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

Prasad RaghavendraUniversity of Washington

Seattle

Optimal Algorithms and Inapproximability Results for

Every CSP?

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.

))()()(( 145532532321 xxxxxxxxxxxx Equivalently the

largest fraction of clauses

Variables : {x1 , x2 , x3 ,x4 , x5} Constraints : 4 clauses

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 ѵ z

))()()(( 145532532321 xxxxxxxxxxxx

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]Pay Off Functions can be Negative

Can model Minimization Problems like Multiway Cut, Min-Uncut.

Objective :

Find an assignment that maximizes the

Average Payoff

Examples of GCSPs

Max-3-SATMax CutMax Di CutMultiway CutMetric Labelling

0-ExtensionUnique Gamesd- to - 1 GamesLabel CoverHorn Sat

Unique GamesA Special Case

E2LIN mod pGiven a set of linear equations of the form:

Xi – Xj = cij 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)

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.

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]

)2/( p)loglog(1 pnO

)log(1 3 nO

))/1log((1 5/12 pO

1log1

Why is UGC important?Problem Best

Approximation Algorithm

NP Hardness Unique Games Hardness

Vertex CoverMax CUTMax 2- SATSPARSEST CUTMax k-CSP

20.878

0.9401

1.360.941

0.95461+ε

20.878

0.9401Every Constant

nlog

kk 2/ kkO 2/2 kkO 2/

UG hardness results are intimately connected to the limitations of Semidefinite Programming

Semidefinite Programming

Max Cut

10

15

3

7

11

Input : a weighted graph G

Find a cut that maximizes the number of crossing edges

Max Cut SDP

Quadratic Program

Variables : x1 , x2 … xn

xi = 1 or -1

Maximize

10

15

3

7

11

1

1

1

-1

-1

-1

-1-1

-1

Eji

jiij xxw),(

2)(4

1

Relax all the xi to be unit vectors instead of {1,-1}. All products are replaced by inner products of vectors

Semidefinite Program

Variables : v1 , v2 … vn

| vi |2 = 1

Maximize

Eji

jiij vvw),(

2||4

1

MaxCut Rounding

v1

v2

v3

v4

v5

Cut the sphere by a random hyperplane, and output the induced graph cut.

- A 0.878 approximation for the problem.

General Boolean 2-CSPs

Total PayOff

In Integral Solutionvi = 1 or -1V0 = 1

Triangle Inequality

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

Total PayOff

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

Results

Two CurvesIntegrality Gap CurveS(c) = smallest value of the integral solution, given SDP value c.

UGC Hardness CurveU(c) = The best polytime computable solution, assuming UGC given an instance with value c.

0 1Optimum Solution

S(c)

U(c)

Fix a GCSP

If UGC is true:U(c) ≥ S(c)

If UGC is false:U(c) is

meaningless!

UG Hardness Result

Roughly speaking,Assuming UGC, the SDP(I), SDP(II),SDP(III) give best

possible approximation for every CSP

c = SDP ValueS(c) = SDP Integrality GapU(c) = UGC Hardness Curve

Theorem 1:For every constant η > 0, and every GCSP Problem,

U(c) < S(c+ η) + η

0 1Optimum Solution

S(c)

U(c)

U(c)

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.

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 ValueS(c) = SDP Integrality GapU(c) = UGC Hardness Curve

0 1Optimum Solution

S(c)

U(c)

0 1Optimum 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?

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) - η0 1Optimum Solution

S(c)

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]

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)

0 1Optimum Solution

S(c)

Explicit bounds on the size of an integrality gap instance for any 2-CSP.

Related WorkProblem Best

Approximation Algorithm

Unique Games Hardness

Vertex CoverMax CUTMax 2- SATSPARSEST CUTMax k-CSP

20.878

0.9401

2 [Khot-Regev] 0.878 [Khot-Kindler-Mossel-O’donnell]0.9401 [Per Austrin]Every Constant [Chawla-Krauthgamer-..] [Trevisan-Samorodnitsky]

kk 2/ kkO 2/nlog

[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

Proof Overview

Dictatorship TestGiven 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 functionF(x1 ,… xR) = xi

F is far from every dictator function

(No influential coordinate)

