1. Subhash Khot Dept of Computer Science NYU-Courant & Georgia Tech
Inapproximability of NP-complete Problems, Discrete Fourier Analysis, Geometry
Subhash Khot
Dept of Computer Science
NYU-Courant & Georgia Tech
TexPoint fonts used in EMF.
Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAAAAA

2. NP-complete Problems Traveling Salesperson MAX-3LIN Sparsest Cut
Kernel Clustering
…………………..
Problem size: n = |G|, # variables, matrix size, ….
Widely believed P ≠ NP hypothesis :
No efficient algorithm can solve NP-complete problems.
Efficient / fast = runs in polynomial in n steps.

3. MAX-3LIN Def: Given a system of linear equations modulo 2,
each equation containing exactly 3 variables,
find an assignment that satisfies maximum #equations.

4. Approximation Algorithms
Def: An approximation algorithm with ratio C = C(n)
is a polynomial time algorithm that on problem instance I,
computes a solution A(I) such that
Maximization problems: A(I) ≥ C ∙ OPT(I) C < 1
Example: MAX-3LIN has approximation algorithm
with ratio ½.
(random assignment satisfies half the equations).
Minimization problems: A(I) ≤ C ∙ OPT(I) C > 1

5. Inapproximability Results

6. This Talk Isoperimetric Problems (Geometry)
Discrete Fourier Problem f: {+1,-1}n  {+1,-1}
(Dictatorship functions versus
Functions far from Dictatorships)
Inapproximability Result

7. Overview of the Talk General framework ((far from) Dictatorships)
MAX-3LIN (Hastad’s Result)
Sparsest Cut (Bourgain’s Noise-Sensitivity Theorem)
(Majority Is Stablest, Borell’s Theorem)
Kernel Clustering (Center of Mass Problem)

8. Dictatorships and Far from Dictatorships

9. Dictatorships and Far from Dictatorships

10. General Framework [Bellare Goldreich Sudan’95, ………]
Intuitively OBJ(f)
captures some
property of dictatorships
Techniques from Probabilistically
Checkable Proofs

11. Overview of the Talk General framework ((far from) Dictatorships)
MAX-3LIN (Hastad’s Result)
Sparsest Cut (Bourgain’s Noise-Sensitivity Theorem)
(Majority Is Stablest, Borell’s Theorem)
Kernel Clustering (Center of Mass Problem)

12. Linearity Test [Blum Luby Rubinfeld ’90]
Dictatorships are stable under noise

13. Linearity Test with Noise

14. Linearity Test with Noise

15. Inapproximability of MAX-3LIN

16. Overview of the Talk General framework ((far from) Dictatorships)
MAX-3LIN (Hastad’s Result)
Sparsest Cut (Bourgain’s Noise-Sensitivity Theorem)
(Majority Is Stablest, Borell’s Theorem)
Kernel Clustering (Center of Mass Problem)

17. Sparsest Cut (Balanced Partitioning)
V\S
Given graph G(V,E), find a partition V = S ﮞ V\S
s.t. |S| ≈ n/2 and minimize E(S, V\S).
Open: Is there approximation algorithm with ratio O(1) ?

18. Noise Sensitivity

19. Bourgain’s Noise-Sensitivity Theorem

20. Negative Type Metrics versus L1

21. Related Results

22. Overview of the Talk General framework ((far from) Dictatorships)
MAX-3LIN (Hastad’s Result)
Sparsest Cut (Bourgain’s Noise-Sensitivity Theorem)
(Majority Is Stablest, Borell’s Theorem)
Kernel Clustering (Center of Mass Problem)

23. Majority Is Stablest (in FarFromDict )

24. Invariance Principle [Rotar’79, MOO’05]

25. Invariance Principle [Rotar’79, MOO’05]

26. Isoperimetric Problem, Borell’s Theorem
[K Kindler Mossel O’donnell ‘04]
Conjectured that Majority is Stablest.
Showed that it implies optimal inapproximability result for
MAX-CUT problem.

27. Overview of the Talk General framework ((far from) Dictatorships)
MAX-3LIN (Hastad’s Result)
Sparsest Cut (Bourgain’s Noise-Sensitivity Theorem)
(Majority Is Stablest, Borell’s Theorem)
Kernel Clustering (Center of Mass Problem)

28. Kernel Clustering (with k clusters)
P.S.D A =

29. Fourier Problem for general k

30. Isoperimetric Problem

31. Observations

32. k = 2, 3, and beyond? k=2 Rn-1 × k=3 Rn-2 × regular k=4 Rn-3 × regular
k=3
Rn-2 ×
regular
k=4
Rn-3 ×
regular
False !

33. Geometric Conjecture One can show that regular k partition for k ≥ 4
is worse than regular partition into 3 parts.
Conjecture: For k ≥ 4, best partition into k parts
is actually partitioning into only 3 regular parts.

34. Conclusion Many more examples ….. 2- ε approximation.
[Friedgut’98] If total influence is k, then f essentially depends on at most 2O(k) co-ordinates.
[K Regev’03] Assuming UGC, Vertex Cover has no
2- ε approximation.
[Green Sanders’06] If spectral norm is k, then f is
± sum of indicators of 22^{O(k^4)} affine subspaces.
Application to inapproximability ?