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Optimal Algorithms and Inapproximability Results for Every CSP?

Optimal Algorithms and Inapproximability Results for Every CSP?. Prasad Raghavendra University of Washington Seattle . Constraint Satisfaction Problem A Classic Example : Max-3-SAT. Given a 3-SAT formula,

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Optimal Algorithms and Inapproximability Results for Every CSP?

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  1. Optimal Algorithms and Inapproximability Results for Every CSP? Prasad Raghavendra University of Washington Seattle

  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 Instance : • Set of variables. • Predicates Pi applied on variables Find an assignment that satisfies the largest fraction of constraints. Problem : Domain : {0,1,.. q-1} Predicates : {P1, P2 , P3 … Pr} Pi : [q]k -> {0,1} Max-3-SAT Domain : {0,1} Predicates : P1(x,y,z) = x ѵ y ѵz Variables : {x1 , x2 , x3 ,x4 , x5} Constraints : 4 clauses

  4. Generalized CSP (GCSP) Replace Predicates by Payoff Functions (bounded real valued) Problem : Domain : {0,1,.. q-1} Pay Offs:{P1, P2 , P3 … Pr} Pi : [q]k -> [-1, 1] Objective : Find an assignment that maximizes the Average Payoff Pay Off Functions can be Negative Can model Minimization Problems like Multiway Cut, Min-Uncut.

  5. Examples of GCSPs Max-3-SAT Max Cut Max Di Cut Multiway Cut Metric Labelling 0-Extension Unique Games d- to - 1 Games Label Cover Horn Sat

  6. 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.

  7. 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.

  8. Unique Games Conjecture A notorious open problem, no general consensus either way. Hardness Results: No constant factor approximation for unique games. [Feige-Reichman]

  9. Why is UGC important? UG hardness results are intimately connected to the limitations of Semidefinite Programming

  10. Semidefinite Programming

  11. Max Cut Input : a weighted graph G Find a cut that maximizes the number of crossing edges 10 15 7 1 1 3

  12. Max Cut SDP Quadratic Program Variables : x1 , x2 … xn xi = 1 or -1 Maximize Semidefinite Program Variables : v1 , v2 … vn • | vi |2= 1 Maximize -1 10 1 -1 15 1 7 1 1 1 -1 -1 -1 3 -1 Relax all the xi to be unit vectors instead of {1,-1}. All products are replaced by inner products of vectors

  13. MaxCut Rounding v2 Cut the sphere by a random hyperplane, and output the induced graph cut. - A 0.878 approximation for the problem. v1 v3 v5 v4

  14. In Integral Solution vi = 1 or -1 V0 = 1 General Boolean 2-CSPs Total PayOff Triangle Inequality

  15. 2-CSP over {0,..q-1} Total PayOff

  16. Arbitrary k-ary GCSP • SDP is similar to the one used by [Karloff-Zwick]Max-3-SAT algorithm. • It is weaker than k-rounds of Lasserre / LS+ heirarchies

  17. Results

  18. Fix a GCSP Two Curves Integrality Gap Curve S(c) = smallest value of the integral solution, given SDP value c. If UGC is true: U(c) ≥ S(c) If UGC is false: U(c) is meaningless! UGC Hardness Curve U(c) = The best polytime computable solution, assuming UGC given an instance with value c. U(c) S(c) 0 Optimum Solution 1

  19. c = SDP Value S(c) = SDP Integrality Gap U(c) = UGC Hardness Curve UG Hardness Result Theorem 1: For every constant η > 0, and every GCSP Problem, U(c) < S(c+ η) + η Roughly speaking, Assuming UGC, the SDP(I), SDP(II),SDP(III) give best possible approximation for every CSP U(c) U(c) S(c) 0 Optimum Solution 1

  20. Consequences If UGC is true, then adding more constraints does not help for any CSP Lovasz-Schriver, Lasserre, Sherali-Adams heirarchies do not yield better approximation ratios for any CSP in the worst case.

