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Constraint Satisfaction over a Non-Boolean Domain Approximation Algorithms and Unique Games Hardness. Venkatesan Guruswami Prasad Raghavendra. University of Washington Seattle, WA. Constraint Satisfaction Problem A Classic Example : Max-3-SAT. Given a 3-SAT formula, - PowerPoint PPT Presentation
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Constraint Satisfaction over a Non-Boolean Domain
Approximation Algorithms and Unique Games Hardness
Venkatesan Guruswami Prasad RaghavendraUniversity of Washington
Seattle, WA
Constraint Satisfaction ProblemA Classic Example : Max-3-SAT
Given a 3-SAT formula,Find an assignment to the variables that satisfies the maximum number of clauses.
))()()(( 145532532321 xxxxxxxxxxxx Equivalently the
largest fraction of clauses
Constraint Satisfaction Problem
General Definition :
Domain : {0,1,.. q-1}Predicates : {P1, P2 , P3 … Pr}
Pi : [q]k -> {0,1}
Arity : Maximum number of variables per constraint (k)
Example : Max-3-SAT
Domain : {0,1}Predicates : P1(x,y,z) = x ѵ y ѵ z
Arity = 3
GOAL : Find an assignment satisfying maximum fraction of constraints
Approximability
Max-3-SAT : 7/8 [Karloff-Zwick],[Hastad]
Most Max-CSP problems are NP-hard to solve exactly.
Different Max-CSP problems are approximable to varying ratios.
3-XOR : ½[Hastad]
Max-Cut : 0.878[Goemans-Williamson],
[Khot-Kindler-Mossel-O’donnel]
Max-2-SAT : 0.94 [Lempel-Livnat-Zwick],
[Austrin]
Question :Which Max-CSP is the hardest to approximate?
Refined Question :Among all Max-CSP problems over domain [q] ={0,..q-1}, and arity k, which is the hardest to approximate?
Clearly, the problems become harder as domain size or the arity grows
PCP Motivation
Probabilistically Checkable Proof
(A string over alphabet {0,1,..q-1})
))()()(( 145532532321 xxxxxxxxxxxx
Verifier
Random bits
ACCEPT/REJECT
Completeness(C) :
If SAT formula is satisfiable, there is a proof that verifier accepts with probability C
Soundness (S):
For an unsatisfiable formula, no proof is accepted with probability more than S
Among all Max-CSP problems over domain [q] ={0,..q-1}, and arity k, which is the hardest to approximate?
PCP Motivation
What is the best possible gap between completeness (c) and soundness (s) for a PCP verifier that makes k queries over an alphabet [q] = {0,1,..q-1} ?
Boolean CSPs
k
k
22 2
k
k22
Hardness:For every k, there is a boolean CSP of arity k, which is NP-hard to approximate better than :
Algorithm:Every boolean CSP of arity k, can be approximated to a factor :
[Samorodnitsky-Trevisan 2000]simplified by [Hastad-Wigderson] k
k
222
[Engebresten-Holmerin]
[Samorodnitsky-Trevisan 2006]
Assuming Unique Games Conjecture
k21
Random Assignment
k22
[Trevisan]
k
kk2log/
[Hast]
k
k288.0
[Charikar-Makarychev-Makarychev]
This Work : Non-Boolean CSPsUG Hardness: Assuming Unique Games Conjecture, For every k, and a prime number q, there is a CSP of arity k over the domain [q] ={0,1,2,..q-1}, which is NP-hard to approximate better than
Algorithm: The algorithm of [Charikar-Makarychev-Makarychev] can be extended to non-boolean domains.
Every CSP of arity k over the domain [q] ={0,1,2,..q-1} can be approximated to a factor
kqkq2
kq
qk 7/
Related Work[Raghavendra 08] “Optimal approximation algorithms and
hardness results for every CSP, assuming Unique Games Conjecture.” – “Every” so applies to the hardest CSPs too.– Does not give explicit example of hardest CSP, nor the explicit value of the
approximation ratio.
[Austrin-Mossel 08] “Assuming Unique Games conjecture, For every prime power q, and k, it is NP-hard to
approximate a certain CSP over [q] to a factor > ”
– Independent work using entirely different techniques(invariance principle)– Show a more general result, that yields a criteria for Approximation Resistance
of a predicate.
kqkqq )1(
Techniques• We extend the proof techniques of [Samorodnitsky-Trevisan
2006] to non- boolean domains.• To this end, we
– Define a subspace linearity test.– Show a technical lemma relating the success probability of a
function F to the Gower’s norm of F (similar to the standard proof relating the number of
multidimensional arithmetic progressions to the Gower’s norm)
• Along the way, we make some minor simplifications to [Samorodnitsky-Trevisan 2006]. – (Remove the need for common influences)
Proof Overview
Given a function F : [q]R [q], • Make at most k queries to F• Based on values of F, Output ACCEPT or REJECT. Distinguish between the following two cases :
Dictatorship Testing Problem
F is a dictator functionF(x1 ,… xR) = xi
F is far from every dictator function
(No influential coordinate)
Pr[ACCEPT ] = Completeness
Pr[ACCEPT ] =Soundness
Goal : Achieve maximum gap between Completeness and Soundness
UG Hardness Proofs
UG Hardness Result:Assuming Unique Games Conjecture, it is NP-hard to approximate a CSP over [q] with
arity k to ratio better than C/S
Using [Khot-Kindler-Mossel-O’Donnell] reduction.
