod q] < 1/qε/4

•KKMO Test with partial active folding for Max-2Lin(Z)

•Pick x ∈ [q]d by random•Get y by rerandomizing each coordinate of x w.p. ε

•Pick c, c' ∈ [q0.5] by random, test

f(x1 - c, ..., xn - c) + c = f(y1 - c', ..., yn - c') + c'

Application to Max-3Lin(Z)

Key Idea in Max-2Lin(Z):"Partial folding" to deal with "wrap-around" event

Håstad's reduction for Max-3Lin(q)•Hastad's Matching Dictatorship Test for

f: [q]L -> Z, g : [q]R -> Z, π : [R] -> [L]

•Pick x ∈ [q]L , y ∈ [q]R, by random

•Let z∈[q]R, s.t. zi = (yi + xπ(i)) mod q

•Rerandomizing each coordinate of x, y, z w.p. ε

•Test f(0, x2 - x1, ..., xn - x1) + x1 + g(y) = g(z) mod q• Completeness: if g is i-th dictator, f is π(i)-th dictator Pr[f, g pass the test] ≥ 1 - 3ε• Soundness: if f and g far from being "matching dictators" Pr[f, g pass the test] < 1/q + δ

(1 - 3ε) vs (1/q + δ) NP-Hardness of Max-3Lin(q)

Our reduction for Max-3Lin(Z)•Matching Dictatorship Test with partial active folding for

f: [q2]L -> Z, g : [q3]R -> Z, π : [R] -> [L]

•Pick x ∈ [q2]L , y ∈ [q3]R, by random

•Let z∈[q3]R, s.t. zi = (yi + xπ(i)) mod q

•Rerandomizing each coordinate of x, y, z w.p. ε

•Pick c ∈ [q] by random•Test f(x1 - c, ..., xn - c) + c + g(y) = g(z)

• Completeness: if g is i-th dictator, f is π(i)-th dictator Pr[f(x1 - c, ..., xn - c) + c + g(y) = g(z)] ≥ 1 - 3ε - 2/q

• Soundness: if f and g far from being "matching dictators" Pr[f(x1 - c, ..., xn - c) + c + g(y) = g(z) mod q] < 1/q + δ

(1-3ε-2/q) vs (1/q+δ) NP-Hardness of Max-3Lin(Z)

The End.

Any questions?