Post on 18-Dec-2015
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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