shorter long codes and applications to unique games
Post on 23-Feb-2016
33 Views
Preview:
DESCRIPTION
TRANSCRIPT
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
6
1. Applications of short code
2. Small set expanders with many large eigenvaluesConstruction and analysis
Application I: Expansion vs Eigenvalue Profiles
7
S1
Expansion: Spectral:Cheeger Inequalities
Small Set Expansion
8
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
11
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
12
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
13
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
14
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
15
MAX-CUT is a UG instance with k = 2
Linear UG with n varsalphabet size k
MAX-CUT of size
Application III: Integrality Gaps
16
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
17
1. Applications of short code
2. Small set expanders with many large eigenvaluesConstruction and analysis
Long Code and Noisy Cube
18
Long code: Longest code imaginableWork with noisy cube – essentially the
same
Eg., is hypercube
Noisy Cube is an SSE
19
Powerful: implies KKL for instance
Our construction “sparsifies” the noisy cube
Thm: Noisy cube is a SSE.
Better SSEs from Noisy Cube
20
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
21
Input:Pick
Accept if
TestingDistance: DQuery Comp.: Good soundness:
Parameters
SSEs from LTCs
22
Given
Thm: Given If
SSEs from LTCs
23
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
24
Thm: Graph has vertices and large eigenvalues and is a SSE.
Vertices: degree poly’s over Edges: if where affine functions.
Analyzing expansion
25
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
26
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
27
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
28
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
29
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
30
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
31
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 ))
top related