Parameterized Complexity

ISBN-13: 9781461267980 / Angielski / Miękka / 2012 / 533 str.

Rodney G. Downey; M. R. Fellows

Opis

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."

Spis treści

1 Computers, Complexity, and Intractability from the Parametric Point of View.- 1.1 Introduction.- 1.2 The Role of Computational Complexity in Modern Science.- 1.3 The Story of Dr.O, Continued.- 1.4 Reworking the Foundations of Computational Complexity.- 1.5 A Deal with the Devil.- 1.6 How Parameters Arise in Practice.- 1.7 A Distinctive Positive Toolkit.- 1.8 O No?.- 1.9 The Barometer of Parametric Intractability.- 1.10 Structural Aspects of Parameterized Complexity.- 1.11 An Overview of Current Research Horizons.- I Parameterized Tractability.- 2 The Basic Definitions.- 2.1 Fixed-Parameter Tractability.- 2.2 The Advice View.- 3 Some Ad Hoc Methods: The Methods of Bounded Search Tree and Problem Kernel.- 3.1 The Method of Bounded Search Trees.- 3.1.1 The Basic Method.- 3.1.2 Heuristic Improvements, Shrinking the Search Tree.- 3.2 The Method of Reduction to a Problem Kernel.- 3.2.1 The Basic Method.- 3.2.2 Hereditary Properties and Leizhen Cai’s Theorem.- 4 Optimization Problems, Approximation Schemes, and Their Relation with FPT.- 4.1 Optimization Problems.- 4.2 How FPT and Optimization Problems Relate.- 4.3 The Classes MAXSNP, MIN F+?1(h), and FPT.- 5 The Advice View Revisited and LOGSPACE.- 6 Methods via Automata and Bounded Treewidth.- 6.1 Classical Automata Theory.- 6.1.1 Deterministic Finite Automata.- 6.1.2 Nondeterministic Finite Automata.- 6.1.3 Regular Languages.- 6.1.4 The Myhill—Nerode Theorem and the Method of Test Sets.- 6.1.5 Classical Tree Automata.- 6.2 Treewidth.- 6.3 Bodlaender’s Theorem.- 6.4 Parse Trees for Graphs of Bounded Treewidth and an Analog of the Myhill—Nerode Theorem.- 6.5 Courcelle’s Theorem.- 6.5.1 The Basic Theorem.- 6.5.2 Implementing Courcelle’s Theorem.- 6.6 Seese’s Theorem.- 6.7 Notes on MS1 Theory.- 7 Well-Quasi-Orderings and the Robertson-Seymour Theorems.- 7.1 Basic WQO Theory.- 7.2 Thomas’ Lemma.- 7.2.1 Thomas’ Lemma Fails for Path Decompositions.- 7.3 The Graph Minor Theorem for Bounded Treewidth.- 7.4 Excluding a Forest.- 7.5 Connections with Automata Theory and Boundaried Graphs.- 7.6 A Sketch of the Proof of the Graph Minor Theorem.- 7.7 Immersions and the Nash-Williams Conjecture.- 7.8 Applications of the Obstruction Principle and WQO’s.- 7.9 Effectivizations of Obstruction-Based Methods.- 7.9.1 Effectivization by Self-Reduction.- 7.9.2 Effectivization by Obstruction Set Computation.- 8 Miscellaneous Techniques.- 8.1 Depth-First Search.- 8.2 Bounded-Width Subgraphs, the Plehn-Voigt Theorem, and Induced Subgraphs.- 8.3 Hashing.- II Parameterized Intractability.- 9 Reductions.- 10 The Basic Class W[1] and an Analog of Cook’s Theorem.- 11 Some Other W[1]-Hardness Results.- 12 The W -Hierarchy.- 13 Beyond W[t]-Hardness.- 14 Fixed Parameter Analogs of PSPACE and k-Move Games.- 15 Provable Intractability: The Class XP.- III Structural and Other Results.- 16 Another Basis for the W -Hierarchy, the Tradeoff-Theorem, and Randomized Reductions.- 17 Relationships with Classical Complexity and Limited Nondeterminism.- 17.1 Classical Complexity.- 17.2 Nondeterminism in P, LOGNP, and the Cai-Chen Model and Other Models.- 18 The Monotone and Antimonotone Collapse Theorems: MONOTONEW[2t + 1] = W[2t] and ANTIMONOTONEW[2t + 2] = W[2t + 1].- 19 The Structure of Languages Under Parameterized Reducibilities.- 19.1 Some Tools.- 19.2 Results.- IV Appendix.- A A Problem Compendium and Guide to W-Hierarchy Completeness, Hardness, and Classification; and Some Research Horizons.- B Research Horizons.- B.2 A Lineup of Tough Customers.- B.3 Connections Between Classical and Parameterized Complexity.- B.4 Classification Gaps.- B.5 Structural Issues and Analogs of Classical Results.- References.