Pr[ACCEPT ] = Completeness

Pr[ACCEPT ] =Soundness

ConnectionsSDP Gap Instance

SDP = 0.9OPT = 0.7

UG Hardness

0.9 vs 0.7

Dictatorship Test

Completeness = 0.9Soundness = 0.7

[Khot-Kindler-Mossel-O’Donnell]

[Khot-Vishnoi]For sparsest cut, max cut.[This Paper]

All these conversions hold for every GCSP

A Dictatorship Test for Maxcut

CompletenessValue of Dictator Cuts

F(x) = xi

SoundnessThe 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

An Example : Maxcutv1

v2

v3

v4

v5

10

15

3

71

1

100 dimensional hypercube

Graph G SDP Solution

CompletenessValue of Dictator Cuts =

SDP Value (G)

SoundnessGiven a cut far from every dictator :It gives a cut on graph G with the same value.

In other words, Soundness ≤ OPT(G)

From Graphs to Tests10

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3

7

11

v1

v2

v3

v4

v5

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

Completeness

Echoice of edge e=(u,v) in G

[EX,Y in 100 dim hypercube with dist |u-v|^2 [ (F(X)-F(Y))2 ] ]

v1

v2

v3

v4

v5

100 dimensional hypercube

-1

-1-1

1

1

1

For each edge e, connect every pair of vertices in hypercube separated by the length of e

Set F(X) = X1

(X1 – Y1)2

X1 is not equal to Y1 with probability |u-v|2 , hence completeness = SDP Value (G)

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 variableshas similar distribution as

Sum of large number of Gaussian random variables.”

From Hypercube to the Sphere

100Dimensional 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]

A Graph on the Sphere10

15

3

7

11

v1

v2

v3

v4

v5

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

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.

Soundnessv1

v2

v3

v4

v5 For each edge e in the graph G connect every pair of vertices in hypercube separated by the length of e

SG

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)

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

Roughly FormallySample 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 vi along all directions g(1), g(2) ,.. g(100)

Yi(j) = v0 v∙ i + (1-ε)(vi – (v0 v∙ i)v0) g∙ (100)

Compute P on projections

Compute xi = P(Yi

(1) , Yi(2) ,.. Yi

(100))Round the output of P

If xi > 1, xi = 1 If xi < -1, xi = -1 If xi is in [-1,1]

xi = 1 with probability (1+xi)/2 -1 with probability (1-xi)/2

Given the Polynomial P(y1,… y100)

Key Lemma

Any CSP Instance

G

DICTGDictatorship Test

on functionsF : {-1,1}n ->{-1,1}

If F is far from a dictator,RoundF (G) ≈ DICTG (F)

1) Tests of the verifier are same as the constraints in instance G2) Completeness = SDP(G)

Any Function

F: {-1,1}n → {-1,1}

RoundFRounding Schemeon CSP Instances G

UG Hardness Result

InstanceSDP = cOPT = s

Dictatorship Test

Completeness = cSoundness <= s

UG Hardness

Completeness = cSoundness <= s

Worst Case Gap Instance

Theorem 1:For every constant η > 0, and every GCSP Problem,

U(c) < S(c+ η) + η

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]

Algorithm

Instance ISDP = cOPT = ?

AnyDictatorship

TestCompleteness = c

UG Hardness

Completeness = c

Soundness of any Dictatorship Test ≥ U(c)

There is some function F : {0,1}R -> {0,1} that hasPr[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

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]

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.

Thank You

Rounding Scheme(For Boolean CSPs)

Rounding Scheme was discovered by the reversing the soundness analysis.This fact was independently observed by Yi Wu

MaxCut Rounding

v1

v2

v3

v4

v5

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 > 0xi = 0 otherwise.

SDP Rounding Schemes

SDP Vectors (v1 , v2 .. vn )

Projections(y1 , y2 .. yn )

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

UG Hardness Results

InstanceSDP = cOPT = s

Dictatorship Test

Completeness = cSoundness <= s

UG Hardness

Completeness = cSoundness <= s

Worst Case Gap Instance

Theorem 1:For every constant η > 0, and every GCSP Problem,

U(c) < S(c+ η) + η

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.

10

15

3

7

11

3-Way Cut:Separate the 3-terminals while

separating the minimum number of edges

[Manokaran, Naor, Schwartz, Raghavendra]

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.

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]

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.

If all this looks deceptively simple, then it is because there was deception

Working with several probability distributions at once.

