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 x i + x j - x k = 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-3LinMax-2Lin over [q] (1 - ε) vs (1/q + ε) NP-hardness [Håstad '01] (1 - ε) vs δ UG-hardness [KKMO '07] over integers/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-3LinMax-2Lin over [q] (1 - ε) vs (1/q + ε) NP-hardness [Håstad '01] (1 - ε) vs δ UG-hardness [KKMO '07] over integers/reals (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(x 1, x 2,..., x d ) = x i (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] < β ε/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(x 1, x 2,..., x n ), return g(x) = f(0, (x 2 - x 1 ) mod q,..., (x n - x 1 ) mod q) + x 1 –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) = (x i - x 1 ) mod q + x 1 when query f(y), get g(y) = (y i - y 1 ) mod q + y 1 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 I Introducing the new "active folding"

The new "active folding" Completeness: Soundness: –Claim. g(x) = f(x 1 - c,..., x n - c) + c is 1/q-balanced –Proof. Pr x,c [f(x 1 - c,..., x n - c) + c = a mod q] = E c [Pr x [f(x 1 - c,..., x n - c) = a - c mod q] ] = E c [Pr x [f(x) = a - c mod q] ] = E x [Pr c [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(x 1 - c,..., x n - c ) + c = f(y 1 - c',..., y n - c') + c' (mod q) mod q

Our method Step II "Partial active folding"

"Partial active folding" Completeness: –f(x 1 - c,..., x n - c) + c = (x i - c) mod q + c = (x i - c) + c = x i w.p /q 0.5 –f(y 1 - c',..., y n - c') + c' = y i w.p /q 0.5 Pr[f(x 1 -c,..., x n -c)+c = f(y 1 -c',..., y n -c')+c'] ≥ 1 - ε - 2/q 0.5 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' ∈ [q 0.5 ] by random, test f(x 1 - c,..., x n - c ) + c = f(y 1 - c',..., y n - c') + c'

"Partial active folding" (cont'd) Completeness: Soundness: –Claim. g(x) = f(x 1 - c,..., x n - c) + c is 1/q 0.5 -balanced –Proof. Pr x,c [f(x 1 - c,..., x n - c) + c = a mod q] = E c [Pr x [f(x 1 - c,..., x n - c) = a - c mod q] ] = E c [Pr x [f(x) = a - c mod q] ] = E x [Pr c [f(x) = a - c mod q] ] ≤ 1/q 0.5 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' ∈ [q 0.5 ] by random, test f(x 1 - c,..., x n - c ) + c = f(y 1 - c',..., y n - c') + c'

"Partial active folding" (cont'd) Completeness: Soundness: –Claim. g(x) = f(x 1 - c,..., x n - c) + c is 1/q 0.5 -balanced –By Majority Is Stablest Theorem, when f is far from dictators Pr[f(x 1 -c,...,x n -c)+c = f(y 1 -c',...,y n -c')+c' mod q] < 1/q ε/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' ∈ [q 0.5 ] by random, test f(x 1 - c,..., x n - c ) + c = f(y 1 - c',..., y n - c') + c'

Application to Max-3Lin(Z) Key Idea in Max-2Lin(Z): "Partial folding" to deal with "wrap-around" event

H åstad'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. z i = (y i + x π(i) ) mod q Rerandomizing each coordinate of x, y, z w.p. ε Test f(0, x 2 - x 1,..., x n - x 1 ) + x 1 + 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: [q 2 ] L -> Z, g : [q 3 ] R -> Z, π : [R] -> [L] Pick x ∈ [q 2 ] L, y ∈ [q 3 ] R, by random Let z ∈ [q 3 ] R, s.t. z i = (y i + x π(i) ) mod q Rerandomizing each coordinate of x, y, z w.p. ε Pick c ∈ [q] by random Test f(x 1 - c,..., x n - c) + c + g(y) = g(z) Completeness: if g is i-th dictator, f is π(i)-th dictator Pr[f(x 1 - c,..., x n - c) + c + g(y) = g(z)] ≥ 1 - 3ε - 2/q Soundness: if f and g far from being "matching dictators" Pr[f(x 1 - c,..., x n - 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?