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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
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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.
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MAX-3LIN Def: Given a system of linear equations modulo 2,
each equation containing exactly 3 variables, find an assignment that satisfies maximum #equations.
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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
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Inapproximability Results
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This Talk Isoperimetric Problems (Geometry)
Discrete Fourier Problem f: {+1,-1}n {+1,-1} (Dictatorship functions versus Functions far from Dictatorships) Inapproximability Result
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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)
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Dictatorships and Far from Dictatorships
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Dictatorships and Far from Dictatorships
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General Framework [Bellare Goldreich Sudan’95, ………]
Intuitively OBJ(f) captures some property of dictatorships Techniques from Probabilistically Checkable Proofs
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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)
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Linearity Test [Blum Luby Rubinfeld ’90]
Dictatorships are stable under noise
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Linearity Test with Noise
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Linearity Test with Noise
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Inapproximability of MAX-3LIN
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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)
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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) ?
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Noise Sensitivity
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Bourgain’s Noise-Sensitivity Theorem
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Negative Type Metrics versus L1
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Related Results
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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)
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Majority Is Stablest (in FarFromDict )
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Invariance Principle [Rotar’79, MOO’05]
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Invariance Principle [Rotar’79, MOO’05]
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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.
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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)
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Kernel Clustering (with k clusters)
P.S.D A =
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Fourier Problem for general k
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Isoperimetric Problem
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Observations
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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 !
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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.
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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 ?
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