Introduction to PCP and Hardness of Approximation

Dana MoshkovitzPrinceton University and

The Institute for Advanced Study

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This Talk

A Groundbreaking Discovery!

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(From 1991-2)

The PCP Theorem and Hardness of Approximation

A Canonical Optimization Problem

MAX-3SAT:Given a 3CNF Á, what fraction of the clauses can

be satisfied simultaneously?

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Á = (x7 : x12 x1) Æ … Æ (:x5 : x9 x28)

x1

x2

x3

x4

x5

x6

x7

x8

xn-3

xn-2

xn-1

xn

. . .

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Good Assignment Exists

Claim: There must exist an assignment that satisfies at least 7/8 fraction of clauses.

Proof: Consider a random assignment.

x1 x2 x3 xn

. . .

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1. Find the Expectation

Let Yi be the random variable indicating whether the i-th clause is satisfied.

For any 1im,

F F F F

F F T T

F T F T

F T T T

T F F T

T F T T

T T F T

T T T T

87

181

0YE i 87

181

0YE i

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1. Find the Expectation

The number of clauses satisfied is a random variable Y=Yi.

By the linearity of the expectation:

E[Y] = E[ Yi] = E[Yi] = 7/8m

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2. Conclude Existence

Thus, there exists an assignment which satisfies at least the expected fraction (7/8) of clauses.

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®-Approximation (Max Version)

OPT

OPT(x)

For every input x, computed value C(x):® ¢ OPT(x) · C(x) · OPT(x)

Corollary: There is an efficient ⅞-approximation algorithm for MAX-3SAT.

Better Approximation?

Fact: An efficient tighter than ⅞-approximation algorithm is not known.

Our Question: Can we prove that if P≠NP such algorithm does not exist?

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Computation Decision

Hardness of distinguishing far off instances Hardness of approximation

A B

gap

OPT(x)

OPT

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Gap Problems (Max Version)

• Instance: …

• Problem: to distinguish between the following two cases:

The maximal solution ≥ B

The maximal solution < A

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Gap NP-Hard Approximation NP-hard

Claim: If the [A,B]-gap version of a problem is NP-hard, then that problem is NP-hard to approximate to

within factor A/B.

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Gap NP-Hard Approximation NP-hard

Proof (for maximization): Suppose there is an approximation algorithm that, for every x, outputs C(x) ≤ OPT so that C(x) ≥ A/B¢OPT.

Distinguisher(x):* If C(x) ≥ A, return ‘YES’* Otherwise return ‘NO’

A B

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(1) If OPT(x) ≥ B (the correct answer is ‘YES’), then necessarily, C(x) ≥ A/B¢OPT(x) ≥ A/B·B = A(we answer ‘YES’)

(2) If OPT(x)<A (the correct answer is ‘NO’), then necessarily, C(x) ≤ OPT(x) < A(we answer ‘NO’).

Gap NP-Hard Approximation NP-hard

New Focus: Gap Problems

Can we prove that gap-MAX-3SAT is NP-hard?

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Connection to Probabilistic Checking of Proofs [FGLSS91,AS92,ALMSS92]

Claim: If [A,1]-gap-MAX-3SAT is NP-hard, then every NP language L has a probabilistically checkable proof (PCP):

There is an efficient randomized verifier that queries 3 proof symbols:

• xL: There exists a proof that is always accepted.• xL: For any proof, the probability to err and

accept is ≤A. Note: Can get error probability ² by making

O(log1/²) queries.

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Probabilistic Checking of xL?

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If yes, all of Á clauses are satisfied. If no, fraction ≤A of Á clauses can be satisfied.

x1

x2

x3

x4

x5

x6

x7

x8

xn-3

xn-2

xn-1

xn

. . .

Prove xL!This assignment satisfies Á! Enough to check a

random clause!

Other Direction: PCP Gap-MAX-3SAT NP-Hard

• Note: Every predicate on O(1) Boolean variables can be written as a conjunction of O(1) 3-clauses on the same variables, as well as, perhaps, O(1) more variables.– If the predicate is satisfied, then there exists an

assignment for the additional variables, so that all 3-clauses are satisfied.

– If the predicate is not satisfied, then for any assignment to the additional variables, at least one 3-clause is not satisfied.

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The PCP Theorem

Theorem […,AS92,ALMSS92]: Every NP language L has a probabilistically checkable proof (PCP):

There is an efficient randomized verifier that queries O(1) proof symbols:

xL: There exists a proof that is always accepted.xL: For any proof, the probability to accept is ≤½.

Remark: Elegant combinatorial proof by Dinur, 05.

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Conclusion

Probabilistic Checking of Proofs (PCP)

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Hardness of Approximation

Tight Inapproximability?

• Corollary: NP-hard to approximate MAX-3SAT to within some constant factor.

• Question: Can we get tight ⅞-hardness?

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The Bellare-Goldreich-Sudan Paradigm, 1995

Projection Games Theorem(aka Hardness of Label-Cover, or low

error two-query projection PCP)

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Tight Hardness of Approximating 3SAT [Håstad97]

Long-code based reduction

The Bellare-Goldreich-Sudan Paradigm, 1995

Projection Games Theorem(aka Hardness of Label-Cover, or low

error two-query projection PCP)

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Tight Hardness of Approximation for Many Problems

Long-code based reduction

e.g., Set-Cover [Feige96]

Projection Games & Label-Cover

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A

B

• Bipartite graph G=(A,B,E) • Two sets of labels §A, §B

• Projections ¼e:§A§B

• Players A & B label vertices• Verifier picks random e=(a,b)2E• Verifier checks ¼e(A(a)) = B(b)

• Value = maxA,BP(verifier accepts)

¼e

Label-Cover: given projection game, compute value.

Equivalent Formulation of PCP Thm

Theorem […,AS92,ALMSS92]: NP-hard to approximate Label-Cover within some constant.

Proof: by reduction to Label-Cover (see picture).

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Verifier randomness

Proof entries

Verifier queries…

Accepting verifier view

Projection =

consistency check

symbol

Projection Games Theorem: Low Error PCP Theorem

Claim: There is an efficient 1/k-approximation algorithm for projection games on k labels (i.e., |§A|,|§B|·k).

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Projection Games TheoremFor every ²>0, there is k=k(²), such that it is NP-

hard to decide for a given projection game on k labels whether its value = 1 or < ².

The Bellare-Goldreich-Sudan Paradigm

Projection Games Theorem(aka Hardness of Label-Cover, or low

error two-query projection PCP)

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Tight Hardness of Approximation for Many Problems

??

How To Prove The Projection Games Theorem?

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Hardness of Approximation

Projection Games Theorem

[AS92,ALMSS92] PCP Theorem

Parallel repetition Theorem [Raz94]

[M-Raz08] Construction

The Khot Paradigm, 2002

Unique Games Conjecture

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Tight Hardness of Approximation for More Problems

e.g., Vertex-Cover [DS02,KR03]

e.g., Max-Cut [KKMO05]

Long-code based reduction

Constraint Satisfaction Problems

[Raghavendra08]

Thank You!

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