Ebook: Sparsity: Graphs, Structures, and Algorithms

This is the first book devoted to the systematic study of sparse graphs and sparse finite structures. Although the notion of sparsity appears in various contexts and is a typical example of a hard to define notion, the authors devised an unifying classification of general classes of structures. This approach is very robust and it has many remarkable properties. For example the classification is expressible in many different ways involving most extremal combinatorial invariants.

This study of sparse structures found applications in such diverse areas as algorithmic graph theory, complexity of algorithms, property testing, descriptive complexity and mathematical logic (homomorphism preservation,fixed parameter tractability and constraint satisfaction problems). It should be stressed that despite of its generality this approach leads to linear (and nearly linear) algorithms.

Jaroslav Nešetřil is a professor at Charles University, Prague; Patrice Ossona de Mendez is a CNRS researcher et EHESS, Paris.

This book is related to the material presented by the first author at ICM 2010.

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This is the first book devoted to the systematic study of sparse graphs and sparse finite structures. Although the notion of sparsity appears in various contexts and is a typical example of a hard to define notion, the authors devised an unifying classification of general classes of structures. This approach is very robust and it has many remarkable properties. For example the classification is expressible in many different ways involving most extremal combinatorial invariants.

This study of sparse structures found applications in such diverse areas as algorithmic graph theory, complexity of algorithms, property testing, descriptive complexity and mathematical logic (homomorphism preservation,fixed parameter tractability and constraint satisfaction problems). It should be stressed that despite of its generality this approach leads to linear (and nearly linear) algorithms.

Jaroslav Ne?et?il is a professor at Charles University, Prague; Patrice Ossona de Mendez is a CNRS researcher et EHESS, Paris.

This book is related to the material presented by the first author at ICM 2010.

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This is the first book devoted to the systematic study of sparse graphs and sparse finite structures. Although the notion of sparsity appears in various contexts and is a typical example of a hard to define notion, the authors devised an unifying classification of general classes of structures. This approach is very robust and it has many remarkable properties. For example the classification is expressible in many different ways involving most extremal combinatorial invariants.

This study of sparse structures found applications in such diverse areas as algorithmic graph theory, complexity of algorithms, property testing, descriptive complexity and mathematical logic (homomorphism preservation,fixed parameter tractability and constraint satisfaction problems). It should be stressed that despite of its generality this approach leads to linear (and nearly linear) algorithms.

Jaroslav Ne?et?il is a professor at Charles University, Prague; Patrice Ossona de Mendez is a CNRS researcher et EHESS, Paris.

This book is related to the material presented by the first author at ICM 2010.

Content:
Front Matter....Pages i-xxiii
Front Matter....Pages 1-1
Introduction....Pages 3-5
A Few Problems....Pages 7-17
Front Matter....Pages 19-19
Prolegomena....Pages 21-60
Measuring Sparsity....Pages 61-88
Classes and Their Classification....Pages 89-114
Bounded Height Trees and Tree-Depth....Pages 115-144
Decomposition....Pages 145-174
Independence....Pages 175-194
First-Order Constraint Satisfaction Problems, Limits and Homomorphism Dualities....Pages 195-226
Preservation Theorems....Pages 227-252
Restricted Homomorphism Dualities....Pages 253-275
Counting....Pages 277-297
Back to Classes....Pages 299-309
Front Matter....Pages 311-311
Classes with Bounded Expansion – Examples....Pages 313-338
Some Applications....Pages 339-361
Property Testing, Hyperfiniteness and Separators....Pages 363-379
Core Algorithms....Pages 381-396
Algorithmic Applications....Pages 397-410
Further Directions....Pages 411-416
Solutions and Hints for some of the Exercises....Pages 417-429
Back Matter....Pages 431-457

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This is the first book devoted to the systematic study of sparse graphs and sparse finite structures. Although the notion of sparsity appears in various contexts and is a typical example of a hard to define notion, the authors devised an unifying classification of general classes of structures. This approach is very robust and it has many remarkable properties. For example the classification is expressible in many different ways involving most extremal combinatorial invariants.

This study of sparse structures found applications in such diverse areas as algorithmic graph theory, complexity of algorithms, property testing, descriptive complexity and mathematical logic (homomorphism preservation,fixed parameter tractability and constraint satisfaction problems). It should be stressed that despite of its generality this approach leads to linear (and nearly linear) algorithms.

Jaroslav Ne?et?il is a professor at Charles University, Prague; Patrice Ossona de Mendez is a CNRS researcher et EHESS, Paris.

This book is related to the material presented by the first author at ICM 2010.

Content:
Front Matter....Pages i-xxiii
Front Matter....Pages 1-1
Introduction....Pages 3-5
A Few Problems....Pages 7-17
Front Matter....Pages 19-19
Prolegomena....Pages 21-60
Measuring Sparsity....Pages 61-88
Classes and Their Classification....Pages 89-114
Bounded Height Trees and Tree-Depth....Pages 115-144
Decomposition....Pages 145-174
Independence....Pages 175-194
First-Order Constraint Satisfaction Problems, Limits and Homomorphism Dualities....Pages 195-226
Preservation Theorems....Pages 227-252
Restricted Homomorphism Dualities....Pages 253-275
Counting....Pages 277-297
Back to Classes....Pages 299-309
Front Matter....Pages 311-311
Classes with Bounded Expansion – Examples....Pages 313-338
Some Applications....Pages 339-361
Property Testing, Hyperfiniteness and Separators....Pages 363-379
Core Algorithms....Pages 381-396
Algorithmic Applications....Pages 397-410
Further Directions....Pages 411-416
Solutions and Hints for some of the Exercises....Pages 417-429
Back Matter....Pages 431-457
....