This book starts by providing the relevant background on computability theory, which is the setting in which Complexity theoretic questions are being studied. Most importantly, this preliminary chapter (i.e., Chapter 1) provides a treatment of central notions, such as search and decision problems, algorithms that solve such problems, and their complexity. Special attention is given to the notion of a universal algorithm.

The main part of this book (i.e., Chapters 2–5) focuses on the P-vs-NP Question and on the theory of NP-completeness. Additional topics covered in this part include the general notion of an efficient reduction (with a special emphasis on reductions of search problems to corresponding decision problems), the existence of problems in NP that are neither NP-complete nor in P, the class coNP, optimal search algorithms, and promise problems. A brief overview of this main part follows.

The P-vs-NP Question. Loosely speaking, the P-vs-NP Question refers to search problems for which the correctness of solutions can be efficiently checked (i.e., there is an efficient algorithm that given a solution to a given instance determines whether or not the solution is correct). Such search problems correspond to the class NP, and the P-vs-NP Question corresponds to whether or not all these search problems can be solved efficiently (i.e., is there an efficient algorithm that given an instance finds a correct solution). Thus, the P-vs-NP Question can be phrased as asking whether finding solutions is harder than checking the correctness of solutions.