UG Hardness Results

InstanceSDP = cOPT = s

Dictatorship Test

Completeness = cSoundness <= s

UG Hardness

Completeness = cSoundness <= s

Worst Case Gap Instance

Best UG Hardness =

Integrality GapU(c) < S(c+η) + η

Algorithm

Instance ISDP = cOPT = ?

AnyDictatorship

TestCompleteness = c

UG Hardness

Completeness = c

Soundness of any Dictatorship Test ≥ U(c)

There is some function F : {0,1}R -> {0,1} that hasPr[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

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

Unconditional Results For 2-CSPs

Dictatorship Test(I)

Completeness = cSoundness = s

UG Hardness

Completeness = cSoundness = s

UG Integrality Gap instance

Integrality Gap instanceSDP = cOPT ≤ sAlgorithm’s performance

matches the integrality gap of the SDP

[Khot-Vishnoi]

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

CompletenessThis time the worst case is along all dictatorship tests on {-1,1}R

- a finite set that can be discretized.

Key Lemma : Through An Example

1

2132

322

21 ||||||3

1vvvvvv

SDP:Variables : v1 , v2 ,v3

|v1|2 = |v2|2 = |v3|2 =1

Maximize2 3

E[a1 a2] = v1 v∙ 2

E[a12] = |v1|2 E[a2

2] = |v2|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

1

32

Fix an edge e = (1,2).

There exists random variables a1 a2 taking values {-1,1} such that:

c = SDP Valuev1 , v2 , v3 = SDP Vectors

A12A13

A23

Dictatorship TestPick an edge (i,j)Generate ai,aj in {-1,1}R as follows:The kth coordinates aik ,ajk come from distribution Aij

Add noise to ai,aj

Accept if F(ai) ≠ F(aj)

c = SDP Valuev1 , v2 , v3 = SDP Vectors

A12,A23,A31 = Local Distributions

1

32

A12

Input Function:F : {-1,1}R -> {-1,1}

Max Cut Instance

AnalysisPick an edge (i,j)Generate ai,aj in {-1,1}R as follows:

The kth coordinates aik,ajk come from distribution Aij

Add noise to ai,aj

Accept if F(ai) ≠ F(aj)

A12,A23,A31 = Local Distributions

1

32

Max Cut Instance

]))()([(

4

1]))()([(

4

1]))()([(

4

1

3

1 213

232

221 312312

aFaFEaFaFEaFaFE AAA

Input Function:F : {-1,1}R -> {-1,1}

]))()([(

4

1]))()([(

4

1]))()([(

4

1

3

1 213

232

221 312312

aFaFEaFaFEaFaFE AAA

CompletenessA12,A23,A31 = Local Distributions

Input Function is a Dictator : F(x) = x1

])[(

4

1])[(

4

1])[(

4

1

3

1 21131

23121

22111 312312

aaEaaEaaE AAA

Suppose (a1 ,a2) is sampled from A12 then :E[a11 a21] = v1 v∙ 2 E[a11

2] = |v1|2 E[a212] = |v2|2

221

221 ||])[(

12vvaaEA

Summing up, Pr[Accept] = SDP Value(v1 , v2 ,v3)

E[b1 b2] = v1 v∙ 2 E[b2 b3] = v2 v∙ 3 E[b3 b1] = v3 v∙ 1

E[b1

2] = |v1|2 E[b22] = |v2|2 E[b3

2] = |v3|2

There is a global distribution B=(b1 ,b2 ,b3) over real numbers such that:All the moments of order at most 2 match the inner products.

Global Random Variablesc = SDP Value

v1 , v2 , v3 = SDP Vectors

g = random Gaussian vector.(each coordinate generated by i.i.d normal variable)

b1 = v1 g∙b2 = v2 g∙b3 = v3 g∙

1

32

B

Rounding with Polynomials

Input Polynomial : F(x1 ,x2 ,.. xR)

Generate b1 = (b11 ,b12 ,… b1R)

b2 = (b21 ,b22 ,… b2R)

b3 = (b31 ,b32 ,… b3R)

with each coordinate (b1t ,b2t ,b3t) according to global distribution B

Compute F(b1),F(b2) ,F(b3)

Round F(b1),F(b2),F(b3) to {-1,1}

Output the rounded solution.