Dictatorship Test Over functions F:[q]R -> [q]
Completeness = C Soundness = S
# of queries = k
For the rest of the talk, we shall focus
on Dictatorship Testing.
Testing Dictatorships by Testing Linearity[Samorodnitsky-Trevisan 2006]
Fix {0,1} : field on 2 elements k = 2d
Given a functionF : {0,1}R -> {0,1}
• Pick a random affine subspace A of dimension d.
• Test if F agrees with some affine linear function on the subspace A.
Every dictator F(x1 , x2 ,.. xR ) = xi is a linear function over vector space
{0,1}R
Random Assignment :
There are 2d+1 different affine linear functions on A.
There are possible functions on A.
So a random function satisfies the test with probability
d22
k
d kd 2
2222
1
Gower’s Norm
For F : {0,1}R -> {0,1}, let f(x) = (-1)F(x) .
dth Gowers Norm Ud(f) =
E [ product of f over C]Expectation over random d-dimensional subcubes C in {0,1}R
},..1{| dSyxCSi
i
x x+y1
x+y2 x+y1+y2
x x+y1
x+y2 x+y1+y2
x x+y1
x+y3+ y2 x+y1+y2+y3
x+y3 x+y1+y3
d-dimensional cube spanned by {x,y1 ,y2 ,.. yd } is
Cubes
More Formally,
Intuitively, the dth Gower’s norm measures the correlation of the function f with degree d-1 polynomials.
Gower’s Norm
Testing Dictatorships by Testing Linearity[Samorodnitsky-Trevisan 2006]
Lemma : If F : {0,1}R -> {0,1} passes the test with probability then f = (-1)F has high dth Gowers Norm. (k=2d)
k
k22
Lemma : If a balanced function f : {0,1}R -> {-1,1} has high dth Gowers Norm, then it has an influential coordinate (k=2d)
Theorem : If a balanced function F : {0,1}R -> {0,1} passes the test with probability then it has an influential coordinate
k
k22
Using Noise sensitivity,There are only a FEW
influential coordinates.
Extending to Larger domains
Fix [q]: field on q elements(q is a prime). k = qd
Given a functionF : [q]R -> [q]
• Pick a random affine subspace A of dimension d.
• Test if F agrees with some affine linear function on the subspace A.
Replace 2 by q in the [Samorodnitsky-
Trevisan] dictatorship test.
The Difficulty
Lemma : If F : {0,1}R -> {0,1} passes the test with probability then f = (-1)F has high dth Gowers Norm. (k=2d)
k
k22
Over {0,1}R, Subcube = Affine subspace.
Testing linearity over a random affine subspace, can be easily related to expectation over a random cube.
Over [q]R , Subcube ≠ Affine subspace. (2R points) (qR points)
Success probability of a function F : {0,1}R -> {0,1}, is related to :let f(x) = (-1)F(x) .
E [ product of f over A]
Expectation over random d-dimensional affine subspace A in [q]R
(Affine subspaces are like multidimensional arithmetic progressions)
x x+y1 x+2y1 x+(q-2)y1x+(q-1)y1
x+y2+y1x+y2+2y1 x+y2+ (q-2)y1 x+y2+ (q-1)y1
x x+y1 x+2y1x+ (q-2)y1 x+ (q-1)y1
x+2y2+y1x+2y2+2y1 x+2y2+ (q-2)y1 x+2y2+ (q-1)y1
x+(q-2)y2+y1 x+(q-2)y2+2y1x+(q-2)y2+ (q-2)y1x+(q-2)y2+ (q-1)y1x+(q-2)y2
x+2y2
x+y2
x+(q-1)y2x+(q-1)y2+y1 x+(q-1)y2+2y1x+(q-1)y2+ (q-2)y1x+(q-1)y2+ (q-1)y1
Multidimensional Progressions
E [ product of f over C]
Expectation over random dq-dimensional subcubes C in [q]R
Alternate Lemma
Lemma : If F : [q]R -> [q] passes the test with probability then f = (-1)F has high dqth Gower’s Norm. (k=qd)
kqkq2
• d-dimensional affine subspace test relates to the dqth Gower’s norm
• The proof is technical and involves repeated use of the Cauchy-Schwartz inequality.
• A special case of a more general result by [Green-Tao][Gowers-Wolf], where they define
“Cauchy-Schwartz Complexity” of a set of linear forms.
Open Questions
CSPs with Perfect Completeness:Which CSP is hardest to approximate, under the promise that the input instance is completely satisfiable?
Approximation Resistance:Characterize CSPs for which the best approximation achievable is given by a random assignment.
Thank You
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.