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How to Round Any CSP Prasad Raghavendra University of Washington, Seattle David Steurer, Princeton University (In Principle)

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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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Theorem: [Raghavendra 08] Assuming Unique Games Conjecture, For every CSP, “a simple semidefinite program (SDP1) gives the best approximation computable efficiently.” [Raghavendra08] A generic rounding scheme for (SDP1) that is optimal for every CSP under UGC. Independent of UGC, for 2CSPs, the generic rounding scheme for (SDP1) achieves an Approximation Ratio ≥ (1- ² ) Integrality Gap of SDP.

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Rounding Algorithm minimum over all instances = value of rounded solution value of SDP solution rounding – ratio A ( ¦ ) (approximation ratio) ≥ (1- ² ) integrality gap ( ¦ ) = value of optimal solution value of SDP solution minimum over all instances For any CSP ¦ and any ² >0, there exists an efficient algorithm A, Unconditionally, the algorithm A as good as all known algorithms for CSPs Very Simple : No Invariance Principle, Dictatorship Tests, Unique Games. Drawbacks Running Time(A) On CSP over alphabet size q, arity k No explicit approximation ratio

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Computing Integrality Gaps Theorem: For any CSP ¦ and any ² >0, there exists an algorithm A to compute integrality gap ( ¦ ) within an accuracy ² Running Time(A) On CSP over alphabet size q, arity k

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Previous Work SDP ALGORITHMS [Charikar-Makarychev-Makarychev 06] MaxCut [Goemans-Williamson] [Charikar-Wirth] [Lewin-Livnat-Zwick] [Charikar-Makarychev-Makarychev 07] [Hast] [Charikar-Makarychev-Makarychev 07] [Frieze-Jerrum] [Karloff-Zwick] [Zwick SODA 98] [Zwick STOC 98] [Zwick 99] [Halperin-Zwick 01] [Goemans-Williamson 01] [Goemans 01] [Feige-Goemans] [Matuura-Matsui] [Trevisan-Sudan-Sorkin-Williamson] [O’Donnell-Wu] Optimal rounding schemes for MaxCut

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ALGORITHM OUTLINE Rounding Any Constraint Satisfaction Problem

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Max Cut Max CUT Input : A weighted graph G Find : A cut with maximum fraction of crossing edges

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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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v1v1 v2v2 v3v3 v4v4 v5v5 MaxCut Rounding Problem Given a graph on the n - dimensional unit ball, Find the maximum cut of the graph.

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Approximation using Finite Models ¦ -CSP Instance = ¦ -CSP Instance = finite variable folding (identifying variables) optimal solution for = finite approximate solution for = unfolding of the assignment constant time Challenge : ensure = finite has a good solution

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Approximation using Finite Models [Frieze-Kannan] For a dense instance =, it is possible to construct finite model = finite OPT( = finite ) ≥ (1-ε) OPT( = ) General Method for CSPs What we will do : SDP value ( = finite ) > (1-ε)SDP value ( = ) PTAS for dense instances

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Analysis of Rounding Scheme ¦ -CSP Instance = ¦ -CSP Instance = finite SDP value ® SDP value > ® - ² OPT value ¯ rounded value ¯ Hence: rounding-ratio for = < (1+ ² ) integrality-ratio for = finite unfolding

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CONSTRUCTING FINITE MODELS (MAXCUT) Rounding Any Constraint Satisfaction Problem

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v1v1 v2v2 v3v3 v4v4 v5v5 STEP 1 : Dimension Reduction Pick d = 1/ Є 4 random Gaussian vectors {G 1, G 2,.. G d } Project the SDP solution along these directions. Map vector V V → V’ = (V∙G 1, V∙G 2, … V∙G d ) v1v1 v3v3 v4v4 v5v5 Constant dimensions STEP 2 : Surgery Scale every vector V’ to unit length STEP 3 : Discretization Pick an Є –net for the d dimensional sphere Move every vertex to the nearest point in the Є –net v2v2 v2v2 FINITE MODEL Graph on Є –net points

