Logic in Algorithmic Graph Structure Theory

Stephan Kreutzer

Technical University Berlin

Bruno’s workshopJune 18-20, 2012, LaBRI, Bordeaux

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Introduction

This talk is motivated by the rôle logic can play in algorithmic graphstructure theory.

We look at standard computational problems on graphs such as:

• Dominating SetFind a min. set of vertices which are neighbours to all others.

• 3-ColourabilityColour the vertices by 3 colours without monochromatic edges.

• Hamiltonian pathFind a path containing every vertex exactly once.

Clearly, all these problems are NP-complete and hence we do not expectthem to be solvable efficiently in general.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 2/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Algorithmic Graph Structure Theory

Restricted classes of input instances.Study these problems on specific, morerestricted classes of inputs where they may become tractable again.

Types of graphs being studied are derived from graph structure theory.

• Planar graphs

• Classes of graphs which are tree-like, e.g. of bounded tree-width

• Classes of “homogeneous” graphs, e.g. of bounded clique-width

• Classes of graphs of bounded degree

• Classes excluding a fixed minor

• Classes locally excluding a fixed minor

• Classes of bounded expansion

Many problems have been shown to be tractable on these types ofgraphs.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 3/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Overview of Graph Classes

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 4/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Overview of Graph Classes

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 4/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Algorithmic Graph Structure TheoryThe general goal of this area is to

1. explore the range and different types of problems that becometractable on any given class or type of graphs and

2. for certain types of problems such as domination problems explorehow far the tractability barrier can be pushed.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 5/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Algorithmic Graph Structure Theory

The general goal of this area is to

1. explore the range and different types of problems that becometractable on any given class or type of graphs and

2. for certain types of problems such as domination problems explorehow far the tractability barrier can be pushed.

Design of algorithms. Much research has gone into developing andimproving algorithms for specific problems on certain classes of graphs.

Meta-theorems.To explore general tractability barriers, results that establishtractability results for a whole range of problems for specific classes ofgraphs are very useful.

All problems satisfying certain criteria are tractableon every class of graphs satisfying a property P.

These results come in different flavours.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 6/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Algorithmic Graph Structure Theory

The general goal of this area is to

1. explore the range and different types of problems that becometractable on any given class or type of graphs and

2. for certain types of problems such as domination problems explorehow far the tractability barrier can be pushed.

Design of algorithms. Much research has gone into developing andimproving algorithms for specific problems on certain classes of graphs.

Meta-theorems.To explore general tractability barriers, results that establishtractability results for a whole range of problems for specific classes ofgraphs are very useful.

All problems satisfying certain criteria are tractableon every class of graphs satisfying a property P.

These results come in different flavours.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 6/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Descriptive Approach to Meta-TheoremsDescriptive approach.

A different approach is to use logic.

Many computational problems on graphs can be described elegantly inlogics such as monadic second-order logic or first-order logic.

Dominating Set. (Parametrized) definable in First-Order Logic (FO)

Is there a set of k vertices such that all others are neighbours of this set?

ϕ(X) := ∀y(

y ∈ X ∨ ∃x ∈ X E(x , y))

∃x1 . . . ∃xk∀y(∨k

i=1 y = xi ∨ E(xi , y))

3-Colourability. definable in Monadic Second-Order Logic (MSO)

Colour the vertices by three colours without monochromatic edges.

∃C1∃C2∃C3

(

∀x∨3

i=1 Ci(x) ∧ ∀x∀y(E(x , y) →∧3

i=1 ¬(Ci(x) ∧ Ci(y))))

Hamiltonian path. definable in Monadic Second-Order Logic (MSO2)

Find a path containing every vertex exactly once.

∃P(Pis a path ∧ ∀y y ∈ V (P))

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 7/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Descriptive Approach to Meta-TheoremsDescriptive approach.

A different approach is to use logic.

Many computational problems on graphs can be described elegantly inlogics such as monadic second-order logic or first-order logic.

Dominating Set. (Parametrized) definable in First-Order Logic (FO)

Is there a set of k vertices such that all others are neighbours of this set?

ϕ(X) := ∀y(

y ∈ X ∨ ∃x ∈ X E(x , y))

∃x1 . . . ∃xk∀y(∨k

i=1 y = xi ∨ E(xi , y))

3-Colourability. definable in Monadic Second-Order Logic (MSO)

Colour the vertices by three colours without monochromatic edges.

