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This text covers the state of the art in multivariate algorithmics and complexity, a vital field with countless applications in modern computing. It describes the latest methods of proving parameterized tractability, including powerful lower-bound techniques.

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This comprehensive and self-contained textbook presents an accessible overview of the state of the art of multivariate algorithmics and complexity. Increasingly, multivariate algorithmics is having significant practical impact in many application domains, with even more developments on the horizon. The text describes how the multivariate framework allows an extended dialog with a problem, enabling the reader who masters the complexity issues under discussion to use the positive and negative toolkits in their own research. Features: describes many of the standard algorithmic techniques available for establishing parametric tractability; reviews the classical hardness classes; explores the various limitations and relaxations of the methods; showcases the powerful new lower bound techniques; examines various different algorithmic solutions to the same problems, highlighting the insights to be gained from each approach; demonstrates how complexity methods and ideas have evolved over the past 25 years.

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Introduction

Part I: Parameterized Tractability

Preliminaries

The Basic Definitions

Part II: Elementary Positive Techniques

Bounded Search Trees

Kernelization

More on Kernelization

Iterative Compression, and Measure and Conquer, for Minimization Problems

Further Elementary Techniques

Colour Coding, Multilinear Detection, and Randomized Divide and Conquer

Optimization Problems, Approximation Schemes, and Their Relation to FPT

Part III: Techniques Based on Graph Structure

Treewidth and Dynamic Programming

Heuristics for Treewidth

Automata and Bounded Treewidth

Courcelle´s Theorem

More on Width-Metrics: Applications and Local Treewidth

Depth-First Search and the Plehn-Voigt Theorem

Other Width Metrics

Part IV: Exotic Meta-Techniques

Well-Quasi-Orderings and the Robertson-Seymour Theorems

The Graph Minor Theorem

Applications of the Obstruction Principle and WQOs

Part V: Hardness Theory

Reductions

The Basic Class W[1] and an Analog of Cook´s Theorem

Other Hardness Results

The W-Hierarchy

The Monotone and Antimonotone Collapses

Beyond W-Hardness

k-Move Games

Provable Intractability: The Class XP

Another Basis

Part VI: Approximations, Connections, Lower Bounds

The M-Hierarchy, and XP-optimality

Kernelization Lower Bounds

Part VII: Further Topics

Parameterized Approximation

Parameterized Counting and Randomization

Part VIII: Research Horizons

Research Horizons

Part IX Appendices

Appendix 1: Network Flows and Matchings

Appendix 2: Menger´s Theorems

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