This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and seminars. More than 300 exercises are included with a selected hint set.

Sanjeev Arora is a professor in the department of computer science at Princeton University. He has done foundational work on probabilistically checkable proofs andapproximability of NP-hardproblems. He is the founding director of the Center for Computational Intractability, which is funded by the National Science Foundation.

Boaz Barak is an assistant professor in the department of computer science at Princeton University. He has done foundational work in computational complexity andcryptography, especially in developing “non-blackbox” techniques.