Parameterized Complexity
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This monograph presents an approach to complexity theory which offers a means of analysing algorithms in terms of their tractability. The authors consider the problem in terms of parameterized languages and taking "k-slices" of the language. In doing so, the reader is introduced to new classes of algorithms which may be analysed more precisely than hereto. The authors have made the book as self-contained as possible and a lot of background material is included. As a result, computer scientists, mathematicians, and graduate students interested in the design and analysis of algorithms will find much of interest in this book.
Contents:
- The Parametric Point of View
- Parameterized Tractability
- The Basic Definitions
- Bounded Search and Problem Kernel
- Optimization Problem, Approximation Schemes and their Relation with FPT
- The Advice View Revisited and LOGSPACE
- Automata and Bounded Treewidth
- WQO and the Robertson-Seymour Theorems
- Miscellaneous Techniques
- Parameterized Intractability
- Reductions
- An Analogue of Cook's Theorem
- Other Hardness Results
- The W-Hierarchy
- Beyond W-Hardness
- k-Move games
- Provable Intractability: the Class XP
- Structural and Other Results
- Another Basis
- Classical Complexity
- The Monotone and Antimonotone Collapses
- Parameterized Reducibilities
- Appendix
- Problem Guide and Compendium
- Research Horizons
- References
- Index
Contents:
- The Parametric Point of View
- Parameterized Tractability
- The Basic Definitions
- Bounded Search and Problem Kernel
- Optimization Problem, Approximation Schemes and their Relation with FPT
- The Advice View Revisited and LOGSPACE
- Automata and Bounded Treewidth
- WQO and the Robertson-Seymour Theorems
- Miscellaneous Techniques
- Parameterized Intractability
- Reductions
- An Analogue of Cook's Theorem
- Other Hardness Results
- The W-Hierarchy
- Beyond W-Hardness
- k-Move games
- Provable Intractability: the Class XP
- Structural and Other Results
- Another Basis
- Classical Complexity
- The Monotone and Antimonotone Collapses
- Parameterized Reducibilities
- Appendix
- Problem Guide and Compendium
- Research Horizons
- References
- Index
The idea for this book was conceived over the second bottle of Villa Maria's Caber net Medot '89, at the dinner of the Australasian Combinatorics Conference held at Palmerston North, New Zealand in December 1990, where the authors first met and discovered they had a number of interests in common. Initially, we embarked on a small project to try to formulate reductions to address the apparent parame terized intractability of DOMINATING SET, and to introduce a structure in which to frame our answers. Having spent several months trying to get the definitions for the reductions right (they now seem so obvious), we turned to our tattered copies of Garey and Johnson's work [239]. We were stunned to find that virtually none of the classical reductions worked in the parameterized setting. We then wondered if we'd be able to find any interesting reductions. Several years, many more bottles, so many papers, and reductions later it [3] seemed that we had unwittingly stumbled upon what we believe is a truly central and new area of complexity theory. It seemed to us that the material would be of great interest to people working in areas where exact algorithms for a small range of parameters are natural and useful (e. g. , Molecular Biology, VLSI design). The tractability theory was rich with distinctive and powerful techniques. The intractability theory seemed to have a deep structure and techniques all of its own.
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