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Venkatesan Guruswami Prasad Raghavendra

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,

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Venkatesan Guruswami Prasad Raghavendra

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  1. Constraint Satisfaction over a Non-Boolean DomainApproximation Algorithms and Unique Games Hardness VenkatesanGuruswami Prasad Raghavendra University of Washington Seattle, WA

  2. 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. Equivalently the largest fraction of clauses

  3. 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 ѵz Arity = 3 GOAL : Find an assignment satisfying maximum fraction of constraints

  4. 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]

  5. Refined Question : Among all Max-CSP problems over domain [q] ={0,..q-1},and arityk, which is the hardest to approximate? Question : Which Max-CSP is the hardest to approximate? Clearly, the problems become harder as domain size or the arity grows

  6. PCP Motivation 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 Probabilistically Checkable Proof (A string over alphabet {0,1,..q-1}) Random bits Verifier ACCEPT/REJECT

  7. 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} ? Among all Max-CSP problems over domain [q] ={0,..q-1},and arityk, which is the hardest to approximate?

  8. [Samorodnitsky-Trevisan 2000] simplified by [Hastad-Wigderson] Boolean CSPs Hardness: For every k, there is a boolean CSP of arityk, which is NP-hard to approximate better than : [Engebresten-Holmerin] Assuming Unique Games Conjecture [Samorodnitsky-Trevisan 2006] [Charikar-Makarychev-Makarychev] Algorithm: Every boolean CSP of arityk, can be approximated to a factor : [Hast] [Trevisan] Random Assignment

  9. This Work : Non-Boolean CSPs UG Hardness: Assuming Unique Games Conjecture, For every k, and a prime numberq, there is a CSP of arityk 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 arityk over the domain [q] ={0,1,2,..q-1} can be approximated to a factor

  10. 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.

  11. 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)

  12. Proof Overview

  13. Dictatorship Testing Problem • 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 : F is a dictator function F(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

  14. UG Hardness Proofs For the rest of the talk, we shall focus on Dictatorship Testing. Dictatorship Test Over functionsF:[q]R -> [q] Completeness = C Soundness = S # of queries = k Using [Khot-Kindler-Mossel-O’Donnell] reduction. 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

  15. Testing Dictatorships by Testing Linearity[Samorodnitsky-Trevisan 2006] 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 Fix {0,1} : field on 2 elements k = 2d Given a function F : {0,1}R -> {0,1} • Pick a random affine subspace A of dimension d. • Test if F agrees with some affinelinear function on the subspace A. Every dictator F(x1 , x2 ,.. xR ) = xi is a linear function over vector space {0,1}R

  16. x+y2 x+y1+y2 x+y3+ y2 Cubes x+y1+y2+y3 x+y2 x+y1+y2 Gower’s Norm x+y3 x x x x+y1 x+y1+y3 x+y1 x+y1 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 d-dimensional cube spanned by {x,y1 ,y2 ,.. yd } is

  17. Gower’s Norm More Formally, Intuitively, the dth Gower’s norm measures the correlation of the function f with degree d-1 polynomials.

  18. 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 dthGowers Norm. (k=2d) Using Noise sensitivity, There are only a FEWinfluential coordinates. Lemma : If a balanced function f : {0,1}R -> {-1,1} has high dthGowers 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

  19. Extending to Larger domains Replace 2 by q in the [Samorodnitsky-Trevisan] dictatorship test. Fix [q]: field on q elements(q is a prime). k = qd Given a function F : [q]R -> [q] • Pick a random affine subspace A of dimension d. • Test if F agrees with some affinelinear function on the subspace A.

  20. The Difficulty Lemma : If F : {0,1}R -> {0,1} passes the test with probability then f = (-1)F has high dthGowers Norm. (k=2d) 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)

  21. Multidimensional Progressions 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+(q-1)y2+y1 x+(q-1)y2+2y1 x+(q-1)y2+ (q-2)y1 x+(q-1)y2+ (q-1)y1 x+(q-1)y2 x+(q-2)y2 x+2y2+ (q-2)y1 x+(q-2)y2+ (q-2)y1 x+y2+ (q-2)y1 x+(q-1)y1 x x x+y2+y1 x+y1 x+y2+2y1 x+2y1 x+ (q-2)y1 x+2y2+y1 x+2y2+2y1 x+(q-2)y1 x+y1 x+2y1 x+(q-2)y2+y1 x+(q-2)y2+2y1 x+2y2 x+2y2+ (q-1)y1 x+y2+ (q-1)y1 x+ (q-1)y1 x+(q-2)y2+ (q-1)y1 x+y2 E[product of f over C] Expectation over random dq-dimensional subcubes C in [q]R

  22. 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) • 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.

  23. 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.

  24. Thank You

  25. x-y = 11 (mod 17) x-z = 13 (mod 17) … …. z-w = 15(mod 17) Unique GamesA Special Case E2LIN mod p Given a set of linear equations of the form: Xi – Xj = cij mod p Find a solution that satisfies the maximum number of equations.

  26. 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-εfraction of the equations. • No assignment satisfies more than εfraction of equations.

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