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To Show: SDP value ( = finite ) > (1-ε)SDP value ( = ) Lemma : “Inner Products are almost preserved under random projections” If V’,U’ are random projections of U, V on 1/ ε 4 directions, Pr [ |V∙U – V’∙U’| > ε] < ε 2

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STEP 1 : Dimension Reduction Project the SDP solution along 1/ Є 4 random directions. STEP 2 : Surgery Scale every vector V’ to unit length STEP 3 : Discretization Pick an Є –net for the d dimensional sphere Move every vertex to the nearest point in the Є –net For SDP value ( = ) Contribution of an edge e = (U,V) |U-V| 2 = 2-2 V∙U To Show: SDP value ( = finite ) > (1-ε)SDP value ( = ) SDP Vectors for = finite = Corresponding vectors in Є –net STEP 1 With probability > 1- Є 2, | |U-V| 2 - |U’-V’| 2 | < 2Є STEP 2 With probability > 1- 2Є 2, 1+ Є < |V’| 2,|U’| 2 < 1- Є, Normalization changes distance by at most 2Є STEP 3 Changes edge length by at most 2Є

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For SDP value ( = ) Contribution of an edge e = (U,V) |U-V| 2 = 2-2 V∙U To Show: SDP value ( = finite ) > (1-ε)SDP value ( = ) SDP Vectors for = finite = Corresponding vectors in Є –net STEP 1 With probability > 1- Є 2, | |U-V| 2 - |U’-V’| 2 | < 2Є STEP 2 With probability > 1- 2Є 2, 1+ Є < |V’| 2,|U’| 2 < 1- Є, Normalization changes distance by at most 2Є STEP 3 Changes edge length by at most 2Є ANALYSIS With probability 1-3Є 2, The contribution of edge e changes by < 6Є In expectation, For (1-3Є 2 ) edges, the contribution of edge e changes by < 6Є SDP value ( = finite ) > SDP value ( = ) - 6Є – 3Є 2

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FINITE MODELS FOR GENERAL CSP Rounding Any Constraint Satisfaction Problem

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Semidefinite Program for CSPs Variables : For each variable X a Vectors {V (a,0), V (a,1) } For each clause P = (x a ν x b ν x c ), Scalar variables μ (P,000), μ (P,001), μ (P,010), μ (P,100), μ (P,011), μ (P,110), μ (P,101), μ (P,111) X a = 1 V (a,0) = 0 V (a,1) = 1 X a = 0 V (a,0) = 1 V (a,1) = 0 If X a = 0, X b = 1, X c = 1 μ (P,000) = 0μ (P,011) = 1 μ (P,001) = 0μ (P,110) = 0 μ (P,010) = 0μ (P,101) = 0 μ (P,100) = 0μ (P,111) = 0 Objective Function :Constraints : For each clause P, 0 ≤μ (P,α) ≤ 1 For each clause P (x a ν x b ν x c ), For each pair X a, X b in P, consitency between vector and LP variables. V (a,0) ∙V (b,0) = μ (P,000) + μ (P,001) V (a,0) ∙V (b,1) = μ (P,010) + μ (P,011) V (a,1) ∙V (b,0) = μ (P,100) + μ (P,101) V (a,1) ∙V (b,1) = μ (P,100) + μ (P,101)

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Semidefinite Relaxation for CSP SDP solution for = : SDP objective: for every constraint Á in = -local distributions ¹ Á over assignments to the variables of Á Example of local distr.: Á = 3XOR(x 3, x 4, x 7 ) x 3 x 4 x 7 ¹ Á … for every variable x i in = -vectors v i,1, …, v i,q constraints (also for first moments) Explanation of constraints : first and second moments of distributions are consistent and form PSD matrix maximize

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Strong and Weak STRENGTH For every clause Á in = -local distributions ¹ Á over assignments to the variables of Á Vector variables v i,a within a clause Á satisfy all valid constraints (like triangle inequality) – the inner products are in the integral hull. WEAKNESS The above hard constraint is only for variables that participate together in a clause