  21. c = SDP Value S(c) = SDP Integrality Gap U(c) = UGC Hardness Curve Efficient Rounding Scheme Roughly speaking, There is a generic polytime rounding scheme that is optimal for every CSP, assuming UGC. U(c) S(c) 0 Optimum Solution 1 Theorem: For every constant η > 0, and every GCSP, there is a polytime rounding scheme that outputs a solution of value U(c-η) – η

  22. If UGC is true, then for every Generalized Constraint Satisfaction Problem : • If UGC is false?? • Hardness result doesn’t make sense. • How good is the rounding scheme? NP-hard U(c) S(c) algorithm 0 Optimum Solution 1

  23. S(c) = SDP Integrality Gap Unconditionally Roughly Speaking, For 2-CSPs, the Approximation ratio obtained is at least the red curve S(c) The rounding scheme achieves the integrality gap of SDP for 2-CSPs (both binary and q-ary cases) Theorem: Let A(c) be rounding scheme’s performance on input with SDP value = c. For every constant η > 0 A(c) > S(c- η) - η S(c) 0 Optimum Solution 1

  24. As good as the best SDP(II) and SDP(III) are the strongest SDPs used in approximation algorithms for 2-CSPs The Generic Algorithm is at least as good as the best known algorithms for 2-CSPs Examples: Max Cut [Goemans-Williamson] Max-2-SAT [Lewin-Livnat-Zwick] Unique Games [Charikar-Makarychev-Makarychev]

  25. Computing Integrality Gaps Theorem: For any η, and any 2-CSP, the curve S(c) can be computed within error η. (Time taken depends on ηand domain size q) S(c) Explicit bounds on the size of an integrality gap instance for any 2-CSP. 0 Optimum Solution 1

  26. Related Work [Austrin 07] Assuming UGC, and a certain additional conjecture: ``For every boolean 2-CSP, the best approximation is given by SDP(III)” [O’Donnell-Wu 08] Obtain matching approximation algorithm, UGC hardness and SDP gaps for MaxCut

  27. Proof Overview

  28. Given a function • F : {-1,1}R {-1,1} • Toss random coins • Make a few queries to F • Output either ACCEPT or REJECT Dictatorship Test 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

  29. Connections SDP Gap Instance SDP = 0.9 OPT = 0.7 [Khot-Vishnoi] For sparsest cut, max cut. [This Paper] UG Hardness 0.9 vs 0.7 Dictatorship Test Completeness = 0.9 Soundness = 0.7 [Khot-Kindler-Mossel-O’Donnell] All these conversions hold for every GCSP

  30. A Dictatorship Test for Maxcut A dictatorship test is a graph G on the hypercube. A cut gives a function F on the hypercube Completeness Value of Dictator Cuts F(x) = xi Soundness The maximum value attained by a cut far from a dictator Hypercube = {-1,1}100

  31. An Example : Maxcut Completeness Value of Dictator Cuts = SDP Value (G) Graph G SDP Solution v2 10 Soundness Given a cut far from every dictator : It gives a cut on graph G with the same value. In other words, Soundness ≤ OPT(G) 15 7 v1 v3 1 1 3 v5 100 dimensional hypercube v4

  32. From Graphs to Tests Graph G (n vertices) v2 H 10 15 Constant independent of size of G 7 v1 v3 1 1 3 100 dimensional hypercube : {-1,1}100 For each edge e, connect every pair of vertices in hypercube separated by the length of e v5 v4 SDP Solution

  33. Completeness For each edge e, connect every pair of vertices in hypercube separated by the length of e Echoice ofedge e=(u,v)inG [EX,Y in 100 dimhypercube with dist |u-v|^2 [ (F(X)-F(Y))2 ] ] v2 v1 -1 v3 Set F(X) = X1 -1 -1 (X1 – Y1)2 X1 is not equal to Y1 with probability |u-v|2, hence completeness = SDP Value (G) 1 v5 1 1 v4 100 dimensional hypercube

  34. The Invariance Principle Invariance Principle for Low Degree Polynomials [Rotar] [Mossel-O’Donnell-Oleszkiewich], [Mossel 2008] “If a low degree polynomial F has no influential coordinate, then F({-1,1}n) and F(Gaussian) have similar distribution.” A generalization of the following fact : ``Sum of large number of {-1,1} random variables has similar distribution as Sum of large number of Gaussian random variables.”

  35. From Hypercube to the Sphere 100 Dimensional hypercube [-1,1] F : Express F as a multilinear polynomial using Fourier expansion, thus extending it to the sphere. 100 dimensional sphere P : Real numbers Nearly always [-1,1] Since F is far from a dictator, by invariance principle, its behaviour on the sphere is similar to its behaviour on hypercube.

