From the beginning of 2006 the authors have decided to get back the rights of the book and to make it freely available on the web. The electronic version is identical to the original printed version (apart, of course, from the cover pages), but it also includes the errata corrige collected since the first printing of the book.
The birth of the theory of computational complexity can be set in the early 1960s when the first users of electronic computers started to pay increasing attention to the performances of their programs. As in the theory of computation, where the concept of a model of computation had led to that of an algorithm and of an algorithmically solvable problem, similarly, in the theory of computational complexity, the concept of resource used by a computation led to that of an efficient algorithm and of a computationally feasible problem.
Since these preliminary stages, many more results have been obtained and, as stated by Hartmanis (1989), ‘the systematic study of computational complexity theory has developed into one of the central and most active research areas of computer science. It has grown into a rich and exciting mathematical theory whose development is motivated and guided by computer science needs and technological advances.’
The aim of this introductory book is to review in a systematic way the most significant results obtained in this new research area. The main goals of computational complexity theory are to introduce classes of problems which have similar complexity with respect to a specific computation model and complexity measure, and to study the intrinsic properties of such classes.
In this book, we will follow a balanced approach which is partly algorithmic and partly structuralist. From an algorithmic point of view, we will first present some ‘natural’ problems and then illustrate algorithms which solve them. Since the aim is merely to prove that the problem belongs to a specific class, we will not always give the most efficient algorithm and we will occasionally give preference to an algorithm which is simpler to describe and analyze.
From a structural point of view, we will be concerned with intrinsic properties of complexity classes, including relationships between classes, implications between several hypotheses about complexity classes, and identification of structural properties of sets that affect their computational complexity.
The reader is assumed to have some basic knowledge of theory of computation (as taught in an undergraduate course on Automata Theory, Logic, Formal Languages Theory, or Theory of Computation) and of programming languages and techniques. Some mathematical knowledge is also required.
The first eight chapters of the book can be taught on a senior undergraduate course. The whole book together with an exhaustive discussion of the problems should be suitable for a postgraduate course. Let us now briefly review the contents of the book and the choices made in selecting the material.
The first part (Chapters 1-3) provide the basic tools which will enable us to study topics in complexity theory. Chapter 1 includes a series of definitions and notations related to classic mathematical concepts such as sets, relationships and languages (this chapter can be skipped and referred to when needed). Chapter 2 reviews some important results of computability theory. Chapter 3 provides the basic tool of complexity theory: dynamic complexity measures are introduced, the concept of classes of languages is presented, the strict correspondence between such classes and decision problems is established, and techniques used to study the properties of such classes are formulated.
The second part (Chapters 4-8) studies, in a detailed way, the properties of some of the most significant complexity classes. Those chapters represent the ‘heart’ of complexity theory: by placing suitable restrictions on the power of the computation model, and thus on the amount of resources allowed for the computation, it becomes possible to define a few fundamental complexity classes and to develop a series of tools enabling us to identify, for most computational problems, the complexity class to which they belong.
The third part (Chapters 9-10) deals with probabilistic algorithms and with the corresponding complexity classes. Probabilistic Turing machines are introduced in Chapter 9 and a few probabilistic algorithms for such machines are analyzed. In Chapter 10, a more elaborate computation model denoted as interactive proof system is considered and a new complexity class based on such a model is studied.
The last part (Chapters 11 and 12) is dedicated to the complexity of parallel computations. As a result of advances in hardware technology, computers with thousands of processors are now available; it thus becomes important, not only from a theoretical point of view but also from a practical one, to be able to specify which problems are best suited to be run on parallel machines. Chapter 11 describes in detail a few important and widely differing models of parallel computers and shows how their performance can be considered roughly equivalent. Chapter 12 introduces the concept of a problem solvable by a fast parallel algorithm and the complementary one of a problem with no fast parallel algorithm, and illustrates examples of both types of problems.
While selecting material to be included in the book, we followed a few guidelines. First, we have focused our attention on results obtained in the past two decades, mentioning without proof or leaving as problems some well-known results obtained in the 1960s. Second, whenever a proof of a theorem uses a technique described in a previous proof, we have provided an outline, leaving the complete proof as a problem for the reader. Finally, we have systematically avoided stating without proof specialistic results in other fields in order to make the book as self-contained as possible.
This book originated from a course on Algorithms and Complexity given at the University of Rome ‘La Sapienza’ by D.P. Bovet and P. Crescenzi since 1986. We would like to thank the students who were exposed to the preliminary versions of the chapters and who contributed their observations to improve the quality of the presentation. We also would like to thank R. Silvestri for pointing out many corrections and for suggesting simpler and clearer proofs of some results.