On the Power of Semidefinite Programming Hierarchies

1. On the Power of Semidefinite Programming Hierarchies Prasad Raghavendra Georgia Institute of Technology, Atlanta, GA. David Steurer Microsoft Research New England Cambridge, MA

2. Overview • Background and Motivation • Introduction to SDP Hierarchies (Lasserre SDP hierarchy) • Rounding SDP hierarchies via Global Correlation. • BREAK • Graph Spectrum and Small-Set Expansion. • Sum of Squares Proofs.

3. Graph Spectrum&Small-Set Expansion

4. A Question in Spectral Graph Theory d-regular graph G A random neighbor of a random vertex in is outside of with probability = Let be the normalized adjacency matrix of the d-regular graph , (all entries are ) vertex set S (say ) Def (Small Set Expander): A regular graph G is a -small set expander if for every set has eigenvalues Question: If is a -small set expander, how many eigenvalues larger than can the adjancency matrix have?

5. Def (Small Set Expander): A regular graph G is a -small set expander if for every set A Question in Spectral Graph Theory d-regular graph G Question: If is a -small set expander, How many eigenvalues larger than can the normalized adjancency matrix have? Intuitively, How many sparse cuts can a graph have without having a unbalanced sparse cut? Cheeger’s inequality: every large eigenvalue a sparse cut in () ( cut of sparsity )

6. Significance of the Question Question: If is a -small set expander, How many eigenvalues larger than can the normalized adjacency matrix have? Def (Small Set Expander): A regular graph G is a -small set expander if for every set of eigenvalues of graph G that are [Barak-Raghavendra-Steurer][Guruswami-Sinop] -round Lasserre SDP solves on graphs with low threshold rank . Graph has small non-expanding sets,  decompose in to smaller pieces and solve each piece. If UGC is true, there must be hard instances of Unique Games that a) Have high threshold rank b) Are Small-Set Expanders.

7. Significance of the Question Question: If is a -small set expander, How many eigenvalues larger than can the normalized adjacency matrix have? Def (Small Set Expander): A regular graph G is a -small set expander if for every set Answer to the above question (a function of ) [Arora-Barak-Steurer 2010] There is a -time algorithm for Gap-Small-Set-Expansion problem – a problem closely related to Unique Games. [Arora-Barak-Steurer 2010] A subexponential-time algorithm for Gap-Small-Set-Expansion problem At the time, Best known lower bound for  (Gap-Small-Set-Expansion problem could be solved in quasipolynomial time.) Gap Small Set Expansion Problem (GapSSE) Given a graph G and constants , Is OR ? where minimum expansion of sets of size

8. Short Code Graph [Barak-Gopalan-Hastad-Meka-Raghavendra-Steurer] For all small constant There exists a graph (the Short Code Graph) that is a small set expander with eigenvalues for some depending on [BGHMRS 11][ABS] • Led to new gadgets for hardness reductions (derandomized Majority is Stablest) and new SDP integrality gaps. • An interesting mix of techniques from, Discrete Fourier analysis, Locally Testable Codes and Derandomization.

9. Overview • Long Code Graph • Eigenvectors • Small Set Expansion • Short Code Graph

10. The Long Code Graph aka Noisy Hypercube n dimensional hypercube : {-1,1}n Noise Graph: Vertices: Edges: Connect every pair of points in hypercube separated by a Hamming distance of 1 1 1 x 1 -1 y Eigenvectors are functions on -1 -1 and differ in coordinates. Dictator cuts: Cuts parallel to the Axis (given by ) The dictator cuts yield sparse cuts in graph

11. 1 1 Sparsity of Dictator Cuts 1 1 Connect every pair of vertices in hypercube separated by Hamming distance of -1 -1 -1 n dimensional hypercube Dictator cuts: -eigenvectors with eigenvalues for graph (Number of vertices N = , so #of eigenvalues = )

12. Eigenfunctions for Noisy Hypercube Graph Eigenfunctions for the Noisy hypercube graph are multilinear polynomials of fixed degree! (Noisy hypercube is a Cayley graph on , therefore its eigen functions are characters of the group ) Eigenfunction Eigenvalue ……………………… Degree d multilinear polynomials

13. Small-Set Expansion (SSE) Given: regular graph with vertex set , parameter Suppose indicator function of a small non-expanding set. S has -fraction of vertices  Fraction of edges inside = = If then, at least a -fraction of edges are inside So, (f is close to the span of eigenvectors of G with eigenvalue ) • Conclusion: • Indicator function of a small non-expanding set f = is a • sparse vector • close to the span of the large eigenvectors of G

14. Hypercontractivity Definition: (Hypercontractivity) A subspace is hypercontractive if for all Projector in to the subspace also called hypercontractive. (No-Sparse-Vectors) Roughly, No sparse vectors in a hypercontractive subspace because, wis -sparse “” Hypercontractivity implies Small-Set Expansion projector into span of eigenvectors of with eigenvalue is hypercontractive  No sparse vector in span of top eigenvectors of G  No small non-expanding set in . (G is a small set expander)

15. Hypercontractivity for Noisy Hypercube Top eigenfunctions of noisy hypercube are low degree polynomials. (Hypercontractivity of Low Degree Polynomials) For a degree multilinear polynomial on , (Noisy Hypercube is a Small-Set Expander) For constant the noisy hypercube is a small-set expander. Moreover, the noisy hypercube has vertices and eigenvalues larger than .

