P, NP, and NP-Completeness The Basics of Computational Complexity

The focus of this book is the P versus NP Question and the theory of NP-completeness. It also provides adequate preliminaries regarding computational problems and computational models. The P versus NP Question asks whether or not finding solutions is harder than checking the correctness of solutions. An alternative formulation asks whether or not discovering proofs is harder than verifying their correctness. It is widely believed that the answer to these equivalent formulations is positive, and this is captured by saying that P is different from NP. Although the P versus NP Question remains unresolved, the theory of NP-completeness offers evidence for the intractability of specific problems in NP by showing that they are universal for the entire class. Amazingly enough, NP-complete problems exist, and furthermore hundreds of natural computational problems arising in many different areas of mathematics and science are NP-complete.

1. Computational tasks and models; 2. The P versus NP Question; 3. Polynomial-time reductions; 4. NP-completeness; 5. Three relatively advanced topics; Epilogue: a brief overview of complexity theory.

This undergraduate introduction to computational complexity gives a wide perspective on two central issues in theoretical computer science. It starts with the relevant background in computability, including Turing machines, search and decision problems, algorithms, circuits, and complexity classes, and then focuses on the P versus NP Question and the theory of NP-completeness.

Oded Goldreich is a Professor of Computer Science at the Weizmann Institute of Science and an incumbent of the Meyer W. Weisgal Professorial Chair. He is an editor for the SIAM Journal on Computing, the Journal of Cryptology, and Computational Complexity and previously authored the books Modern Cryptography, Probabilistic Proofs and Pseudorandomness, the two-volume work Foundations of Cryptography, and Computational Complexity: A Conceptual Perspective.