1

32

B

]))()([(

4

1]))()([(

4

1]))()([(

4

1

3

1 213

232

221 bFbFEbFbFEbFbFE BBB

Invariance

Suppose F is far from every dictator then since A12 and B have same first two moments,

F(a1),F(a2) has nearly same distribution as F(b1),F(b2)

•

• F(b1), F(b2) are close to {-1,1}

]))()([(4

1]))()([(

4

1 221

22112

bFbFEaFaFE BA

From Gap instances to Gap instances

InstanceSDP = cOPT = s

Dictatorship Test

Completeness = cSoundness = s

UG Hardness

Completeness = cSoundness = s

UG Gap instance for a

Strong SDP

A Gap Instance for the Strong SDP for

CSP

For each variable u in CSP,Introduce q variables : {u0 , u1 ,.. uq-1 }

uc = 1,

ui = 0 for i≠c

Payoff for u,v :P(u,v) = ∑a ∑b P(a,b)ua vb

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

u = c

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

Total PayOff

Arbitrary k-ary GCSP

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

Rounding Scheme(For Boolean CSPs)

Rounding Scheme was discovered by the reversing the soundness analysis.This fact was independently observed by Yi Wu

SDP Rounding Schemes

SDP Vectors (v1 , v2 .. vn )

Projections(y1 , y2 .. yn )

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

Roughly FormallySample 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 vi along all directions w(1), w(2) ,.. w(R)

Yi(j) = v0 v∙ i + (1-ε)(vi – (v0 v∙ i)v0) w∙ (j)

Compute P on projections

Compute xi = P(Yi

(1) , Yi(2) ,.. Yi

(R))Round the output of P

If xi > 1, xi = 1 If xi < -1, xi = -1 If xi is in [-1,1]

xi = 1 with probability (1+xi)/2 -1 with probability (1-xi)/2

Rounding By Polynomial P(y1,… yR)

Algorithm

Solve SDP(III) to obtain vectors (v1 ,v2 ,… vn )

Smoothen the SDP solution (v1 ,v2 ,… vn )

For all multlinear polynomials P(y1 ,y2, .. yR) do

Round using P(y1 ,y2, .. yR)

Output the best solution obtained

R is a constant parameter

“For all multilinear polynomials P(y1 ,y2, .. yR) 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

Smoothening SDP Vectors

Let u1 ,u2 .. un denote the SDP vectors corresponding to the following distribution over integral solutions:``Assign each variable uniformly and independently at random”

Substitute vi

* v∙ j* = (1-ε) (vi v∙ j) + ε (ui u∙ j)

Non-Boolean CSPs

There will be q rounding polynomials instead of one polynomial.

Projection is in the same fashion: Yi

(j) = v0 v∙ i + (1-ε)(vi – (v0 v∙ i)v0) w∙ (j)

To Round the Output of the polynomial, do the following:

From Gap instances to Gap instances

InstanceSDP = cOPT = s

Dictatorship Test

Completeness = cSoundness = s

UG Hardness

Completeness = cSoundness = s

UG Gap instance for a

Strong SDP

A Gap Instance for the Strong SDP for

CSPWorst Case

Instance

Backup Slides

Rounding for larger domains

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).

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.

“Sample R independent vectors : w1, w2 ,.. wR each with i.i.d Gaussian components.For all multlinear polynomials P(y1 ,y2, .. yR) do

Compute xi = P(vi w∙ 1 , vi w∙ 2 ,.. vi w∙ R)”

Goemans-Williamson rounding uses one single random projection, this algorithm uses a constant number of random projections.

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

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?

Max Cut

10

15

3

7

11

Input : a weighted graph G

Find a cut that maximizes the number of crossing edges

Max Cut SDP

Quadratic Program

Variables : x1 , x2 … xn

xi = 1 or -1

Maximize

10

15

3

7

11

1

1

1

-1

-1

-1

-1-1

-1

Eji

jiij xxw),(

2)(4

1

Relax all the xi to be unit vectors instead of {1,-1}. All products are replaced by inner products of vectors

Semidefinite Program

Variables : v1 , v2 … vn

| vi |2 = 1

Maximize

Eji

jiij vvw),(

2||4

1

Semidefinite Program

Variables : v1 , v2 … vn

| vi |2 = 1

Maximize

Max Cut SDP

10

15

3

7

11

1

1

1

-1

-1

-1

-1-1

-1

Eji

jiij vvw),(

2||4

1