1. 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)

2. 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

3. 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!

4. Our Result 4 Main: Exponentially more efficient “replacement” for long code. Not necessarily a blackbox replacment. Preserves main properties: Fourier analysis, dictatorship testing etc.

5. 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 …

6. Outline of Talk 6 1. Applications of short code 2. Small set expanders with many large eigenvalues Construction and analysis

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

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

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

10. 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

11. Small Set Expansion 11 Question: How many large eigenvalues can a SSE have? Small set  Small sets expand  “Many” bad balanced cuts BAD CUT

12.  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.

13. Application II: Efficient Alphabet Reduction 13  Goemans-Williamson: 0.878 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 -> 0.878 tight!

14. 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 -> 0.878 tight!

15. 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

16. 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.

17. Outline of Talk 17 1. Applications of short code 2. Small set expanders with many large eigenvalues Construction and analysis

18. Long Code and Noisy Cube 18  Long code: Longest code imaginable  Work with noisy cube – essentially the same Eg., is hypercube

19. Noisy Cube is an SSE 19  Powerful: implies KKL for instance  Our construction “sparsifies” the noisy cube Thm: Noisy cube is a SSE.

20. 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

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

22. SSEs from LTCs 22 Given

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

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

25. Analyzing expansion 25 When do small sets expand? Need: Indicators of small sets are far from span of top eigenvectors  First analyze noisy cube.

26. Analyzing expansion for noisy cube 26  Is (essentially) a Cayley graph.  Eigenvectors: Characters of Hamming weight Eigenvalues 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!

27. 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

28.  N eigenvalues  Threshold decay: Large eig. -> “weight” small 28  Edges of :  A Cayley graph! Eigenvalues 0 1 2 Proof of Expansion Smoothness, low query com. of Smoothness, low query com. of Soundness of Soundness of

29. 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.

30. 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!

31. Open Problems 31 Is ABS bound for SSE tight? Need better LTCs

32. 32 Thank you

33. 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” …

34. 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. 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.

36. 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