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Throwing away constraints {v i,a } { μ …} -Infeasible SDP solution for a instance =, it does not satisfy the consistency for a clause P. Consider instance = ‘ = = - {P} Now {v i,a } { μ … } is a good SDP solution for = ‘ Throw away clauses from CSP Throw away constraints from SDP relaxation

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v1v1 v2v2 v3v3 v4v4 v5v5 v1v1 v3v3 v4v4 v5v5 Constant dimensions v2v2 v2v2 FINITE MODEL CSP on Є –net points STEP 1 : Dimension Reduction Project the SDP solution along d =1/ Є 4 random directions. STEP 3 : Discretization Pick an Є –net for the d dimensional sphere Move every variable to the nearest point in the Є –net = finite = discretized instance STEP 2 : Throw away Discard clauses for which the corresponding inner products are not preserved within Є. = ‘ = New instance

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To Show: SDP value ( = finite ) > (1-ε)SDP value ( = ) SDP Vectors for = finite = Corresponding vectors in Є –net LP variables { μ …}? Problem : The inner products of vectors corresponding to a clause P might not be in the integral hull. ( For example : 3 arbitrary vectors in a Є –net are not guaranteed to satisfy triangle inequality ) The initial SDP solution satisfied all the constraints

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STEP 1 : Dimension Reduction Project the SDP solution along d =1/ Є 4 random directions. STEP 3 : Discretization Pick an Є –net for the d dimensional sphere Move every variable to the nearest point in the Є –net = finite = discretized instance STEP 2 : Throw away Discard clauses for which the corresponding inner products are not preserved within Є. = ‘ = New instance From STEP 2, We have discarded clauses for which inner products are not preserved within Є Discarding a clause P Forget about constraints corresponding to P Discretization changes inner product by Є For every remaining clause, all inner products are within 2Є of what it was.

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Smoothing Operation Canonical SDP Solution Uniform Distribution over all Integral solutions. Example: V a,0 ∙V a,0 = V a,1 ∙V a,1 = ½ V a,0 ∙V b,0 = V a,0 ∙V b,1 = V a,1 ∙V b,0 = V a,1 ∙V b,1 = 1/4 Є –net Solution SDP Vectors for = finite = Corresponding vectors in Є –net (1-Є) X + = Final SDP solution Integral Hull Є X Є Consider the inner products corresponding to a single clause P SDP Objective value remains roughly the same.

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Conclusions Rounding stronger SDPs. More efficient rounding? Can this SDP be solved in constant dimensional space directly? Integrality gaps for stronger SDP relaxation of Unique Games

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Thank You

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Good finite Models from SDP solutions – Dimension Reduction & Discretization ¦ -CSP Instance = ¦ -CSP Instance = finite SDP solution for = compute Dimension Reduction Project on random low dimensional subspace almost SDP solution for = Discretize Move vectors to closest point on ² -net almost SDP solution for = RnRn RdRd identify variables with same vectors Theorem: SDP value ( = finite ) > SDP value ( = ) Theorem: SDP value ( = finite ) > SDP value ( = ) Idea : use almost SDP solution and do surgery finite number of different vectrs

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Constraint Satisfaction Problems (CSP) CSP ¦ finite set of allowed types of constraints Á : [q] k {0,1} (alphabet [q], arity k) e.g. ¦ = { 3XOR, 3SAT, 3NAE} ¦ -CSP Instance = -variables x 1,…,x n -list of constraints Á of type ¦ on subsets of variables ¦ -CSP Instance = -variables x 1,…,x n -list of constraints Á of type ¦ on subsets of variables Goal : Find assignment x 2 [q] n so as to maximize fraction of satisfied constraints opt( = ) Examples: Max-Cut, Max-3SAT,… PCP Theorem: NP-hard to distinguish opt( = )=1 and opt( = )<0.9 (even for constant k and q) Approximation Algorithms: Goemans-Williamson, Zwick, CMM, …

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