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Parameterized Complexity Theory

Parameterized complexity theory is a recent branch of computational complexity theory that provides a framework for a refined analysis of hard algorithmic problems. The central notion of the theory, fixed-parameter tractability, has led to the development of various new algorithmic techniques and a whole new theory of intractability. This book is a state-of-the-art introduction to both algorithmic techniques for fixed-parameter tractability and the structural theory of parameterized complexity classes, and it presents detailed proofs of recent advanced results that have not appeared in book form before. Several chapters are each devoted to intractability, algorithmic techniques for designing fixed-parameter tractable algorithms, and bounded fixed-parameter tractability and subexponential time complexity. The treatment is comprehensive, and the reader is supported with exercises, notes, a detailed index, and some background on complexity theory and logic. The book will be of interest to computer scientists, mathematicians and graduate students engaged with algorithms and problem complexity.

Parameterized complexity theory provides a framework for a refined analysis of hard algorithmic problems.

Classical complexity theory analyzes and classifies problems by the amount of a resource, usually time or space, that is required by algorithms solving them. It was a fundamental idea, going back to the work of Hartmanis and Stearns in the early 1960s, to measure the required amount of the resource as a function of the size of the input. This has led to a manageable variety of complexity classes and a clean-cut theory of intractability. However, measuring complexity only in terms of the input size means ignoring any structural information about the input instances in the resulting complexity theory. Sometimes, this makes problems appear harder than they typically are. Parameterized complexity theory takes a step backwards and measures complexity not only in terms of the input size, but in addition in terms of a parameter, which is a numerical value that may depend on the input in an arbitrary way. The main intention is to address complexity issues in situations where we know that the parameter is comparatively small.



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