E3-LIN-2 is hard to approximate

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E3-LIN-2 is hard to approximate
Hastad
Speaker Guy Kindler
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An m-SAT instance ?

• Variables (not necessarily Boolean).
• Constraints Each over m variables.
• Example
• For all x,y in (Z2)n,
• A(x)A(y)A(xy)

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An m-CSP instance ?

• Variables (not necessarily Boolean).
• Constraints Each over m variables.
• S(?) maximum fraction of satisfied constraints
  .

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Cook-Levin

• In 3-SAT, it is hard to distinguish between the
  cases
• S(?)1
• S(?)lt1

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PCP Theorem

• In 3-SAT, it is hard to distinguish between
  S(?)1 and S(?)lt1-?

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PCP Theorem

• GAP(1-? ,1) 3-SAT, is NP-hard.

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Our Goal

• Show hardness for GAP(1/2 ?,1- ?) 3-CSP
• Where each constraint is a linear equation over Z2

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The Scheme

1. From gap 3-SAT to gap(?,1) 2-CSP
2. V ? long-code table,constraint(V,U) ? linear
   equations.

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par(?,k)

• ? variables x, constraints c
• V(x1,x2,..,xk) (k-tuple)
• U(c1,..,ck)
• V?U if xi?ci
• One constraint per V?U

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Parallel Repetition Raz

• If ? is in gap(1-? ,1) 3-sat, then
• par(?,k) has gap (g(?)k,1)
• Exercise par(?,1) has gap (1-?/3 ,1)

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par(?,k) Forgetful Functor

• V-variables over v
• U-variables over u
• A constraint over U,V a function ?u--gtv
• If U is assigned i, V should get j?(i)
• Either all constraints are satisfiable,or not
  even an ?-fraction.
• Holds for random-assignments as well.

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Final System - Variables

• For each V A(y) for every y in (Z2)v.
• For each U B(x) for every x in (Z2)u.
• If V is assigned j A(y) is assigned yj
• If U is assigned i B(x) is assigned xi

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Final System - Tests

• Pick V,U at random
• Pick x in (Z2)u and y in (Z2)v.
• Pick ?-noize z in (Z2)u
• Verify B(x)A(y)B(xyz)