two query pcp with subconstant error dana moshkovitz princeton university and the institute for...
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Two Query PCP with Subconstant Error
Dana MoshkovitzPrinceton University and
The Institute for Advanced Study
Ran RazThe Weizmann Institute
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This Talk
Hardness of Approximation & PCPs (Probabilistically Checkable Proofs)
How we can construct PCPs that are useful for hardness of approximation.
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Hardness of Approximation
The 3SAT Maximization Problem:Given a 3CNF Á, how many clauses can be
satisfied simultaneously?
x1 x2 x3 x4 x5 x6 xn
0 0 1 0 1 1 . . . 1
Á = (x7 : x12 x1) Æ … Æ (:x5 : x9 x28)
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Hardness of Approximating 3SAT
Theorem (Håstad97): For any constant >0, 3SAT is NP-hard to
approximate within ⅞ + .
This work: Improving Håstad97 to =o(1), and many more results!
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The Bellare-Goldreich-Sudan Paradigm
Projection Games Theorem(aka Hardness of Label-Cover, or
two-query projection PCP)
Hardness of Approximating 3SAT
Long-code based reduction
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The Bellare-Goldreich-Sudan Paradigm
Projection Games Theorem(aka Hardness of Label-Cover, or
two-query projection PCP)
Hardness of Approximating Constraint Satisfaction Problems
… and many more problems!
Long-code based reduction e.g., Vertex-Cover [DS02]
e.g., Set-Cover [Feige96]
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Projection Games
?
?
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 of game = maxA,BP(verifier accepts)
¼e
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Projection Games Theorem
Projection Games Theorem There exists 0<c<1, s.t.For every ²¸1/nc, 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.
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How To Prove The Projection Games Theorem?
??
Hardness of Approximation
Projection Games Theorem
[AS92,ALMSS92] PCP Theorem + [Raz94] Parallel Repetition
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Caveat in Parallel Repetition
• Parallel repetition blows-up size to nθ(log 1=²:– Proves Quasi-NP-hardness – NP-hardness only for constant ².
• [Feige, Kilian, 95]: No “de-randomization”!
Projection Games Theorem There exists 0<c<1, s.t.For every ²¸1/nc, 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.
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Subconstant Error for Projection Games?
• [RS97, AS97, DFKRS99, MR07]: subconstant error ²=²(n), as low as ²=2-(logn)1® for all ®>0.
• More than two queries! Not projection game! Much less useful for hardness of approx.
• Folklore: three queries for error ²=2-(logn)® for some ®>0.
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Our Work
Hardness of Approximation
Projection Games Theorem
[AS92,ALMSS92] PCP Theorem + [Raz94]Parallel RepetitionNew construction with almost-linear size
n1+o(1)poly(1/²)
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Caveat in Our Work
• Many labels: k=2poly(1/²)
• “Sliding-Scale Conjecture” [BGLR93]: k=poly(1/²) • k = poly(n) only for ²¸1/(logn)¯ for some ¯>0
Projection Games Theorem There exists 0<c<1, s.t.For every ²¸1/nc, 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.
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Implications
Improving Håstad: NP-hard to approximate 3SAT on inputs of size N within 7/8+1/(loglogN) for some constant >0 (blow-up N=n1+o(1)).
Similarly, improvements to: 3LIN [Håstad,97], amortized query complexity and free bit complexity [Samorodnitsky-Trevisan,00],
…
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Starting Point
Projection Games Theorem with many labelsFor every ², there is k=k(n,²)=2poly(no(1),1/²), such that
it is NP-hard to decide for a given projection game on k labels whether its value = 1 or < ².
The reduction is almost-linear n1+o(1)poly(1/²).
• Construction is algebraic, based on low degree testing theorem with low error [AS97,RS97].
• Almost-linear size by [MR06,MR07].
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Composition
Reduce the number of labelsk=k(n,²)=2poly(no(1),1/²) k=k(²)=2poly(1/²)
by composition
Previously: Either:1) Increase in # queries [AS92...BGHSV04]2) Two queries, but error ²¼1 [DR04]
Recently: Generalization by [Dinur-Harsha, 09]
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Code Concatenation (Forney, 1966)
• Can iterate. When ni·logn, can use Hadamard (exponential length).
. . .
poly(n±,1/²)
. . .
poly(n±2,1/²)
n1+o(1)poly(1/²)
• Length: multiplies• Distance: multiplies
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Analogy to Codes
. . .B
A . . .labels to B codewordA vertex constraint: B neighborhood
consistent with label to a
value maxA Ea[% consistent neighbors]
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Change Perspective: Switch Sides!
• Associate each A vertex with its B neighborhood.• View B vertices as posing constraints: consistency
among containing neighborhoods.
. . .B
A. . . . . . . . . . .
. . .
. . . . . . . . . . .
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Sunflowers• Label to B vertex = “sunflower” of labels to A neighbors• log|§new| = Bdegree· log|§old| = poly(1/²)· log|§old|
. . .B
A. . . . . . . . . . .
Will reduce this!
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The Key Idea
Label to B vertex = a sunflower of sub-petalsEncode A labels so can locally decode/reject center
. . .B
A. . . . . . . . . . .
CompositionEncode each neighborhood with LDRC
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. . .B
A. . .
... .... . .Binner
Ainner. . . . . . . . . . . .
Binner
Ainner