The PCP theorem implies that there exists an ε > 0 such that -approximation of MAX-3SAT is NP-hard. Proof: Any NP-complete problem by the PCP theorem. For x ∈ L, a 3-CNF formula Ψx is constructed so that
x ∈ L ⇒ Ψx is satisfiable
x ∉ L ⇒ no more than m clauses of Ψx are satisfiable.
The VerifierV 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 Ri ∈ V, we obtain poly strings since the length of each string.
For each Ri
* V chooses q positions i1,...,iq and a Boolean functionfR: q-> and accepts if and only iffR. 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,...,xl, 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 fR using 2qSAT clauses. Clauses of length q are converted to length 3 by adding new variables e.g. x2 ∨ x10 ∨ x11 ∨ x12 = ∧. This requires a maximum of q2q3-SAT clauses.
If z ∈ L then
* there is a proof π such that Vπ accepts for every Ri.
* All clauses are satisfied if xi = π and the auxiliary variables are added correctly.
If input z ∉ L then
* For every assignment to x1,...,xl and yR's, the corresponding proof π = xi causes the Verifier to reject for half of all R ∈ r.
** For each R, one clause representing fR 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.
Theorem 2
Håstad demonstrates a tighter result than Theorem 1 i.e. the best known value for ε. He constructs a PCP Verifier for 3-SAT that reads only 3 bits from the Proof.
For every ε > 0, there is a PCP-verifier M for 3-SAT that reads a random string r of length and computes query positions ir, jr, kr in the proof π and a bitbr. It accepts if and only if
The Verifier has completeness and soundness 1/2 + ε. The Verifier satisfies If the first of these two equations were equated to "=1" as usual, one could find a proof π by solving a system of linear equations implying P = NP.
If z ∈ L, a fraction ≥ of clauses are satisfied.
If z ∉ L, then for a fraction of R, 1/4 clauses are contradicted.
MAX-3SAT is the restricted special case of MAX-3SAT where every variable occurs in at most B clauses. Before the PCP theorem was proven, Papadimitriou and Yannakakis showed that for some fixed constant B, this problem is MAX SNP-hard. Consequently with the PCP theorem, it is also APX-hard. This is useful because MAX-3SAT can often be used to obtain a PTAS-preserving reduction in a way that MAX-3SAT cannot. Proofs for explicit values of B include: all B ≥ 13, and all B ≥ 3. Moreover, although the decision problem 2SAT is solvable in polynomial time, MAX-2SAT is also APX-hard. The best possible approximation ratio for MAX-3SAT, as a function of B, is at least and at most, unless NP=RP. Some explicit bounds on the approximability constants for certain values of B are known. Berman, Karpinski and Scott proved that for the "critical" instances of MAX-3SAT in which each literal occurs exactly twice, and each clause is exactly of size 3, the problem is approximation hard for some constant factor. MAX-EkSAT is a parameterized version of MAX-3SAT where every clause has exactly literals, for k ≥ 3. It can be efficiently approximated with approximation ratio using ideas from coding theory. It has been proved that random instances of MAX-3SAT can be approximated to within factor.
