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

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Constraint Satisfaction Problem A 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

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Variables : {x 1, x 2, x 3,x 4, x 5 } Constraints : 4 clauses Constraint Satisfaction Problem Instance : Set of variables. Predicates P i applied on variables Find an assignment that satisfies the largest fraction of constraints. Problem : Domain : {0,1,.. q-1} Predicates :{P 1, P 2, P 3 … P r } P i : [q] k -> {0,1} Max-3-SAT Domain : {0,1} Predicates : P 1 (x,y,z) = x ѵ y ѵ z

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Generalized CSP (GCSP) Replace Predicates by Payoff Functions (bounded real valued) Problem : Domain : {0,1,.. q-1} Pay Offs: {P 1, P 2, P 3 … P r } P i : [q] k -> [-1, 1] Pay Off Functions can be Negative Can model Minimization Problems like Multiway Cut, Min-Uncut. Objective : Find an assignment that maximizes the Average Payoff

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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

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Unique Games A Special Case E2LIN mod p Given a set of linear equations of the form: X i – X j = c ij mod p Find a solution that satisfies the maximum number of equations. x-y = 11 (mod 17) x-z = 13 (mod 17) … …. z-w = 15(mod 17)

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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.

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Unique Games Conjecture A notorious open problem, no general consensus either way. Hardness Results: No constant factor approximation for unique games. [Feige- Reichman] Algorithm On (1-Є) satisfiable instances [Khot 02] [Trevisan] [Gupta-Talwar] 1 – O(ε logn) [Charikar-Makarychev-Makarychev] [Chlamtac-Makarychev-Makarychev] [Arora-Khot-Kolla-Steurer-Tulsiani-Vishnoi]

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Why is UGC important? ProblemBest Approximation Algorithm NP HardnessUnique Games Hardness Vertex Cover Max CUT Max 2- SAT SPARSEST CUT Max k-CSP ε Every Constant UG hardness results are intimately connected to the limitations of Semidefinite Programming

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Semidefinite Programming

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Max Cut Input : a weighted graph G Find a cut that maximizes the number of crossing edges

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Max Cut SDP Quadratic Program Variables : x 1, x 2 … x n x i = 1 or -1 Maximize Relax all the x i to be unit vectors instead of {1,-1}. All products are replaced by inner products of vectors Semidefinite Program Variables : v 1, v 2 … v n | v i | 2 = 1 Maximize

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MaxCut Rounding v1v1 v2v2 v3v3 v4v4 v5v5 Cut the sphere by a random hyperplane, and output the induced graph cut. - A approximation for the problem.

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General Boolean 2-CSPs Total PayOff In Integral Solution v i = 1 or -1 V 0 = 1 Triangle Inequality

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

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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

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Results

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Two Curves Integrality Gap Curve S(c) = smallest value of the integral solution, given SDP value c. UGC Hardness Curve U(c) = The best polytime computable solution, assuming UGC given an instance with value c. 01 Optimum Solution S(c) U(c) Fix a GCSP If UGC is true: U(c) ≥ S(c) If UGC is false: U(c) is meaningless!

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UG Hardness Result Roughly speaking, Assuming UGC, the SDP(I), SDP(II),SDP(III) give best possible approximation for every CSP c = SDP Value S(c) = SDP Integrality Gap U(c) = UGC Hardness Curve Theorem 1: For every constant η > 0, and every GCSP Problem, U(c) < S(c+ η) + η 01 Optimum Solution S(c) U(c)

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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.

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Efficient Rounding Scheme Roughly speaking, There is a generic polytime rounding scheme that is optimal for every CSP, assuming UGC. 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 c = SDP Value S(c) = SDP Integrality Gap U(c) = UGC Hardness Curve 01 Optimum Solution S(c) U(c)

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01 Optimum Solution S(c) U(c) NP-hard algorithm 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?

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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) S(c) = SDP Integrality Gap 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 01 Optimum Solution S(c)

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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]

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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 η and domain size q) 01 Optimum Solution S(c) Explicit bounds on the size of an integrality gap instance for any 2-CSP.

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Related Work ProblemBest Approximation Algorithm Unique Games Hardness Vertex Cover Max CUT Max 2- SAT SPARSEST CUT Max k-CSP [Khot-Regev] [Khot-Kindler-Mossel-O’donnell] [Per Austrin] Every Constant [Chawla-Krauthgamer-..] [Trevisan-Samorodnitsky] [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

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Proof Overview

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Dictatorship Test Given a function F : {-1,1} R {-1,1} Toss random coins Make a few queries to F Output either ACCEPT or REJECT F is a dictator function F(x 1,… x R ) = x i F is far from every dictator function (No influential coordinate) Pr[ACCEPT ] = Completeness Pr[ACCEPT ] = Soundness

