Invitation to fixed-parameter algorithms

Classic complexity theory indicates that a large number of natural combinatorial problems are inherently hard to solve algorithmically (in technical terms, they are NP-hard and, thus, it is not expected that they admit a polynomial time algorithm). The parameterization idea tries to cope with this rather discouraging fact by asking whether it is possible to to design algorithms where the non-polynomial part of their running times depends exclusively on some parameter of the problem input. The idea in this approach is that the chosen parameterization reflects a good part of real instances of the problem. Parameterized complexity and algorithms have developed rapidly during the last three decades. Since the fundamental work of Downey and Fellows in [1,3–5], parameterized complexity theory introduced numerous innovative ideas in algorithmic design and offered insightful results in almost all disciplines of theoretical computer science. The first monograph in this field was the book: [R.G. Downey and M.R. Fellows. Parameterized complexity. Monographs in Computer Science. Springer-Verlag, New York, 1999.] The next two monographs in the field appeared during 2006. One was the book: [Rolf Niedermeier. Invitation to fixed-parameter algorithms, volume 31 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2006] and the other was the book: [J. Flum and M. Grohe. Parameterized complexity theory. Texts in Theoretical Computer Science. An EATCS Series. Springer-Verlag, Berlin, 2006]. Also, surveys on the topic appeared in [6–12]. In this review we will focus on the book of Rolf Niedermeier [14].