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Reviewer: Alexander T Tzanov

The topic of parameterized algorithms is one of the main subjects in modern computer science. In this type of algorithm analysis, the running time is a function of the size of the input data and of a set of one or more parameters. The theory of parameterized algorithms originated in late 1980 when the first systematic explorations of this class of algorithms began. However, the fast development of the field caused many classic texts (on the subject) to age quickly, as they did not cover most modern algorithm results, tools, and techniques. In this sense Parameterized algorithms aims to discuss some of the most advanced and novel algorithms in the field such as algorithms for important separators, algorithms for linear programming, algorithms based on representative families of matroids, algorithms related to use of the strong exponential time hypothesis (SETH), and algorithms for faster tree decomposition. The covered material is divided into three parts. The first part, "Basic Toolbox" (chapters 1 to 7), is introductory, presenting the main algorithmic techniques. In other words, it is designed to be used as an introductory text on the topic of fixed-parameter tractability. In this part, each chapter discusses one algorithm in fair detail. It is very significant to mention, and is a big plus to the book, that each chapter ends with an abundance of exercises (both hard and easy). These, along with hints and some bibliographic notes, make the book self-contained and suitable for self-study for advanced undergraduates. The second part of the book (chapters 8 to 12) covers advanced algorithmic techniques, which may interest a reader who wants a quick introduction to the most novel algorithms and techniques. As such, the second part may serve advanced users well. Because of its advanced level, each chapter has fewer but more difficult exercises, along with more hints, to provoke independent problem solving. The last part (chapters 13 to 15) is devoted to special topics. Here, the interested reader could find the evidence that certain problems are not fixed parameter tractable, along with techniques and tools used to show the lower bounds for algorithms discussed before. The style of the book is clear, and the material is well positioned to be accessible by graduate students and advanced undergraduate students. The exercises and hints provide a good ground for self-study, while bibliographic notes point to original papers and related work. Overall, this is an excellent book that can be useful to graduate and advanced undergraduate students either as a self-study text or as part of a course. Online Computing Reviews Service

Reviewer: Soubhik Chakraborty

There are 36 definitions of complexity like the 36 flavors of ice cream! In simplest terms, complexity refers to the amount of resources consumed to execute an algorithm (resources here imply running time, space, computational effort, and so on). While classical algorithmic complexity is expressed as a function of the size of the input, parameterized complexity is its refined version, in which the complexity is expressed in terms of multiple parameters of the input, the input size being only one of them. For example, the parameters of the input probability distribution can be the other factors. The first systematic work on parameterized complexity was done by R. G. Downey and M. R. Fellows. [1] The present work serves a twofold objective. First, it introduces the reader to those problems that can be subjected to parameterized complexity analysis. Second, it teaches how to do such analysis, describing some of the latest methods in this area. The book is organized in three parts. Part 1 gives the basic toolbox. Chapter 1 gives a formal introduction where the technical terms are defined and explained. In the subsequent six chapters, the authors proceed to explain kernelization, bounded search trees, iterative compression, randomized methods in parameterized algorithms, and so on. Part 2 is on advanced algorithmic techniques and contains five chapters on advanced kernelization techniques, algebraic techniques like sieves, convolutions and polynomials, improved dynamic programming applied to tree decomposition, matroids, and so on. Part 3, "Lower Bounds," contains topics such as fixed parameter intractability, lower bounds based on the exponential time hypothesis, and lower bounds for kernelization. I enjoyed reading this book, which is a good textbook for graduate and advanced undergraduate students of computer science. Each chapter contains sufficient exercises with hints whenever necessary and helpful bibliographic notes. I found the references quite comprehensive, and the index was quite useful. The authors have, however, missed the important role that factorial experiments (as used by statisticians like me) can play in studying the interactive effects of the input parameters, when these are allowed to simultaneously vary, on the complexity of the underlying algorithm. Nevertheless, this is the best book I have seen on the topic. I strongly recommend it. Online Computing Reviews Service