16. Short Code Graph [Barak-Gopalan-Hastad-Meka-Raghavendra-Steurer 2009] For all small constant There exists a graph (the Short Code Graph) that is a small set expander with eigenvalues for some depending on

17. Short Code Graph Noise Graph: Vertices: Edges: Connect every pair of points in hypercube separated by a Hamming distance of 1 x -1 Has sparse cuts, but vertices -- too many vertices! Idea: Pick a subset of vertices of the long code graph, and their induced subgraph. The dictator cuts still yield -sparse cuts The subgraph is a small-set expander! If we reduce to then, the number of eigenvalues will be Choice: Reed Muller Codewords of large constant degree.

18. 1 1 Sparsity of Dictator Cuts 1 1 Connect every pair of vertices in hypercube separated by Hamming distance of -1 -1 -1 n dimensional hypercube Easy: Pretty much for any reasonable subset of vertices, dictators will be sparse cuts.

19. Preserving Small Set Expansion Top eigenfunctions of noisy hypercube are low degree polynomials. + (Hypercontractivity of Low Degree Polynomials) For a degree multilinear polynomial on , (Noisy Hypercube is a Small-Set Expander) For constant the noisy hypercube is a small-set expander.

20. Preserving Small Set Expansion (Hypercontractivity of Low Degree Polynomials) For a degree multilinear polynomial on , For a degree polynomial By hypercontractivityover hypercube, We picked a subset and so we want, is degree d, so and are degree . If S is a 4d-wise independent set then,

21. Preserving Small Set Expansion Top eigenfunctions of noisy hypercube are low degree polynomials. We Want: Only top eigenfunctions on the subgraph of noisy hypercube are also low degree polynomials. Connected to local-testability of the dual of the underlying code S! We appeal to local testability result of Reed-Muller codes [Bhattacharya-Kopparty-Schoenebeck-Sudan-Zuckermann]

22. Applications of Short Code [Barak-Gopalan-Hastad-Meka-Raghavendra-Steurer 2009] `Majority is Stablest’ theorem holds for the short code. -- More efficient gadgets for hardness reductions. -- Stronger integrality gaps for SDP relaxations. [Kane-Meka] A SDP gap with triangle inequalities for Balanced Separator Recap: ``How many large eigenvalues can a small set expander have?’’ -- A graph construction led to better hardness gadgets and SDP integrality gaps.

23. On the Power of Sum-of-Squares Proof

24. SoS hierarchy is a natural candidate algorithm for refuting UGC Should try to prove that this algorithm fails on some instances Only candidate instances were based on long-code or short-code graph Result: Level-8 SoS relaxation refutes UG instances based on long-code and short-code graphs We don’t know any instances on which this algorithm could potentially fail!

25. Result: Level-8 SoS relaxation refutes UG instances based on long-code and short-code graphs How to prove it? (rounding algorithm?) Interpret dual as proof system Show in this proof system that no assignments for these instances exist We already know “regular” proof of this fact! (soundness proof) Try to lift this proof to the proof system qualitative difference to other hierarchies: basis independence

26. Sum-of-Squares Proof System (informal) Axioms derive ( bounded-degree polynomials) … Rules Polynomial operations “Positivstellensatz” [Stengel’74] for any polynomial Intermediate polynomials have bounded degree (c.f. bounded-width resolution, but basis independent)

27. Example Axiom: Derive: (non-negativity of squares) (axiom)

28. Components of soundness proof (for known UG instances) Non-serious issues: Cauchy–Schwarz / Hölder Influence decoding Serious issues: can use variant of inductive proof, works in Fourier basis Hypercontractivity Invariance Principle typically uses bump functions, but for UG, polynomials suffice

29. long-code graph where projector into span of eigenfunctions of with eigenvalue SoS proof of hypercontractivity: is a sum of squares

30. long-code graph where projector into span of eigenfunctions of with eigenvalue SoS proof of hypercontractivity: difference is sum of squares For long-code graph, projects into Fourier polynomials with degree Stronger ind.Hyp.: where is a generic degree-Fourier polynomial and is a generic degree- Fourier polynomial

31. long-code graph where projector into span of eigenfunctions of with eigenvalue SoS proof of hypercontractivity: For long-code graph, projects into Fourier polynomials with degree Stronger ind.Hyp.: where is a generic degree-Fourier polynomial and is a generic degree- Fourier polynomial Write and (degrees of smaller than ) … ( (ind.hyp.)

32. Open Questions Does 8 rounds of Lasserre hierarchy disprove UGC? Can we make the short code, any shorter? (applications to hardness gadgets) Subexponential time algorithms for MaxCut or Vertex Cover (beating the current ratios) Thanks!