PCP Theorem — Approximation Hardness

Why it matters

PCP underlies many hardness results. Understanding shapes complexity theory.

The architecture

NP = PCP(log n, O(1)).

Verifier reads few bits.

Enables gap-preserving reductions.

How it works end to end

Verifier: O(log n) random bits, reads O(1) bits.

Completeness + soundness.

Enables MAX-3SAT hardness.