MAX-3SAT - Theorem 1 (Inapproximability)

Theorem 1 (Inapproximability)

The PCP theorem implies that there exists an ε > 0 such that (1-ε)-approximation of MAX-3SAT is NP-hard.

Proof:

Any NP-complete problem L ∈ PCP(O(log (n)), O(1)) by the PCP theorem. For x ∈ L, a 3-CNF formula Ψ_x is constructed so that

• x ∈ L ⇒ Ψ_x is satisfiable
• x ∉ L ⇒ no more than (1-ε)m clauses of Ψ_x are satisfiable.

The Verifier V reads all required bits at once i.e. makes non-adaptive queries. This is valid because the number of queries remains constant.

• Let q be the number of queries.
• Enumerating all random strings R_i ∈ V, we obtain poly(x) strings since the length of each string r(x) = O(log |x|).
• For each R_i
  • V chooses q positions i₁,...,i_q and a Boolean function f_R: {0,1}q->{0,1} and accepts if and only if f_R(π(i₁,...,i_q)). Here π refers to the proof obtained from the Oracle.

Next we try to find a Boolean formula to simulate this. We introduce Boolean variables x₁,...,x_l, where l is the length of the proof. To demonstrate that the Verifier runs in Probabilistic polynomial-time, we need a correspondence between the number of satisfiable clauses and the probability the Verifier accepts.

• For every R, add clauses representing f_R(x_i1,...,x_iq) using 2q SAT clauses. Clauses of length q are converted to length 3 by adding new (auxiliary) variables e.g. x₂ ∨ x₁₀ ∨ x₁₁ ∨ x₁₂ = ( x₂ ∨ x₁₀ ∨ y_R) ∧ ( ∨ x₁₁ ∨ x₁₂). This requires a maximum of q2q 3-SAT clauses.
• If z ∈ L then
  • there is a proof π such that Vπ (z) accepts for every R_i.
  • All clauses are satisfied if x_i = π(i) and the auxiliary variables are added correctly.
• If input z ∉ L then
  • For every assignment to x₁,...,x_l and y_R's, the corresponding proof π(i) = x_i causes the Verifier to reject for half of all R ∈ {0,1}r(|z|).
    • For each R, one clause representing f_R fails.
    • Therefore a fraction of clauses fails.

It can be concluded that if this holds for every NP-complete problem then the PCP theorem must be true.