∃C1∃C2∃C3

(

∀x∨3

i=1 Ci(x) ∧ ∀x∀y(E(x , y) →∧3

i=1 ¬(Ci(x) ∧ Ci(y))))

Hamiltonian path. definable in Monadic Second-Order Logic (MSO2)

Find a path containing every vertex exactly once.

∃P(Pis a path ∧ ∀y y ∈ V (P))

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 7/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Descriptive Approach to Meta-TheoremsDescriptive approach.

A different approach is to use logic.

Many computational problems on graphs can be described elegantly inlogics such as monadic second-order logic or first-order logic.

Dominating Set. (Parametrized) definable in First-Order Logic (FO)

Is there a set of k vertices such that all others are neighbours of this set?

ϕ(X) := ∀y(

y ∈ X ∨ ∃x ∈ X E(x , y))

∃x1 . . . ∃xk∀y(∨k

i=1 y = xi ∨ E(xi , y))

3-Colourability. definable in Monadic Second-Order Logic (MSO)

Colour the vertices by three colours without monochromatic edges.

∃C1∃C2∃C3

(

∀x∨3

i=1 Ci(x) ∧ ∀x∀y(E(x , y) →∧3

i=1 ¬(Ci(x) ∧ Ci(y))))

Hamiltonian path. definable in Monadic Second-Order Logic (MSO2)

Find a path containing every vertex exactly once.

∃P(Pis a path ∧ ∀y y ∈ V (P))

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 7/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Descriptive Approach to Meta-TheoremsDescriptive approach.

A different approach is to use logic.

Many computational problems on graphs can be described elegantly inlogics such as monadic second-order logic or first-order logic.

Dominating Set. (Parametrized) definable in First-Order Logic (FO)

Is there a set of k vertices such that all others are neighbours of this set?

ϕ(X) := ∀y(

y ∈ X ∨ ∃x ∈ X E(x , y))

∃x1 . . . ∃xk∀y(∨k

i=1 y = xi ∨ E(xi , y))

3-Colourability. definable in Monadic Second-Order Logic (MSO)

Colour the vertices by three colours without monochromatic edges.

∃C1∃C2∃C3

(

∀x∨3

i=1 Ci(x) ∧ ∀x∀y(E(x , y) →∧3

i=1 ¬(Ci(x) ∧ Ci(y))))

Hamiltonian path. definable in Monadic Second-Order Logic (MSO2)

Find a path containing every vertex exactly once.

∃P(Pis a path ∧ ∀y y ∈ V (P))

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 7/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Algorithmic Meta-Theorems

Use logics to specify algorithmic problems and show that all problemsdefinable in a logic can be solved efficiently on certain graph classes.

First Algorithmic Meta-Theorem. (Courcelle ’90)Every graph property definable in monadic second-order logic (MSO2)can be decided in linear time on any class of structures of boundedtree-width.

Courcelle’s theorem has found numerous applications:

• As starting point of a whole theory of algorithmic meta-theorems.

• Used in various algorithms to solve the bounded tree-width casequickly. In some it is used as an integral part.

• Essential part of techniques such as meta-kernelization.

• There are surprisingly efficient implementations available(Rossmanith’s group)

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 8/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Algorithmic Meta-Theorems

Use logics to specify algorithmic problems and show that all problemsdefinable in a logic can be solved efficiently on certain graph classes.

First Algorithmic Meta-Theorem. (Courcelle ’90)Every graph property definable in monadic second-order logic (MSO2)can be decided in linear time on any class of structures of boundedtree-width.

Courcelle’s theorem has found numerous applications:

• As starting point of a whole theory of algorithmic meta-theorems.

• Used in various algorithms to solve the bounded tree-width casequickly. In some it is used as an integral part.

• Essential part of techniques such as meta-kernelization.

• There are surprisingly efficient implementations available(Rossmanith’s group)

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 8/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Algorithmic Meta-Theorems

Use logics to specify algorithmic problems and show that all problemsdefinable in a logic can be solved efficiently on certain graph classes.

First Algorithmic Meta-Theorem. (Courcelle ’90)Every graph property definable in monadic second-order logic (MSO2)can be decided in linear time on any class of structures of boundedtree-width.

Courcelle’s theorem has found numerous applications:

• As starting point of a whole theory of algorithmic meta-theorems.