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

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

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An Example : Maxcut v1v1 v2v2 v3v3 v4v4 v5v dimensional hypercube Graph G SDP Solution Completeness Value of Dictator Cuts = SDP Value (G) 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)

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From Graphs to Tests v1v1 v2v2 v3v3 v4v4 v5v5 Graph G (n vertices) 100 dimensional hypercube : {-1,1} 100 SDP Solution For each edge e, connect every pair of vertices in hypercube separated by the length of e Constant independent of size of G H

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Completeness E choice of edge e=(u,v) in G [ E X,Y in 100 dim hypercube with dist |u-v|^2 [ (F(X)-F(Y)) 2 ] ] v1v1 v2v2 v3v3 v4v4 v5v5 100 dimensional hypercube For each edge e, connect every pair of vertices in hypercube separated by the length of e Set F(X) = X 1 (X 1 – Y 1 ) 2 X 1 is not equal to Y 1 with probability |u-v| 2, hence completeness = SDP Value (G)

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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.”

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

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A Graph on the Sphere v1v1 v2v2 v3v3 v4v4 v5v5 Graph G (n vertices) 100 dimensional sphere SDP Solution For each edge e, connect every pair of vertices in sphere separated by the length of e S

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

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Soundness v1v1 v2v2 v3v3 v4v4 v5v5 For each edge e in the graph G connect every pair of vertices in hypercube separated by the length of e S G 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. Hence MaxCut value of F on H is at most the max cut value of G. Soundness ≤ MaxCut(G)

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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 G c of G inside the sphere graph S Output the solution assigned by P to G C

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RoughlyFormally Sample R Random Directions Sample R independent vectors : g (1), g (2),.. g (100) Each with i.i.d Gaussian components. Project along them Project each v i along all directions g (1), g (2),.. g (100) Y i (j) = v 0 ∙v i + (1-ε)(v i – (v 0 ∙v i )v 0 ) ∙ g (100) Compute P on projections Compute x i = P(Y i (1), Y i (2),.. Y i (100) ) Round the output of P If x i > 1, x i = 1 If x i < -1, x i = -1 If x i is in [-1,1] x i = 1 with probability (1+x i )/2 -1 with probability (1-x i )/2 Given the Polynomial P(y 1,… y 100 )

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

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

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Generic Rounding Scheme Solve SDP(III) to obtain vectors (v 1,v 2,… v n ) Add little noise to SDP solution (v 1,v 2,… v n ) For all multlinear polynomials P(y 1,y 2,.. y 100 ) do Round using P(y 1,y 2,.. y 100 ) Output the best solution obtained P is Multilinear polynomial in 100 variables with coefficients in [-1,1]

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Algorithm Instance I SDP = c OPT = ? Any Dictatorship Test Completeness = c UG Hardness Completeness = c Soundness of any Dictatorship Test ≥ U(c) 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) Dictatorship Test (I) Completeness = c

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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]

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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.

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Thank You

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

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MaxCut Rounding v1v1 v2v2 v3v3 v4v4 v5v5 Cut the sphere by a random hyperplane, and output the induced graph cut. Equivalently, Pick a random direction g. For each vector v i, project v i along g y i = v i. g Assign x i = 1 if y i > 0 x i = 0 otherwise.

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SDP Rounding Schemes SDP Vectors (v 1, v 2.. v n ) Projections (y 1, y 2.. y n ) Assignment Random Projection Process the projections 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

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

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Multiway Cut and Labelling Problems Theorem: Assuming Unique Games Conjecture, The earthmover linear program gives the best approximation. Theorem: Unconditionally, the simple SDP does not give better approximations than the LP Way Cut: Separate the 3-terminals while separating the minimum number of edges [Manokaran, Naor, Schwartz, Raghavendra]

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Maximum Acyclic Subgraph Given a directed graph, order the vertices to maximize the number of forward edges. [Guruswami,Manokaran, Raghavendra] Theorem: Assuming Unique Games Conjecture, The best algorithm’s output is as good as a random ordering. Theorem: Unconditionally, the simple SDP does not give better approximations than random.

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The Grothendieck Constant The Grothendieck constant is the smallest constant k(H) for which the following inequality holds for all matrices : The constant is just the integrality gap of the SDP for bipartite quadratic optimization. Value of the constant is between 1.6 and 1.7 but is unknown yet. [Raghavendra,Steurer]

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Grothendieck Constant [Raghavendra,Steurer] Theorem: There is an algorithm to compute arbitrarily good approximations to the Grothendieck constant. Theorem: There is an efficient algorithm that solves the bipartite quadratic optimization problem to an approximation equal to Grothendieck constant.

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If all this looks deceptively simple, then it is because there was deception Working with several probability distributions at once.

