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How to Round Any CSP

How to Round Any CSP. (In Principle). Prasad Raghavendra University of Washington, Seattle David Steurer , Princeton University. Constraint Satisfaction Problem A Classic Example : Max-3-SAT. Given a 3-SAT formula,

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How to Round Any CSP

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

  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. 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.

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

  6. 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

  7. 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] Previous Work [O’Donnell-Wu] Optimal rounding schemes for MaxCut

  8. Rounding Any Constraint Satisfaction Problem AlGORITHM OUTLINE

  9. Max Cut Max CUT Input : A weighted graph G Find : A cut with maximum fraction of crossing edges 10 15 7 1 1 3

  10. Max Cut SDP 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

  11. v2 MaxCutRounding Problem Given a graph on the n - dimensional unit ball, Find the maximum cut of the graph. v1 v3 v5 v4

  12. Approximation using Finite Models 1 -1 10 approximate solution for = ¦-CSP Instance = 1 -1 15 7 -1 1 1 1 -1 1 variable folding unfolding of the assignment -1 (identifying variables) 3 -1 1 optimal solution for =finite ¦-CSP Instance =finite 1 constant time -1 Challenge: ensure =finite has a good solution

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

  14. Analysis of Rounding Scheme ¦-CSP Instance = ¦-CSP Instance =finite SDP value > ® - ² SDP value ® unfolding 010001001 rounded value ¯ 010001001 OPT value ¯ Hence: rounding-ratio for = < (1+²) integrality-ratio for =finite

  15. Rounding Any Constraint Satisfaction Problem Constructing FINITE MODELS (MAXCUT)

  16. STEP 1 : Dimension Reduction • Pick d = 1/ Є4 random Gaussian vectors {G1 , G2 , .. Gd} • Project the SDP solution along these directions. • Map vector V • V → V’ = (V∙G1 , V∙G2 , … V∙Gd) v2 v1 v3 v2 STEP 2 : Surgery Scale every vector V’ to unit length v2 v1 v3 • STEP 3 : Discretization • Pick an Є–netfor the • d dimensional sphere • Move every vertex to the nearest point in the Є–net v4 v5 v5 Constant dimensions v4 FINITE MODEL Graph on Є–net points

  17. 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/ε4 directions, Pr [ |V∙U – V’∙U’| > ε] < ε2

  18. To Show:SDP value (=finite) > (1-ε)SDP value (=) For SDP value (=) Contribution of an edge e = (U,V) |U-V|2 = 2-2 V∙U SDP Vectors for =finite = Corresponding vectors in Є–net • STEP 1 : Dimension Reduction • Project the SDP solution along 1/ Є4 random directions. STEP 1 With probability > 1- Є2 , ||U-V|2 - |U’-V’|2 | < 2Є STEP 2 With probability > 1- 2Є2 , 1+ Є< |V’|2 ,|U’|2 < 1- Є, Normalization changes distance by at most 2Є STEP 2 : Surgery Scale every vector V’ to unit length • STEP 3 : Discretization • Pick an Є–netfor the • d dimensional sphere • Move every vertex to the nearest point in the Є–net STEP 3 Changes edge length by at most 2Є

  19. To Show:SDP value (=finite) > (1-ε)SDP value (=) For SDP value (=) Contribution of an edge e = (U,V) |U-V|2 = 2-2 V∙U SDP Vectors for =finite = Corresponding vectors in Є–net ANALYSIS With probability 1-3Є2, The contribution of edge e changes by < 6Є In expectation, For (1-3Є2) edges, the contribution of edge e changes by < 6Є SDP value (=finite) > SDP value (=) - 6Є – 3Є2 STEP 1 With probability > 1- Є2 , ||U-V|2 - |U’-V’|2 | < 2Є STEP 2 With probability > 1- 2Є2 , 1+ Є< |V’|2 ,|U’|2 < 1- Є, Normalization changes distance by at most 2Є STEP 3 Changes edge length by at most 2Є

  20. Rounding Any Constraint Satisfaction Problem FINITE MODELS FOR GENERAL CSP

  21. SemidefiniteProgram for CSPs Constraints : For each clause P, 0 ≤μ(P,α)≤ 1 For each clause P (xaνxbνxc), For each pair Xa,Xb 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) Variables : For each variable Xa Vectors {V(a,0) , V(a,1)} For each clause P = (xaνxbνxc), 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) Objective Function : Xa = 1 V(a,0) = 0 V(a,1) = 1 Xa = 0 V(a,0) = 1 V(a,1) = 0 If Xa = 0, Xb = 1, Xc = 1 μ(P,000) = 0 μ(P,011) = 1 μ(P,001) = 0 μ(P,110) = 0 μ(P,010) = 0 μ(P,101) = 0 μ(P,100) = 0 μ(P,111) = 0

  22. Semidefinite Relaxation for CSP SDP solution for =: Example of local distr.: Á = 3XOR(x3, x4, x7) • for every constraint Á in = • local distributions ¹Á over assignments to the variables of Á x3x4x7 ¹Á 0 0 0 0.1 0 0 1 0.01 0 1 0 0 … 1 1 1 0.6 • for every variable xi in = • vectors vi,1 , … , vi,q Explanation of constraints: first and second moments of distributions are consistent and form PSD matrix constraints (also for first moments) SDP objective: maximize

  23. Strong and Weak STRENGTH For every clause Á in = • local distributions ¹Á over assignments to the variables of Á Vector variables vi,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

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

  25. STEP 1 : Dimension Reduction • Project the SDP solution along • d =1/ Є4 random directions. v2 STEP 2 : Throw away Discard clauses for which the corresponding inner products are not preserved within Є. =‘ = New instance v1 v3 v2 v2 v1 • STEP 3 : Discretization • Pick an Є–netfor the • d dimensional sphere • Move every variable to the nearest point in the Є–net • =finite = discretized instance v3 v4 v5 v5 Constant dimensions v4 FINITE MODEL CSP on Є–net points

  26. 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

  27. 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 • STEP 1 : Dimension Reduction • Project the SDP solution along • d =1/ Є4 random directions. STEP 2 : Throw away Discard clauses for which the corresponding inner products are not preserved within Є. =‘ = New instance • STEP 3 : Discretization • Pick an Є–netfor the • d dimensional sphere • Move every variable to the nearest point in the Є–net • =finite = discretized instance Discretization changes inner product by Є For every remaining clause, all inner products are within 2Єof what it was.

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

  29. 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

  30. Thank You

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

  32. 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} Examples: Max-Cut, Max-3SAT,… PCP Theorem: NP-hard to distinguish opt(=)=1 and opt(=)<0.9 (even for constant k and q) ¦-CSP Instance = • variables x1,…,xn • list of constraints Á of type ¦ on subsets of variables Approximation Algorithms: Goemans-Williamson, Zwick, CMM, … Goal: Find assignment x 2 [q]n so as to maximize fraction of satisfied constraints opt(=)

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