Shorter Long Codes and Applications to Unique Games

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Boaz Barak (MSR, New England)Parikshit Gopalan (MSR, SVC)

Johan Håstad (KTH)Prasad Raghavendra (GA Tech)

David Steurer (MSR, New England)

Raghu Meka (IAS, Princeton)

Is Unique Games Conjecture true?

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Settles longstanding open problems in approximation algorithmsE.g., Max-Cut, vertex cover

Interesting even if notIntegrality gaps: Khot-Vishnoi’04.

UGC ~ Hardness of a certain CSP

Is Unique Games Conjecture true?

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Fastest algorithm [ABS10]: .

Best evidence: lowerbound in certain models. [Khot-Vishnoi’04, Khot-Saket’09, Raghavendra-Steurer’09]Captures ABS algorithm – BRS11,

GS11.Best algorithms for most problems!

E.g., Max-Cut, Sparsest-Cut.

Huge Gap!Source of gap: Long code is

actually quite long!

Our Result

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Main: Exponentially more efficient “replacement” for long

code.

Not necessarily a blackbox replacment.

Preserves main properties: Fourier analysis, dictatorship testing etc.

Is Unique Games Conjecture true?

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Fastest algorithm [ABS10]: .

This Work: Near quasi-polynomial lowerbounds in certain models.

Smaller gap …

Outline of Talk

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1. Applications of short code

2. Small set expanders with many large eigenvaluesConstruction and analysis

Application I: Expansion vs Eigenvalue Profiles

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S1

Expansion: Spectral:Cheeger Inequalities

Small Set Expansion

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Complete graph

Complete graph

Dumbell graph: not expanding … Is it really?Small sets expand!

When is a graph SSE?Interesting by itselfClosely tied to Unique

Games – RS10

Small Set Expansion (SSE)

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S1

Spectral:

???

Core of ABS algorithm for Unique Games

Small Set Expansion (SSE)

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S

Arora-Barak-Steurer’10Spectral:Atmost

eigenvalues larger than .

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Small Set Expansion

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Question: How many large eigenvalues can a SSE have?

Small set

Small sets expand “Many” bad

balanced cutsBAD CUT

BAD CUT

BAD CUT

Previous best: Noisy cube – .

Small Set Expansion

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Question: How many large eigenvalues can a SSE have?

Our Result: A SSE with large eigenvalues.

Corollary: Rules out quasi-polynomial run time for ABS algorithm.

Application II: Efficient Alphabet Reduction

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Goemans-Williamson: 0.878 approximation

MAX-CUTGiven G find S maximizing E(S,Sc)

KKMO’04 + MOO’05: UGC true -> 0.878 tight!

Are we done? (Short of proving UGC …)

Application II: UGC hardness for Max-CUT

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UGC with n varsalphabet size k

MAX-CUT of sizeKKMO+MOO

KKMO’04 + MOO’05: UGC true -> 0.878 tight!

Application II: Efficient Alphabet Reduction

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MAX-CUT is a UG instance with k = 2

Linear UG with n varsalphabet size k

MAX-CUT of size

Application III: Integrality Gaps

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SDP Hierarchies: Powerful paradigm for optimization problems.

Which level suffices?

Basic SDP

Optimal Solution

No. Variable Levels

Eg: SDP+SA, LS, LS+, Lasserre, …SDP + SA

KV04: UG, Max-Cut, Sparsest Cut not in O(1) levels.

KS-RS09: UG, Max-Cut, Sparsest Cut not in levels.

This work: UG, Max-Cut, not in levels.

Outline of Talk

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1. Applications of short code

2. Small set expanders with many large eigenvaluesConstruction and analysis

Long Code and Noisy Cube

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Long code: Longest code imaginableWork with noisy cube – essentially the

same

Eg., is hypercube

Noisy Cube is an SSE

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Powerful: implies KKL for instance

Our construction “sparsifies” the noisy cube

Thm: Noisy cube is a SSE.

Better SSEs from Noisy Cube

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Idea: Find a subgraph of the noisy cube.

Natural approach: Random subsetComplete failure: No edges!

Our Approach: pick a linear codeNeed: bad rate, not too good distance!But not too bad… want local testablity of dual

Locally Testable Codes

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Input:Pick

Accept if

TestingDistance: DQuery Comp.: Good soundness:

Parameters

SSEs from LTCs

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Given

Thm: Given If

SSEs from LTCs

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Symmetry across coordinates.Fraction of non-

zero coordinates.

Instantiate with Reed-Muller (RM) CodesC = RM code of degree Dual = RM of degree Testability: Batthacharya-Kopparty-

Schoenbeck-Sudan-Zuckerman’10

SSEs from RM Codes

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Thm: Graph has vertices and large eigenvalues and is a SSE.

Vertices: degree poly’s over Edges: if where affine functions.

Analyzing expansion

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When do small sets expand?

Need: Indicators of small sets are far from span of top eigenvectors

First analyze noisy cube.

Analyzing expansion for noisy cube

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Is (essentially) a Cayley graph.

Eigenvectors: Characters of

Hamming weight

Eige

nval

ues

0 1 2

N eigenvalues Exponential decay: Large eig. -> weight small

Need: Indicators of small sets far from span of low-weight characters

Follows from (2,4)-hypercontractivity!

SSEs from LTCs

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Eigenvecs -> CharactersLarge eval -> low-weight(2,4)-Hypercontractivity

Cayley GraphLocal TestabilityK-wise independence

SSE for Noisy Cube SSE for

N eigenvalues Threshold decay: Large eig. -> “weight” small

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Edges of :

A Cayley graph!

Eige

nval

ues

0 1 2

Proof of Expansion

Smoothness, low query com. of Soundness of

𝜒𝑤 (𝑥 )= (−1 )⟨𝑤 , 𝑥 ⟩

Proof of Expansion

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Eigenvecs -> CharactersLarge eval -> low-weight(2,4)-Hypercontractivity

Cayley GraphLocal TestabilityK-wise independence

SSE for Noisy Cube SSE for

Fact: is (D-1)-wise independent. QED.

Open Problems

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Prove/refute the UGCProof: Larger alphabets?Refute: Need new algorithmic ideas or maybe

stronger SDP hierarchies

Question: Integrality gaps for rounds of Lasserre hierarchy?

Very recent work - Barak, Harrow, Kelner, Steurer, Zhou : Lasserre(8) breaks current instances!

Open Problems

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Is ABS bound for SSE tight?Need better LTCs

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Thank you

Long Code d-Short Code

Dict. testing: Noisy cube Dict. testing: RM testerAnalysis: Maj. is stablest Analysis: SSE, Maj. is

stablest

Take Home …

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Using Long code? Try the “Short code” …

Sketch for Other Applicatons

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Dictatorship testing for long code/noisy cube[Kahn-Kalai-Linial’88, Friedgut’98,

Bourgain’99, Mossel-O’Donnel-Oleszkiewicz’05], ...

Focus on MOO: Majority is StablestInvariance principle for low-degree

polynomials

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P multilinear, no variable influential.

MOO’05: Invariance principle for Polynomials

Need . Can’t prove in general … … but true for RM code!RM codes fool polynomial threshold functions

PRG for PTFs [M., Zuckerman 10].

Corollary: Majority is stablest over RM codes.Corollary: Alphabet reduction with quasi-polynomial blowup.

Integrality Gaps for Unique Games, MAX-CUT

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Idea: Noisy cube -> RM graph in [Khot-Vishnoi’04], [KKMO’05] etc.,

Analyze via Raghavendra-Steurer’09Thm: vertex Max-Cut instance resisting:

rounds in SDP+SA (compare to ))