PCP Characterization of NP: Towards Polynomially Small ErrorProbability

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PCP Characterization of NP Towards Polynomially
Small Error-Probability
Irit Dinur
Eldar Fischer
Guy Kindler
Ran Raz
Shmuel Safra
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The class NP
?L?
Membership in L is verifiable in polynomial time,
given a correct membership proof .
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Cooks Theorem3-SAT?NPC
Input
LocalTests
VariableSet
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?L
Gap-SAT
yes Local-tests are satisfiable no At most
? are satisfiable
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Gap-SATD,V,?
BooleanFunctions

• Given
• ? A set of n local-tests over ?D variables.
• Variable range size2V
• Distinguish between
• ? is satisfiable.
• ? is not even ? satisfiable.

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ThmAS,ALMSS-92
Gap-SATO(1) , 1 , 1/2 is NP-hard.
ConjBGLR-93
Gap-SATO(1) , f , 2-f for any f(n)?O(log
n)is NP-hard.
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ThmRaSa-97
Gap-SATO(1) , f , 2-f for f(n)log? n,
?constlt1/4 is NP-hard.
Our Result
Gap-SATO(1) , f , 2-f for f(n)log? n, ?any
constlt1 is NP-hard.
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Gap-QSD,F,?

• Given
• ? A set of n Quadratic-Equations over ?D
  variables.
• Variables range in the field F.
• Distinguish between
• ? is satisfiable.
• ? is not even ? satisfiable.

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Main Theorem
Gap-QSO(1) , F , 2/F for F2log? n, ?any
constlt1 is NP-hard.
XY2
X2-YZ
ZXY
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ThmGJ
Exact-QS is NP-hard (for any F).
Simple
ThmHPS-93
Gap-QSn, F, 2/F is NP-hard.
Our Goal
Gap-QSO(1) , F , 2/F
Constant numberof variables in each equation
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Typical Proof Structure
Low Degree TestD,V,?
Gap-SATD,V,?
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Low-Degree TestD,V,?
r,d-LDF (low-degree function)
A degree r polynomial PFd?F
Representation
A set of variables, fixing P induces an
assignment for each. Variable-range size is 2v.
Example
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Low-Degree Test

• A set of local-tests, each accessing D variables.
• Local-tests can accept or reject.

LocalTests
RepresentationVariables
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D,V, ?-test
? Assignment ? P1,,P1/?2
A randomly chosen ?(xi1,..,xiD) satisfies
P1
P8
P5
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D,V, ?-test
? Assignment ? P1,,P1/?2
A randomly chosen ?(xi1,..,xiD) satisfies
P1
P8
P5
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xi1,..,xiD agree with no Pi yet ? accepts lt ?
probability
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History of Tests

• BFL,BFLS,BLR,FGLSS
• RuSu - r , logF , 1/2
• ASa,ALMSS - O(1), logF , 1/2
• RaSa - O(1), r2logF , F-c
• ASu - O(1) , r logF , F-c
• Our Test - O(1) , logF , F-c

? is exponentially small in the number of bits
accessed.
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Typical Proof Structure
Low Degree TestD1 , V1 , ?1
CompositionRecursion
Gap-SATD1 , V1 , ?1
Low Degree TestD,V, ?
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The RaSa Test

• Representation Variable per point,
  Variable per plane.
• Local-Tests Compare a point against a plane
  containing it.
• Parameters 2 , r2logF , F-c

r,d
r,2
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The Embedding Technique
Y
X
r,2
b,2logbr
b3
Y3Y3 Y9Y9 Y27Y27
X3X3 X9X9 X27X27
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The Embedding
X12Y56(X3)1(X9)1(Y)2(Y27)2
Y3Y3 Y9Y9 Y27Y27
X3X3 X9X9 X27X27
(X3)1(X9)1(Y)2(Y27)2
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Dimension Reduction
Degree Reduction
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Future Work?

• Full BGLR conjectureGap-SAT(O(1) , f , 2-f )
  for f(n)log n, is NP-hard.
• Gap-QS analogue gap-QS(O(1) , F , 2/F)For
  F?n, is NP-hard.
• Getting D2.
• Showing hardness for other approximation problems
  by similar techniques - SSAT