The Basic Class W[1] and an Analog of Cook’s Theorem

Abstract

In the last chapter, we looked at one of the ingredients of classical hardness results: the notion of a reducibility. The other basic ingredient is the identification of classes of intractable problems. In the case of NP-completeness, recall that a language L is a member of the class NP iff there is a polynomial-time relation R and a polynomial p such that x ∈ L iff ∃y[|y| ≤ p(|x|) ∧ R(x, y) holds] For instance, if L is the collection of satisfiable formulas, then a formula x is satisfiable iff there is some satisfying assignment y of the variables. Recall that L is NP-complete iff L ∈ NP, and for any language L¹ ∈ N P, L¹ ≤ ^p _m L. The hidden beauty and power of these definitions comes from the profound discovery that there are literally thousands of important and natural NP-complete problems. If any of them were solvable in polynomial-time, then all of them would be. Moreover, other than a few exceptions, if a natural problem is in NP and is apparently not in P, then the problem seems to always be NP-complete.