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Reviewer: Burkhard Englert

In their new book, Flum and Grohe provide an excellent introduction to parameterized complexity theory. Parameterized complexity theory is a relatively new branch of complexity theory first developed by Downey and Fellows in several groundbreaking papers in the early 1990s. Classical complexity theory analyzes and classifies problems solely by the amount of a resource required by an algorithm to solve the given problem. Parameterized complexity theory furthermore considers additional numerical values, so-called parameters that depend on the input in an arbitrary way. For example, in a database query, the size of the query may be a natural parameter for a parameterized complexity analysis. Parameterized complexity theory has grown out of the experience that since classical complexity theory does not pay attention to any structural information about the input of a problem, it sometimes makes a problem appear harder than it really is. In other words, if a parameter is relatively small, a problem might still have a tractable solution. Hence, parameterized complexity theory relaxes the classical notion of tractability to fixed parameter tractability. It allows algorithms whose nonpolynomial behavior is restricted by a parameter. Moreover, the study of parameterized problems from different areas led researchers to develop new general algorithmic techniques, such as bounded search trees, kernelization, color coding, and dynamic programming on tree decompositions. The book begins in chapter 1 with an introduction of the central notion of fixed-parameter tractability and an explanation of the method of bounded search trees, the most basic algorithmic technique for designing fixed parameter tractable algorithms. In this first chapter, the outstanding care and effort with which the two authors treat their subject becomes evident. Even though parameterized complexity theory requires a rigorous treatment involving complex notations, the authors never leave it at that. Especially in the early chapters, they continuously step aside and slow down the rigorous treatment to provide the reader with intuitive explanations and helpful examples. Notations are always defined and explained, never assumed, making this book a pleasure to read. All chapters end with some helpful exercises of various difficulty, ranging from simple reinforcement to more challenging problems; they also include some chapter notes providing references to the reader who wants to explore the subject in more depth. In chapters 2 through 8, the theory of fixed parameter tractability is laid out, ranging from the definition of an appropriate notion of reduction to the definition of the appropriate complexity classes. Chapters 9 through 14 are devoted to the development of some advanced algorithmic techniques for designing fixed parameter tractable algorithms. In the final two chapters, 15 and 16, two examples of bounded fixed parameter tractability are studied. The book requires familiarity with classical complexity theory, and the basic notions of logic and discrete mathematics. As such, it is probably best suited for advanced graduate students who have already taken a course in complexity theory, or researchers in computer science or mathematics that would like to learn about this relatively new area. Overall, the book is very well written and designed. Many theoretical textbooks are either awash in unexplained formalism, a self-indulgent style that cares little about the readers, or simply only scratch the surface of the area they explore, leaving more advanced readers unsatisfied. This book strikes a perfect balance. It allows newcomers to learn about its subject in a thorough manner, but is also attractive to more experienced readers. I highly recommend it as an invaluable resource for anybody interested in parameterized complexity theory, and as an excellent example of good theoretical textbook writing. Online Computing Reviews Service