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P robabilistically C heckable P roofs (and inapproximability)

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Probabilistically Checkable Proofs(and inapproximability)

Irit Dinur, Weizmann

open day, May 1st 2009

P NP (12th Revision)

By

Ayror Sappen

# Pages to follow: 15783

(slide by Madhu Sudan)

- 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?

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

- 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]

G is 3col

H is 3col

G is not 3col

H is <90% 3col

we want an“error”-amplifyingreduction…

err-amp

H

G

every 3-col of H’s vertices

violates > 10% edges

err-amp

we want an“error”-amplifyingreduction…

without looking…

(similar to error correcting codes)

Interactive Proofs, Cryptography

Finite fields, Reed Muller &

Reed Solomon codes,

low degree curves

Expanders and

pseudorandom

objects

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.

Approx

4

5

7

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

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.

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.

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

MIN

Approx

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

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?

VC(G) = k

VC(H) = k’

VC(G) > k

VC(H) > (2-²) k’

we want a“gap”-amplifyingreduction…

gap-amp

H

G

we want a“gap”-amplifyingreduction…

gap-amp

H

G

G is 3col

H is 3col

G is not 3col

H is <90% 3col

- “error”-amplifying reductions
… are inapproximability results!

&

… are PCPs!

PCP

Prob.Checkable.Proof

G is 3col

H is 3col

Verifier

G is not 3col

H is <90% 3col

x 2? L

imability

[FGLSS, ALMSS]

( x G H )

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

max-cut

3-SAT

vertex-cover

coloring

- 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

thank you!