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Shorter Long Codes and Applications to Unique Games. Raghu Meka (IAS, Princeton). Boaz Barak (MSR, New England) Parikshit Gopalan (MSR, SVC) Johan Håstad (KTH) Prasad Raghavendra (GA Tech) David Steurer (MSR, New England). Is Unique Games Conjecture true?. - PowerPoint PPT Presentation
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Shorter Long Codes and Applications to Unique Games
1
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?
2
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?
3
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
4
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?
5
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)
9
S1
Spectral:
???
Core of ABS algorithm for Unique Games
Small Set Expansion (SSE)
10
S
Arora-Barak-Steurer’10Spectral:Atmost
eigenvalues larger than .
1
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
32
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 …
33
Using Long code? Try the “Short code” …
Sketch for Other Applicatons
34
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
35
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
36
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 ))