• Used in various algorithms to solve the bounded tree-width casequickly. In some it is used as an integral part.

• Essential part of techniques such as meta-kernelization.

• There are surprisingly efficient implementations available(Rossmanith’s group)

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 8/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Algorithmic Meta-Theorems

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 9/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Algorithmic Meta-Theorems

We are interested in results of the following form.

Every problem definable in a given logic L is tractable on any class ofgraphs satisfying a certain property.

Results of this form are usually referred to as algorithmic meta-theorems.

Algorithmic meta-theorems.

1. Provide a uniform explanation why natural classes of problems aretractable on a class of graphs.

2. Establish general algorithmic techniques for solving them.

3. Corresponding intractability results for logics exhibit naturalboundaries beyond which these techniques fail.

In this talk.Present some of the result obtained in this area focussing on the logicaltools available.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 10/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Algorithmic Meta-Theorems

We are interested in results of the following form.

Every problem definable in a given logic L is tractable on any class ofgraphs satisfying a certain property.

Results of this form are usually referred to as algorithmic meta-theorems.

Algorithmic meta-theorems.

1. Provide a uniform explanation why natural classes of problems aretractable on a class of graphs.

2. Establish general algorithmic techniques for solving them.

3. Corresponding intractability results for logics exhibit naturalboundaries beyond which these techniques fail.

In this talk.Present some of the result obtained in this area focussing on the logicaltools available.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 10/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Rephrasing everything in terms of logic

Let L be a logic and C be a class of structures.

The Model-Checking ProblemMC(L, C):

Given : Finite structure A := (A, σ) ∈ C, formula ϕ ∈ L

Problem: Decide A |= ϕ?

We write MC(L) if C is the class of all structures over some signature.

Note. We only consider model-checking for formulas without free variables.

With this terminology, we rephrase algorithmic meta-theorems as follows.

Algorithmic Meta-Theorems.For a logic L, we are interested in understanding on which classes C ofgraphs the problem MC(L, C) is tractable.

That is, we will study the model-checking complexity for standard logics,in particular first-order logic and variants of monadic second-order logic.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 11/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Rephrasing everything in terms of logic

Let L be a logic and C be a class of structures.

The Model-Checking ProblemMC(L, C):

Given : Finite structure A := (A, σ) ∈ C, formula ϕ ∈ L

Problem: Decide A |= ϕ?

We write MC(L) if C is the class of all structures over some signature.

Note. We only consider model-checking for formulas without free variables.

With this terminology, we rephrase algorithmic meta-theorems as follows.

Algorithmic Meta-Theorems.For a logic L, we are interested in understanding on which classes C ofgraphs the problem MC(L, C) is tractable.

That is, we will study the model-checking complexity for standard logics,in particular first-order logic and variants of monadic second-order logic.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 11/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Parametrized Complexity

The complexity theoretical framework we use is the framework ofparameterized complexity introduced by Downey and Fellows.

Fixed-Parameter tractability. A model-checking problem is fixed-parametertractable (fpt) if it can be solved in time

f (|ϕ|) · |A|c , (or e.g. f (|ϕ|+ tw(A)) · |A|c)

where c is a constant and f is a computable function.

Similarly, problems such as Dominating Set are fixed-parameter tractableon a class C of graphs if, on input G ∈ C and k , it can be decided in timef (k) · |G|c whether G contains a dominating set of size k .

FPT is the class of all fixed-parameter tractable problems.

Comparable to PTIME in classical complexity.

The rôle of NP is played by a hierarchy of classes W[1], W[2], ...

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 12/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Parametrized Complexity

The complexity theoretical framework we use is the framework ofparameterized complexity introduced by Downey and Fellows.

Fixed-Parameter tractability. A model-checking problem is fixed-parametertractable (fpt) if it can be solved in time

f (|ϕ|) · |A|c , (or e.g. f (|ϕ|+ tw(A)) · |A|c)

where c is a constant and f is a computable function.

Similarly, problems such as Dominating Set are fixed-parameter tractableon a class C of graphs if, on input G ∈ C and k , it can be decided in timef (k) · |G|c whether G contains a dominating set of size k .

FPT is the class of all fixed-parameter tractable problems.

Comparable to PTIME in classical complexity.

The rôle of NP is played by a hierarchy of classes W[1], W[2], ...

