Book Review:
Book Description
Computational complexity theory has developed rapidly in the past three decades. The list of surprising and fundamental results proved since 1990 alone could fill a book: these include new probabilistic definitions of classical complexity classes (IP = PSPACE and the PCP Theorems) and their implications for the field of approximation algorithms; Shor\'s algorithm to factor integers using a quantum computer; an understanding of why current approaches to the famous P versus NP will not be successful; a theory of derandomization and pseudorandomness based upon computational hardness; and beautiful constructions of pseudorandom objects such as extractors and expanders.
This book aims to describe such recent achievements of complexity theory in the context of the classical results. It is intended to be a text and as well as a reference for selfstudy. This means it must simultaneously cater to many audiences, and it is carefully designed with that goal. The book will explain the context in which a certain notion is useful, and why things are defined in a certain way. Examples and solved exercises accompany key definitions.
The book has three parts and an appendix. Part I covers basic complexity classes; it provides a broad introduction to the field and covers basically the same ground as Papadimitriou\'s text from the early 1990s  but more quickly. Part II covers lower bounds for concrete computational models; it concerns lower bounds on resources required to solve algorithmic tasks on concrete models such as circuits, decision trees, etc. Part III covers advanced topics; this constitutes the latter half of the book and is largely devoted to developments since the late 1980s. It includes average case complexity, derandomization and pseudorandomness, the PCP theorem and hardness of approximation, proof complexity and quantum computing. Finally, the Appendix outlines mathematical ideas that may be useful for following certain chapters, especially in parts II and III.
Intended Audience
This book assumes essentially no computational background (though a slight exposure to computing may help) and very little mathematical background apart from the ability to understand proofs and some elementary probability on finite sample spaces. A typical undergraduate course on "Discrete Math" taught in many math and CS departments should suffice (together with the Appendix).
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