Lecture 9: Dinur’s Proof of the PCP Theorem

The theory of NP-completeness, as developed by Cook, Levin, and Karp, states that any language, L in NP is reducible to the Boolean satisfiability problem, 3SAT. By this, we mean that for every instance, x of the language L, we can obtain a satisfiability instance, φ such that x ∈ L if and only if φ is satisfiable. Thus, 3SAT is at least as hard as any other problem in NP. Karp further showed that 3SAT can be reduced to other problems such as CLIQUE and 3-COLORABILITY and hence that these problems are at least as hard as any problem in NP. In other words, solving these problems optimally is as hard as solving any other problem in NP optimally. However the question of the hardness of approximation was left open. For instance, can the following be true – finding a satisfying assignment for 3SAT is NP-hard, however it is easy to find an assignment that satisfies 99% of the clauses. Questions such as Other examples: can we approximate the clique size in a graph? Or, can we obtain a 3-coloring that satisfies 99% of the edge constraints? In other words, is the approximation version of some of NP-hard problems easier than the optimization versions. The PCP Theorem [FGLSS, AS, ALMSS] states that this is not the case – for several of the NP-hard problems, the approximation version is just as hard as the optimization version. The PCP theorem can be viewed as a strengthening of Karp reductions. It provides a reduction from a 3SAT instance φ to another 3SAT instance ψ such that if φ ∈ 3SAT , then ψ ∈ 3SAT and if φ 6∈ 3SAT , then any assignment to ψ violates at least α fraction of the clauses. This provides a hardness of approximation result for MAX-3SAT (i.e., the problem of finding the assignment that satisfies the most number of clauses in a given 3CNF formula).