Notes for Lecture 3 Some Consequences of the Pcp Theorem

Let now φ be an instance of 3SAT with n variables x1, . . . , xn and m clauses. For each variable xi, let occi be the number of occurrences of xi, that is, the number of clauses that involve the literal xi or the literal x̄i. We write xi ∈ Cj if the variable xi occurs in clause Cj . Notice that ∑n i=1 occi = 3m. For each i, construct a 1-expander graph Gi = (Vi, Ei) where Vi has occi vertices, one for each occurrence of xi in φ. We denote the vertices of Vi as pairs [i, j] such that xi occurrs in Cj . Each of these graphs has constant degree d.