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

Solving linear equationsGiven a set of linear equations over reals, is there a solution satisfying all the equations?Easy : Gaussian elimination.

Noisy versionGiven 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. [Hstad '01] Given a set of linear equations modulo q, it is NP-hard to distinguish betweenthere is a solution satisfying (1 - )-fraction of the equations no solution satisfies more than (1/q + )-fraction of the equations

Equations are sparse, and are of the form xi + xj - xk = c (mod q)(1 - ) vs (1/q + ) approx. for Max-3Lin(q) is NP-Hard

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

Sparser equations: Max-2Lin(q)

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

overintegers/reals??

Max-3LinMax-2Linover [q](1 - ) vs (1/q + ) NP-hardness[Hstad '01](1 - ) vs UG-hardness[KKMO '07]

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 [Hstad '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 , > 0, it is NP-Hard to (1 - ) vs approximate Max-3Lin(Z) 3-query PCP over integersImplies the hardness for Max-3Lin(R)

Proof follows [Hstad '01], but much more involvedderandomized Long Code testingFourier analysis with respect to an exponential distribution on Z+

Max-3LinMax-2Linover [q](1 - ) vs (1/q + ) NP-hardness[Hstad '01](1 - ) vs UG-hardness[KKMO '07]over integers/reals(1 - ) vs NP-hardness[GR '07]?

Unique Games over Integers?Can we use the techniques in [Guruswami-Raghavendra '07] prove a (1 - ) vs 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 , > 0, it is UG-Hard to (1 - ) vs 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 , > 0, it is UG-Hard to (1 - ) vs approximate Max-2Lin(Z)

By [KKMO '07], only need to design a (1 - ) vs 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 - Soundness: for functions far from dictators, Pr[equation holds] <

(1 - ) vs hardness of Max-2Lin(q)

Dictatorship Test over IntegersA distribution over equations f(x) - f(y) = cCompleteness: for dictators, Pr[f(x) - f(y) =c] 1 - Soundness: for functions far from dictators, Pr[f(x) - f(y) = c mod q] <

It is UG-Hard to distinguish betweena Max-2Lin(Z) instance is (1 - )-satisfiablethe instance is not -satisfiable even when the the equations are modulo q

Recap of KKMO Dictatorship Test

Back to KKMO Dictatorship TestDictatorship test over [q]: a distribution over equations f(x) - f(y) = c (mod q)Completeness: for dictators, Pr[equation holds] 1 - Soundness: for functions far from dictators, Pr[equation holds] < KKMO Test

Pick x [q]d by randomGet y by rerandomizing each coordinate of x w.p. Test f(x) - f(y) = 0 (mod q)

Back to KKMO Dictatorship Test (cont'd)Soundness analysis

"Majority Is Stablest" Theorem [MOO '05]If f is far from dictators and "-balanced", then Pr[f passes the test] < /2

f is -balanced : Pr[f(x) = a mod q] < for all 0 a < qKKMO Test

Pick x [q]d by randomGet y by rerandomizing each coordinate of x w.p. Test f(x) - f(y) = 0 (mod q)

Back to KKMO Dictatorship Test (cont'd)Soundness analysis"Folding" trick: to make sure f is -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 analysisg(x) is 1/q-balancedKKMO Test

Pick x [q]d by randomGet y by rerandomizing each coordinate of x w.p. Test f(x) - f(y) = 0 (mod q)

Dictatorship Test for Max-2Lin(Z)A distribution over equations f(x) - f(y) = cCompleteness: for dictators, Pr[f(x) - f(y) =c] 1 - Soundness: for functions far from dictators, Pr[f(x) - f(y) = c mod q] < If we use KKMO test...Soundness: the same,Completeness does not hold, becausewhen query f(x), get g(x) = (xi - x1) mod q + x1when query f(y), get g(y) = (yi - y1) mod q + y1

Max-2Lin(q): Pr[g(x) - g(y) = 0 mod q] 1 - 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-balancedProof. 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/qKKMO Test with active folding

Pick x [q]d by randomGet y by rerandomizing each coordinate of x w.p.

Pick c, c' [q] by random, test f(x1 - c, ..., xn - c) + c = f(y1 - c', ..., yn - c') + c' (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/q0.5f(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 - - 2/q0.5KKMO Test with partial active folding for Max-2Lin(Z)

Pick x [q]d by randomGet y by rerandomizing each coordinate of x w.p.

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-balancedProof. 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/q0.5KKMO Test with partial active folding for Max-2Lin(Z)

Pick x [q]d by randomGet y by rerandomizing each coordinate of x w.p.

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-balancedBy 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/q/4KKMO Test with partial active folding for Max-2Lin(Z)

Pick x [q]d by randomGet y by rerandomizing each coordinate of x w.p.

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

Hstad's reduction for Max-3Lin(q)Completeness: if g is i-th dictator, f is (i)-th dictator Pr[f, g pass the test] 1 - 3Soundness: if f and g far from being "matching dictators" Pr[f, g pass the test] < 1/q +

(1 - 3) vs (1/q + ) NP-Hardness of Max-3Lin(q)Hastad's Matching Dictatorship Test for f: [q]L -> Z, g : [q]R -> Z, : [R] -> [L]

Pick x [q]L , y [q]R, by random Let z[q]R, s.t. zi = (yi + x(i)) mod q Rerandomizing each coordinate of x, y, z w.p.

Test f(0, x2 - x1, ..., xn - x1) + x1 + g(y) = g(z) mod q

Our reduction for Max-3Lin(Z)Completeness: if g is i-th dictator, f is (i)-th dictator Pr[f(x1 - c, ..., xn - c) + c + g(y) = g(z)] 1 - 3 - 2/qSoundness: if f and g far from being "matching dictators" Pr[f(x1 - c, ..., xn - c) + c + g(y) = g(z) mod q] < 1/q +

(1-3-2/q) vs (1/q+) NP-Hardness of Max-3Lin(Z)Matching Dictatorship Test with partial active folding for f: [q2]L -> Z, g : [q3]R -> Z, : [R] -> [L] Pick x [q2]L , y [q3]R, by random Let z[q3]R, s.t. zi = (yi + x(i)) mod q Rerandomizing each coordinate of x, y, z w.p. Pick c [q] by randomTest f(x1 - c, ..., xn - c) + c + g(y) = g(z)

The End.

Any questions?