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

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


1
Inapproximability of NP-complete Problems,
Discrete Fourier Analysis,
Geometry
  • Subhash Khot
  • Dept of Computer Science
  • NYU-Courant Georgia Tech

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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 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
5
Inapproximability Results
6
This Talk
  • Isoperimetric Problems (Geometry)
  • Discrete Fourier Problem f 1,-1n ? 1,-1
  • (Dictatorship functions versus
  • Functions far from Dictatorships)
  • Inapproximability Result

7
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)

8
Dictatorships and Far from Dictatorships
9
Dictatorships and Far from Dictatorships
10
General Framework Bellare Goldreich Sudan95,

Techniques from Probabilistically Checkable
Proofs
11
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)

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 (Hastads Result)
  • Sparsest Cut (Bourgains
    Noise-Sensitivity Theorem)
  • (Majority Is
    Stablest, Borells Theorem)
  • Kernel Clustering (Center of Mass
    Problem)

17
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) ?
18
Noise Sensitivity
19
Bourgains Noise-Sensitivity Theorem
20
Negative Type Metrics versus L1
21
Related Results
22
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)

23
Majority Is Stablest (in FarFromDict )
24
Invariance Principle Rotar79, MOO05
25
Invariance Principle Rotar79, MOO05
26
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.
27
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)

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?
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 .. 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 ?
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