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P robabilistically C heckable P roofs (and inapproximability). Irit Dinur, Weizmann. open day, May 1 st 2009. P  NP (12 th Revision) By Ayror Sappen # Pages to follow: 15783. How Efficiently Can Proofs Be Checked ?. (slide by Madhu Sudan). our real interest: NP proofs.

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P robabilistically c heckable p roofs and inapproximability

Probabilistically Checkable Proofs(and inapproximability)

Irit Dinur, Weizmann

open day, May 1st 2009


How efficiently can proofs be checked

P  NP (12th Revision)

By

Ayror Sappen

# Pages to follow: 15783

How Efficiently Can Proofs Be Checked?

(slide by Madhu Sudan)


Our real interest np proofs
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?


Randomizing proof access
Randomizing proof access

3-colorability

  • One proof for 3-colorability is a 3-coloring:

  • We can verify it edge by edge

  • Murphy’s law! we detect an “error” only on the last clause (no abundance of errors)

  • How can we gain by randomizing? (ask for another proof!)


Prob.Checkable.Proof: 

Add randomness, allow errors (ideas coming from interactive proofs and cryptography)

Randomizing proof access

  • If x2 L then9proof, Pr[Ver accepts (x,)] = 1

  • If x L then 8proof, Pr[Ver accepts (x,)] < s < 1

Possible gain:

read fewer proof bits

Verifier

Input: x


Restricting proof access
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]


How can this be done

G is 3col

H is 3col

G is not 3col

H is <90% 3col

How can this be done ???

we want an“error”-amplifyingreduction…

err-amp

H

G

every 3-col of H’s vertices

violates > 10% edges


How can this be done1

err-amp

How can this be done ???

we want an“error”-amplifyingreduction…

without looking…

(similar to error correcting codes)


Approaches

Interactive Proofs, Cryptography

Finite fields, Reed Muller &

Reed Solomon codes,

low degree curves

Expanders and

pseudorandom

objects

approaches


Approximation

and

Inapproximability


4

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 that’s “nearly” minimal!

Minimum Vertex Cover

Vertex-Cover: Given a graph find the smallest set of vertices that touch all edges.


Approximation

4

5

7

What do we mean by approximation?

Each instance has many solutions, each has a value.

In optimization, we are seeking the minimal.


Approximation

Approx

4

5

7

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


Approximation1

This is a solution: all edges are covered

How big is it?

No more than twice the minimum!

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.


Approximation2

How big is it?

No more than twice the minimum!

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.

We’ve seen an approximation algorithm for Vertex-Cover, with approximation factor 2.


Approximation3

MIN

Approx

No, assuming very very strong PCP conjecture (“unique games”)

Approximation

x 4/3

x 3/2

x 1.99

x 2

We’ve 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)


ma

hakesher?


How does one prove inapproximability

VC(G) = k

VC(H) = k’

VC(G) > k

VC(H) > (2-²) k’

How does one prove inapproximability?

we want a“gap”-amplifyingreduction…

gap-amp

H

G


How does one prove inapproximability1
How does one prove inapproximability?

we want a“gap”-amplifyingreduction…

gap-amp

H

G

G is 3col

H is 3col

G is not 3col

H is <90% 3col


The fglss connection
The [FGLSS] connection

  • “error”-amplifying reductions

    … are inapproximability results!

    &

    … are PCPs!


Pcp inapprox

PCP

Prob.Checkable.Proof

G is 3col

H is 3col

Verifier

G is not 3col

H is <90% 3col

x 2? L

PCP & Inapprox

imability

[FGLSS, ALMSS]

( x  G  H )


Getting tight results

Metric Embedding,

Semi-definite programming

Discrete Fourier Analysis

Complexity of

Boolean functions, Influences

Extremal set theory, EKR intersection theorems

Probability and

Noise correlation, Invariance principles

Getting tight results

max-cut

3-SAT

vertex-cover

coloring


Summary
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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