Probabilistically Checkable Proofs (and inapproximability)

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Probabilistically Checkable Proofs(and
inapproximability)

• Irit Dinur, Weizmann

open day, May 1st 2009
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How Efficiently Can Proofs Be Checked?
(slide by Madhu Sudan)
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our real interest NP proofs

• NP class of problems with efficiently
  verifiable solutions
• Examples 3-colorability, Satisfiability,
  Clique, etc.
• Theory of NP-completeness provides enormous
  collection of new formats for writing proofs.
• Strange, but just as valid (every thm has proof,
  but no false thm has one). Possibly new formats
  give more power? new features?

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Randomizing proof access
3-colorability

• One proof for 3-colorability is a 3-coloring
• We can verify it edge by edge
• Murphys law! we detect an error only on the
  last clause (no abundance of errors)
• How can we gain by randomizing? (ask for another
  proof!)

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Add randomness, allow errors (ideas coming from
interactive proofs and cryptography)
Randomizing proof access
Possible gain read fewer proof bits
Verifier
Input x

• If x2 L then 9proof ?, PrVer accepts (x,?) 1
• If x? L then 8proof ?, PrVer accepts (x,?) lt s
  lt 1

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Restricting proof access

• How much of the proof must the Verifier read?
• stage 1 proof-bit-queries logarithmic in
  proof length
• stage 2 proof-bit-queries absolute constant
  !! ? The PCP Theorem Arora-Safra,
  Arora-Lund-Motwani-Sudan-Szegedy 92
• stage 3 proof-bit-queries 3 Hastad 97

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How can this be done ???
we want an error-amplifying reduction
err-amp
H
G
every 3-col of Hs vertices violates gt 10 edges
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How can this be done ???
we want an error-amplifying reduction
without looking
(similar to error correcting codes)
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approaches
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• Approximation
• and
• Inapproximability

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Optimization Problems finding nearly optimal
solutions

• Example the Minimum Vertex Cover problem
• Facts 1. Best algorithm runs in time (1.21)n
  Robson 86
• 2. VC is NP-hard. Karp 72
• What about approximation.. Output a vertex cover
  thats nearly minimal!

Minimum Vertex Cover
Vertex-Cover Given a graph find the smallest
set of vertices that touch all edges.
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Approximation
What do we mean by approximation? Each instance
has many solutions, each has a value. In
optimization, we are seeking the minimal.
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Approximation
An approximation algorithm finds a solution
within a certain neighborhood of MIN

• Example An algorithm for Approximating Vertex
  Cover
• Given G, find a maximal set of edges that do not
  touch each
  other.
• Add both vertices of each edge to the vertex
  cover.

MIN
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Approximation
This is a solution all edges are covered
How big is it? No more than twice the minimum!
An approximation algorithm finds a solution
within a certain neighborhood of MIN

• Example An algorithm for Approximating Vertex
  Cover
• Given G, find a maximal set of edges that do not
  touch each
  other.
• Add both vertices of each edge to the vertex
  cover.

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Approximation
How big is it? No more than twice the minimum!
An approximation algorithm finds a solution
within a certain neighborhood of MIN

• Example An algorithm for Approximating Vertex
  Cover
• Given G, find a maximal set of edges that do not
  touch each
  other.
• Add both vertices of each edge to the vertex
  cover.

Weve seen an approximation algorithm for
Vertex-Cover, with approximation factor 2.
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Approximation
x 2
x 3/2
x 4/3
x 1.99
Weve seen a factor 2 algorithm. Q Is there a
factor 1.99 algorithm? 3/2 ? 4/3 ?
No, due to PCP thm (and more work)
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• ma
• hakesher?

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How does one prove inapproximability?
we want a gap-amplifying reduction
gap-amp
H
G
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How does one prove inapproximability?
we want a gap-amplifying reduction
gap-amp
H
G
G is 3col
H is 3col
G is not 3col
H is lt90 3col
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The FGLSS connection

• error-amplifying reductions
• are inapproximability results!
• are PCPs!

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PCP Inapprox
imability
FGLSS, ALMSS
( x ? G ? H )
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Getting tight results
max-cut
3-SAT
vertex-cover
coloring
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summary

• Probabilistically Checkable Proofs
• randomize proof access ? gain locality
• how? by amplifying errors in false proofs
• like in error correcting codes
• Hardness of approximation
• vertex cover
• amplifying gaps
• towards tight results
• Connections

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• thank you!