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]
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-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/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 Max3Lin(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 I
Introducing the new "active folding"
The new "active folding"
•KKMO Test with active folding
mod q
•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)
• 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
Our method
Step II
"Partial active folding"
"Partial active folding"
•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'
• Completeness:
– f(x1 - c, ..., xn - c) + c = (xi - c) mod q + c
= (xi - c) + c = xi
– f(y1 - c', ..., yn - c') + c' = yi
w.p. 1 - 1/q0.5
w.p. 1 - 1/q0.5
Pr[f(x1-c, ..., xn-c)+c = f(y1-c', ..., yn-c')+c'] ≥ 1 - ε - 2/q0.5
"Partial active folding" (cont'd)
•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'
• 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
"Partial active folding" (cont'd)
•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'
• 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
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?