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UG Hardness Results Instance SDP = c OPT = s Dictatorship Test Completeness = c Soundness <= s UG Hardness Completeness = c Soundness <= s Worst Case Gap Instance Best UG Hardness = Integrality Gap U(c) < S(c+η) + η

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Algorithm Instance I SDP = c OPT = ? Any Dictatorship Test Completeness = c UG Hardness Completeness = c Soundness of any Dictatorship Test ≥ U(c) 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) Dictatorship Test (I) Completeness = c

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On some instance I with SDP value = c, algorithm outputs a solution with value s. For every function F far from dictator, Performance of F in rounding I ≤ s By Key Lemma, For every such F Pr[ F is accepted by Dict(I) ] ≤ s Thus the Dict(I) is a test with soundness s. Unconditional Results For 2-CSPs

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Dictatorship Test(I) Completeness = c Soundness = s UG Hardness Completeness = c Soundness = s UG Integrality Gap instance Integrality Gap instance SDP = c OPT ≤ s Algorithm’s performance matches the integrality gap of the SDP [Khot-Vishnoi]

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Computing Integrality Gaps Integrality gap of a SDP relaxation = Worst case ratio of Integral Optimum SDP Optimum Worst Case over all instances - an infinite set Due to tight relation of integrality gaps/ dictatorship tests for 2-CSPs Integrality gap of a SDP relaxation = Worst case ratio of Soundness Completeness This time the worst case is along all dictatorship tests on {-1,1} R - a finite set that can be discretized.

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Key Lemma : Through An Example 1 SDP: Variables : v 1, v 2,v 3 |v 1 | 2 = |v 2 | 2 = |v 3 | 2 =1 Maximize 2 3

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E[a 1 a 2 ] = v 1 ∙ v 2 E[a 1 2 ] = |v 1 | 2 E[a 2 2 ] = |v 2 | 2 For every edge, there is a local distribution over integral solutions such that: All the moments of order at most 2 match the inner products. Local Random Variables Fix an edge e = (1,2). There exists random variables a 1 a 2 taking values {-1,1} such that: c = SDP Value v 1, v 2, v 3 = SDP Vectors A 12 A 13 A 23

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Dictatorship Test Pick an edge (i,j) Generate a i,a j in {-1,1} R as follows: The k th coordinates a ik,a jk come from distribution A ij Add noise to a i,a j Accept if F(a i ) ≠ F(a j ) c = SDP Value v 1, v 2, v 3 = SDP Vectors A 12,A 23,A 31 = Local Distributions A 12 Input Function: F : {-1,1} R -> {-1,1} Max Cut Instance

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Analysis Pick an edge (i,j) Generate a i,a j in {-1,1} R as follows: The k th coordinates a ik,a jk come from distribution A ij Add noise to a i,a j Accept if F(a i ) ≠ F(a j ) A 12,A 23,A 31 = Local Distributions Max Cut Instance Input Function: F : {-1,1} R -> {-1,1}

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Completeness A 12,A 23,A 31 = Local Distributions Input Function is a Dictator : F(x) = x 1 Suppose (a 1,a 2 ) is sampled from A 12 then : E[a 11 a 21 ] = v 1 ∙ v 2 E[a 11 2 ] = |v 1 | 2 E[a 21 2 ] = |v 2 | 2 Summing up, Pr[Accept] = SDP Value(v 1, v 2,v 3 )

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E[b 1 b 2 ] = v 1 ∙ v 2 E[b 2 b 3 ] = v 2 ∙ v 3 E[b 3 b 1 ] = v 3 ∙ v 1 E[b 1 2 ] = |v 1 | 2 E[b 2 2 ] = |v 2 | 2 E[b 3 2 ] = |v 3 | 2 There is a global distribution B=(b 1,b 2,b 3 ) over real numbers such that: All the moments of order at most 2 match the inner products. Global Random Variables c = SDP Value v 1, v 2, v 3 = SDP Vectors g = random Gaussian vector. (each coordinate generated by i.i.d normal variable) b 1 = v 1 ∙ g b 2 = v 2 ∙ g b 3 = v 3 ∙ g B

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Rounding with Polynomials Input Polynomial : F(x 1,x 2,.. x R ) Generate b 1 = (b 11,b 12,… b 1R ) b 2 = (b 21,b 22,… b 2R ) b 3 = (b 31,b 32,… b 3R ) with each coordinate (b 1t,b 2t,b 3t ) according to global distribution B Compute F(b 1 ),F(b 2 ),F(b 3 ) Round F(b 1 ),F(b 2 ),F(b 3 ) to {-1,1} Output the rounded solution B

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Invariance Suppose F is far from every dictator then since A 12 and B have same first two moments, F(a 1 ),F(a 2 ) has nearly same distribution as F(b 1 ),F(b 2 ) F(b 1 ), F(b 2 ) are close to {-1,1}

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From Gap instances to Gap instances Instance SDP = c OPT = s Dictatorship Test Completeness = c Soundness = s UG Hardness Completeness = c Soundness = s UG Gap instance for a Strong SDP A Gap Instance for the Strong SDP for CSP

