Inapproximability of NP-complete Problems, Discrete Fourier Analysis, Geometry

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Inapproximability of NP-complete Problems,
Discrete Fourier Analysis,
Geometry

• Subhash Khot
• Dept of Computer Science
• NYU-Courant Georgia Tech

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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 lt 1 Example MAX-3LIN
has approximation algorithm with
ratio ½. (random assignment
satisfies half the equations).
Minimization problems A(I) C OPT(I)
C gt 1
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Inapproximability Results
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This Talk

• Isoperimetric Problems (Geometry)
• Discrete Fourier Problem f 1,-1n ? 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 (Hastads Result)
• Sparsest Cut (Bourgains
  Noise-Sensitivity Theorem)
• (Majority Is
  Stablest, Borells 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 Sudan95,

Techniques from Probabilistically Checkable
Proofs
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Overview of the Talk

• General framework ((far from)
  Dictatorships)
• MAX-3LIN (Hastads Result)
• Sparsest Cut (Bourgains
  Noise-Sensitivity Theorem)
• (Majority Is
  Stablest, Borells 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 (Hastads Result)
• Sparsest Cut (Bourgains
  Noise-Sensitivity Theorem)
• (Majority Is
  Stablest, Borells Theorem)
• Kernel Clustering (Center of Mass
  Problem)

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Sparsest Cut (Balanced Partitioning)
Given graph G(V,E), find a partition V S
? 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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Bourgains 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 (Hastads Result)
• Sparsest Cut (Bourgains
  Noise-Sensitivity Theorem)
• (Majority Is
  Stablest, Borells Theorem)
• Kernel Clustering (Center of Mass
  Problem)

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Majority Is Stablest (in FarFromDict )
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Invariance Principle Rotar79, MOO05
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Invariance Principle Rotar79, MOO05
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Isoperimetric Problem, Borells Theorem
K Kindler Mossel Odonnell 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 (Hastads Result)
• Sparsest Cut (Bourgains
  Noise-Sensitivity Theorem)
• (Majority Is
  Stablest, Borells 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?
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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 .. Friedgut98 If
total influence is k, then f essentially depends
on at most 2O(k) co-ordinates. K Regev03
Assuming UGC, Vertex Cover has no
2- e approximation. Green
Sanders06 If spectral norm is k, then f is
sum of indicators of 22O(k4) affine
subspaces. Application to inapproximability ?