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 algorithms E.g., Max-Cut, vertex cover  Interesting even if not Integrality 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] 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. 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! 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. 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 eigenvalues Construction and analysis

Application I: Expansion vs Eigenvalue Profiles 7 S S 1 Expansion: Spectral: Cheeger Inequalities

Small Set Expansion 8 Complete graph Dumbell graph: not expanding … Is it really? Small sets expand!

When is a graph SSE? Interesting by itself Closely tied to Unique Games – RS10 Small Set Expansion (SSE) 9 S S 1 Spectral: ??? Spectral: ???

Core of ABS algorithm for Unique Games Small Set Expansion (SSE) 10 S S Arora-Barak-Steurer’10 Spectral: Atmost eigenvalues larger than. Spectral: 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 cuts 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. 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: approximation MAX-CUT Given G find S maximizing E(S,S c ) MAX-CUT Given G find S maximizing E(S,S c ) KKMO’04 + MOO’05: UGC true -> tight!

Are we done? (Short of proving UGC …) Application II: UGC hardness for Max-CUT 14 UGC with n vars alphabet size k MAX-CUT of size KKMO+MOO KKMO’04 + MOO’05: UGC true -> tight!

Application II: Efficient Alphabet Reduction 15  MAX-CUT is a UG instance with k = 2 Linear UG with n vars alphabet 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.

Outline of Talk Applications of short code 2. Small set expanders with many large eigenvalues Construction and analysis

Long Code and Noisy Cube 18  Long code: Longest code imaginable  Work 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 subset Complete failure: No edges! Our Approach: pick a linear code Need: bad rate, not too good distance! But not too bad… want local testablity of dual

Locally Testable Codes 21 Input: Pick Accept if Testing Distance: D Query Comp.: Good soundness: Parameters

SSEs from LTCs 22 Given

Thm: Given If Thm: Given If SSEs from LTCs 23 Symmetry across coordinates. Fraction of non-zero coordinates.

SSEs from RM Codes 24 Thm: Graph has vertices and large eigenvalues and is a SSE.

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 Eigenvalues  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 -> Characters Large eval -> low-weight (2,4)-Hypercontractivity Cayley Graph Local Testability K-wise independence SSE for Noisy CubeSSE for

 N eigenvalues  Threshold decay: Large eig. -> “weight” small 28  Edges of :  A Cayley graph! Eigenvalues Proof of Expansion Smoothness, low query com. of Smoothness, low query com. of Soundness of Soundness of

Proof of Expansion 29 Eigenvecs -> Characters Large eval -> low-weight (2,4)-Hypercontractivity Cayley Graph Local Testability K-wise independence SSE for Noisy CubeSSE for  Fact: is (D-1)-wise independent.  QED.

Open Problems 30 Prove/refute the UGC Proof: Larger alphabets? Refute: Need new algorithmic ideas or maybe stronger SDP hierarchies 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 Coded-Short Code Dict. testing: Noisy cubeDict. testing: RM tester Analysis: Maj. is stablestAnalysis: 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 Stablest Invariance 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! 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’09