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For each variable u in CSP, Introduce q variables : {u 0, u 1,.. u q-1 } u c = 1, u i = 0 for i≠c Payoff for u,v : P(u,v) = ∑ a ∑ b P(a,b)u a v b 2-CSP over {0,..q-1} u = c

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

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Arbitrary k-ary GCSP SDP is similar to the one obtained by k-rounds of Lasserre

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

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SDP Rounding Schemes SDP Vectors (v 1, v 2.. v n ) Projections (y 1, y 2.. y n ) Assignment Random Projection Process the projections For any CSP, it is enough to do the following: Instead of one random projection, pick sufficiently many projections Use a multilinear polynomial P to process the projections

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RoughlyFormally Sample R Random Directions Sample R independent vectors : w (1), w (2),.. w (R) Each with i.i.d Gaussian components. Project along them Project each v i along all directions w (1), w (2),.. w (R) Y i (j) = v 0 ∙v i + (1-ε)(v i – (v 0 ∙v i )v 0 ) ∙ w (j) Compute P on projections Compute x i = P(Y i (1), Y i (2),.. Y i (R) ) Round the output of P If x i > 1, x i = 1 If x i < -1, x i = -1 If x i is in [-1,1] x i = 1 with probability (1+x i )/2 -1 with probability (1-x i )/2 Rounding By Polynomial P(y 1,… y R )

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Algorithm Solve SDP(III) to obtain vectors (v 1,v 2,… v n ) Smoothen the SDP solution (v 1,v 2,… v n ) For all multlinear polynomials P(y 1,y 2,.. y R ) do Round using P(y 1,y 2,.. y R ) Output the best solution obtained R is a constant parameter

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“For all multilinear polynomials P(y 1,y 2,.. y R ) do” - All multilinear polynomials with coefficients bounded within [-1,1] - Discretize the set of all such multi-linear polynomials There are at most a constant number of such polynomials. Discretization

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Smoothening SDP Vectors Let u 1,u 2.. u n denote the SDP vectors corresponding to the following distribution over integral solutions: ``Assign each variable uniformly and independently at random” Substitute v i * ∙ v j * = (1-ε) (v i ∙ v j ) + ε (u i ∙ u j )

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Non-Boolean CSPs There will be q rounding polynomials instead of one polynomial. Projection is in the same fashion: Y i (j) = v 0 ∙v i + (1-ε)(v i – (v 0 ∙v i )v 0 ) ∙ w (j) To Round the Output of the polynomial, do the following:

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From Gap instances to Gap instances Instance SDP = c OPT = s Dictatorship Test Completeness = c Soundness = s UG Hardness Completeness = c Soundness = s UG Gap instance for a Strong SDP A Gap Instance for the Strong SDP for CSP Worst Case Instance

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Backup Slides

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Rounding for larger domains

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Remarks For every CSP and every ε > 0, there is a large enough constant R such that Approximation achieved is within ε of optimal for all CSPs if Unique Games Conjecture is true. For 2-CSPs, the approximation ratio is within ε of the integrality gap of the SDP(I).

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Rounding Schemes Very different rounding schemes for every CSP. with often complex analysis. Max Cut - Random hyperplane cutting Multiway cut - Complicated Cutting the simplex. Our algorithm is a generic rounding procedure. Analysis does not compute the approximation factor, but indirectly shows that it is equal to the integrality gap.

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“Sample R independent vectors : w 1, w 2,.. w R each with i.i.d Gaussian components. For all multlinear polynomials P(y 1,y 2,.. y R ) do Compute x i = P(v i ∙ w 1, v i ∙ w 2,.. v i ∙ w R )” Goemans-Williamson rounding uses one single random projection, this algorithm uses a constant number of random projections.

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Semidefinite Programming Linear program over the inner products Strongest algorithmic tool in approximation algorithms Used in a large number of algorithms. Integrality gap of a SDP relaxation = Worst case ratio of Integral Optimum SDP Optimum

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More Constraints? Most SDP algorithms use simple relaxations with few constraints. [Arora-Rao-Vazirani] used the triangle inequalities to get sqrt(log n) approximation for sparsest cut. Can the stronger SDPs yield better approximation ratios for problems of interest?

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Max Cut Input : a weighted graph G Find a cut that maximizes the number of crossing edges

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Max Cut SDP Quadratic Program Variables : x 1, x 2 … x n x i = 1 or -1 Maximize Relax all the x i to be unit vectors instead of {1,-1}. All products are replaced by inner products of vectors Semidefinite Program Variables : v 1, v 2 … v n | v i | 2 = 1 Maximize

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Semidefinite Program Variables : v 1, v 2 … v n | v i | 2 = 1 Maximize Max Cut SDP

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