Chapter outline

This chapter studies the aspects of NP-completeness that bear most directly on the analysis of algorithms. In Section 34.1, we formalize our notion of "problem" and define the complexity class P of polynomial-time solvable decision problems. We also see how these notions fit into the framework of formal-language theory. Section 34.2 defines the class NP of decision problems whose solutions can be verified in polynomial time. It also formally poses the P ≠ NP question.

Section 34.3 shows how relationships between problems can be studied via polynomial-time "reductions." It defines NP-completeness and sketches a proof that one problem, called "circuit satisfiability," is NP-complete. Having found one NP-complete problem, we show in Section 34.4 how other problems can be proven to be NP-complete much more simply by the methodology of reductions. The methodology is illustrated by showing that two formula-satisfiability problems are NP-complete. A variety of other problems are shown to be NP-complete in Section 34.5.