E3-LIN-2 is hard to approximate Hastad Speaker : Guy Kindler

An m-SAT instance   Variables (not necessarily Boolean).  Constraints: Each over m variables.  Example: For all x,y in (Z 2 ) n, A(x)A(y)=A(xy)

An m-CSP instance   Variables (not necessarily Boolean).  Constraints: Each over m variables.  S(  ) – maximum fraction of satisfied constraints.

Cook-Levin  In 3-SAT, it is hard to distinguish between the cases: –S(  )=1 –S(  )<1

PCP LL LL

PCP Theorem PCP LL LL  In 3-SAT, it is hard to distinguish between: S(  )=1 and S(  )<1- 

PCP Theorem PCP LL LL  GAP(1- ,1) 3-SAT, is NP-hard.

Our Goal  Show hardness for GAP(1/2+ ,1-  ) 3-CSP  Whereeach constraint is a linear equation over Z 2

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

par( ,k)   : variables x, constraints c  V=(x 1,x 2,..,x k ) (k-tuple)  U=(c 1,..,c k )  V  U : if x i  c i  One constraint per V  U

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)

par( ,k) : Forgetful Functor  V-variables over [v]  U-variables over [u]  A constraint over U,V: a function  :[u]-->[v]  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.

Final System - Variables  For each V: A(y) for every y in (Z 2 ) v.  For each U: B(x) for every x in (Z 2 ) u.  If V is assigned j: A(y) is assigned y j  If U is assigned i: B(x) is assigned x i

Final System - Tests  Pick V,U at random  Pick x in (Z 2 ) u and y in (Z 2 ) v.  Pick  -noize z in (Z 2 ) u  Verify: B(x)A(y)=B(xyz)