Introduction%20to%20PCP%20and%20Hardness%20of%20Approximation

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Introduction to PCP and Hardness of Approximation

• Dana Moshkovitz
• Princeton University and
• The Institute for Advanced Study

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This Talk

• A Groundbreaking Discovery!

(From 1991-2)
The PCP Theorem and Hardness of Approximation
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A Canonical Optimization Problem

• MAX-3SAT
• Given a 3CNF Á, what fraction of the clauses can
  be satisfied simultaneously?

Á (x7 ? x12 ? x1) Æ Æ (x5 ? x9 ? x28)
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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.

. . .
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1. Find the Expectation

• Let Yi be the random variable indicating whether
  the i-th clause is satisfied.
• For any 1?i?m,

? ? ? ?????
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
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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
• EY E? Yi ? EYi 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)
For every input x, computed value C(x)
OPT(x) C(x) OPT(x)
OPT
OPT(x)
Corollary There is an efficient ?-approximation
algorithm for MAX-3SAT.
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Better Approximation?

• Fact An efficient tighter than ?-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

OPT
OPT(x)
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Gap Problems (Max Version)

• Instance
• Problem to distinguish between the following two
  cases
• The maximal solution B
• The maximal solution lt A

YES
NO
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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/BOPT.
• Distinguisher(x)
• If C(x) A, return YES
• Otherwise return NO

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Gap NP-Hard ? Approximation NP-hard

• (1) If OPT(x) B (the correct answer is YES),
  then necessarily, C(x) A/BOPT(x) A/BB
  A(we answer YES)
• (2) If OPT(x)ltA (the correct answer is NO),
  then necessarily, C(x) OPT(x) lt A(we answer
  NO).

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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
• x?L There exists a proof that is always
  accepted.
• x?L 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 x?L?

If yes, all of Á clauses are satisfied. If no,
fraction A of Á clauses can be satisfied.
Prove x?L!
This assignment satisfies Á!
Enough to check a random clause!
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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
• x?L There exists a proof that is always
  accepted.
• x?L 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)

Hardness of Approximation
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Tight Inapproximability?

• Corollary NP-hard to approximate MAX-3SAT to
  within some constant factor.
• Question Can we get tight ?-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)

Long-code based reduction
Tight Hardness of Approximating 3SAT Håstad97
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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)

Long-code based reduction
e.g., Set-Cover Feige96

• Tight Hardness of Approximation for Many Problems

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Projection Games Label-Cover
A

• Bipartite graph G(A,B,E)
• Two sets of labels A, B
• Projections ¼eA?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)

?







B


?







¼e
Label-Cover given projection game, compute value.
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Equivalent Formulation of PCP Thm
Verifier randomness

• Theorem ,AS92,ALMSS92 NP-hard to approximate
  Label-Cover within some constant.
• Proof by reduction to Label-Cover (see picture).

Proof entries













Verifier queries










Projection consistency check
symbol
Accepting verifier view
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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,Bk).

Projection Games Theorem For every ²gt0, there is
kk(²), such that it is NP-hard to decide for a
given projection game on k labels whether its
value 1 or lt ².
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The Bellare-Goldreich-Sudan Paradigm

• Projection Games Theorem
• (aka Hardness of Label-Cover, or low error
  two-query projection PCP)

Tight Hardness of Approximation for Many Problems
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How To Prove The Projection Games Theorem?
AS92,ALMSS92 PCP Theorem
Parallel repetition Theorem Raz94
M-Raz08 Construction
??
Projection Games Theorem
Hardness of Approximation
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The Khot Paradigm, 2002

• Unique Games Conjecture

Long-code based reduction
Constraint Satisfaction Problems Raghavendra08
e.g., Max-Cut KKMO05
e.g., Vertex-Cover DS02,KR03
Tight Hardness of Approximation for More Problems
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Thank You!