  36. A Graph on the Sphere Graph G (n vertices) v2 S 10 15 7 v1 v3 1 1 3 100dimensional sphere For each edge e, connect every pair of vertices in sphere separated by the length of e v5 v4 SDP Solution

  37. Hypercube vs Sphere S H P : sphere -> Nearly {-1,1} Is the multilinear extension of F F:{-1,1}100 -> {-1,1} is a cut far from every dictator. By Invariance Principle, MaxCut value of F on H≈Maxcut value of P on S.

  38. Soundness For each edgee in the graph Gconnect every pair of vertices in hypercube separated by the length of e G v2 S v1 v3 Alternatively, generate S as follows: Take the union of all possible rotations of the graph G S consists of union of disjoint copies of G. Thus, MaxCut Value of S < Max cut value of G. v5 Hence MaxCut value of F on H is at most the max cut value of G. Soundness ≤ MaxCut(G) v4

  39. Algorithmically, Given a cut F of the hypercube graph H • Extend F to a function P on the sphere using its Fourier expansion. • Pick a random rotation of the SDP solution to the graph G • This gives a random copy Gc of G inside the sphere graph S • Output the solution assigned by P to GC

  40. Given the Polynomial P(y1,… y100)

  41. 1) Tests of the verifier are same as the constraints in instance G 2) Completeness = SDP(G) Key Lemma DICTG Dictatorship Test on functions F : {-1,1}n ->{-1,1} Any CSP Instance G RoundF Rounding Scheme on CSP Instances G Any Function F: {-1,1}n→ {-1,1} If F is far from a dictator, RoundF (G) ≈ DICTG (F)

  42. UG Hardness Result Instance SDP = c OPT = s Worst Case Gap Instance Dictatorship Test Completeness = c Soundness <= s UG Hardness Completeness = c Soundness <= s Theorem 1: For every constant η > 0, and every GCSP Problem, U(c) < S(c+ η) + η

  43. Generic Rounding Scheme Solve SDP(III) to obtain vectors (v1 ,v2 ,… vn) Add little noise to SDP solution (v1 ,v2 ,… vn) For all multlinear polynomials P(y1 ,y2, .. y100) do Round using P(y1 ,y2, .. y100) Output the best solution obtained P is Multilinear polynomial in 100 variables with coefficients in [-1,1]

  44. Soundness of any Dictatorship Test ≥ U(c) Algorithm Any Dictatorship Test Completeness = c Dictatorship Test (I) Completeness = c UG Hardness Completeness = c Instance I SDP = c OPT = ? There is some function F : {0,1}R -> {0,1} that has Pr[F is accepted] ≥ U(c) By Key Lemma, Performance of F as rounding polynomial on instance I = Pr[F is accepted] > U(c)

  45. Related Developments • Multiway Cut and Metric Labelling problems. • Maximum Acyclic Subgraph problem • Bipartite Quadratic Optimization Problem (Computing the Grothendieck constant) [Manokaran, Naor, Schwartz, Raghavendra] [Guruswami,Manokaran, Raghavendra] [Raghavendra,Steurer]

  46. Conclusions Unique Games and Invariance Principle connect : Integrality Gaps, Hardness Results ,Dictatorship tests and Rounding Algorithms. These connections lead to new algorithms, and hardness results unifying several known results.

  47. Thank You

  48. Rounding Scheme(For Boolean CSPs) Rounding Scheme was discovered by the reversing the soundness analysis. This fact was independently observed by Yi Wu

  49. MaxCut Rounding • Cut the sphere by a random hyperplane, and output the induced graph cut. • Equivalently, • Pick a random direction g. • For each vector vi , project vi along g • yi = vi . g • Assign • xi = 1 if yi > 0 • xi = 0 otherwise. v2 v1 v3 v5 v4

  50. SDP Rounding Schemes For any CSP, it is enough to do the following: Instead of one random projection, pick sufficiently many (say 100) projections Use a multi linear polynomial P to process the projections SDP Vectors (v1 , v2 .. vn ) Random Projection Projections (y1 , y2 .. yn ) Process the projections Assignment

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