Many natural computational problems have input consisting of two or more parts, one of which may be considered a parameter. For example, there are many problems for which the input consists of a graph and a positive integer. A number of results are presented concerning parameterized problems that can be solved in complexity classes below P, given a single word of advice for each parameter value. Different ways in which the word of advice can be employed are considered, and it (...) is shown that the class FPT of tractable parameterized problems has interesting and natural internal structure. (shrink)

A completeness theory for parameterized computational complexity has been studied in a series of recent papers, and has been shown to have many applications in diverse problem domains including familiar graph-theoretic problems, VLSI layout, games, computational biology, cryptography, and computational learning [ADF,BDHW,BFH, DEF,DF1-7,FHW,FK]. We here study the parameterized complexity of two kinds of problems: (1) problems concerning parameterized computations of Turing machines, such as determining whether a nondeterministic machine can reach an accept state in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} (...) \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $k$\end{document} steps (the Short TM Computation Problem), and (2) problems concerning derivations and factorizations, such as determining whether a word \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $x$\end{document} can be derived in a grammar \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $G$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $k$\end{document} steps, or whether a permutation has a factorization of length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $k$\end{document} over a given set of generators. We show hardness and completeness for these problems for various levels of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $W$\end{document} hierarchy. In particular, we show that Short TM Computation is complete for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $W[1]$\end{document}. This gives a new and useful characterization of the most important of the apparently intractable parameterized complexity classes. (shrink)