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 12/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Parametrized Complexity

The complexity theoretical framework we use is the framework ofparameterized complexity introduced by Downey and Fellows.

Fixed-Parameter tractability. A model-checking problem is fixed-parametertractable (fpt) if it can be solved in time

f (|ϕ|) · |A|c , (or e.g. f (|ϕ|+ tw(A)) · |A|c)

where c is a constant and f is a computable function.

Similarly, problems such as Dominating Set are fixed-parameter tractableon a class C of graphs if, on input G ∈ C and k , it can be decided in timef (k) · |G|c whether G contains a dominating set of size k .

FPT is the class of all fixed-parameter tractable problems.

Comparable to PTIME in classical complexity.

The rôle of NP is played by a hierarchy of classes W[1], W[2], ...

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 12/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Structural Characterisation of Model-Checking Problems

In the terminology of parametrized complexity:

MSO-model-checking is fpt on any class of graphs of bounded tree-width.

Rephrasing algorithmic meta-theorems.What are the largest/most generalclasses of graphs on which MSO becomes tractable?

And the same question applies to first-order logic.

Research programme.For each of the natural logics L such as FO or MSO,identify a structural property P of classes C of graphs such that MC(L, C)is tractable if, and only if, C has the property P

We may not always get an exact characterisation, there may be gaps.

But such a characterisation would give an easy tool to assess whetherMSO-model-checking is tractable on some class.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 13/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Structural Characterisation of Model-Checking Problems

In the terminology of parametrized complexity:

MSO-model-checking is fpt on any class of graphs of bounded tree-width.

Rephrasing algorithmic meta-theorems.What are the largest/most generalclasses of graphs on which MSO becomes tractable?

And the same question applies to first-order logic.

Research programme.For each of the natural logics L such as FO or MSO,identify a structural property P of classes C of graphs such that MC(L, C)is tractable if, and only if, C has the property P

We may not always get an exact characterisation, there may be gaps.

But such a characterisation would give an easy tool to assess whetherMSO-model-checking is tractable on some class.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 13/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Structural Characterisation of Model-Checking Problems

In the terminology of parametrized complexity:

MSO-model-checking is fpt on any class of graphs of bounded tree-width.

Rephrasing algorithmic meta-theorems.What are the largest/most generalclasses of graphs on which MSO becomes tractable?

And the same question applies to first-order logic.

Research programme.For each of the natural logics L such as FO or MSO,identify a structural property P of classes C of graphs such that MC(L, C)is tractable if, and only if, C has the property P under suitable complexitytheoretical assumptions.

We may not always get an exact characterisation, there may be gaps.

But such a characterisation would give an easy tool to assess whetherMSO-model-checking is tractable on some class.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 13/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Structural Characterisation of Model-Checking Problems

In the terminology of parametrized complexity:

MSO-model-checking is fpt on any class of graphs of bounded tree-width.

Rephrasing algorithmic meta-theorems.What are the largest/most generalclasses of graphs on which MSO becomes tractable?

And the same question applies to first-order logic.

Research programme.For each of the natural logics L such as FO or MSO,identify a structural property P of classes C of graphs such that MC(L, C)is tractable if, and only if, C has the property P under suitable complexitytheoretical assumptions.

We may not always get an exact characterisation, there may be gaps.

But such a characterisation would give an easy tool to assess whetherMSO-model-checking is tractable on some class.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 13/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Structural Characterisation of Model-Checking Problems

To achieve such a characterisation we need

• upper bounds: tractability of model-checking on specific classes ofgraphs.

Such results are known as algorithmic meta-theorems

• lower bounds: results establishing intractability of model-checkingproblems if certain structural parameters are not given.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 14/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Structural Characterisation of Model-Checking Problems

To achieve such a characterisation we need

• upper bounds: tractability of model-checking on specific classes ofgraphs.

Such results are known as algorithmic meta-theorems

Part I of this talk

• lower bounds: results establishing intractability of model-checkingproblems if certain structural parameters are not given.

Part II of this talk

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 14/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Upper Bounds on the Complexity of Model-Checking Problems

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Overview of Algorithmic Meta-Theorems

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

A High Level DescriptionEssentially, all algorithmic meta-theorems known to date are based onthe following simple idea.

Given a graph G and a formula ϕ, we recursively decompose the graphinto smaller, or simpler, subgraphs until we reach graphs of constant size.

The simplest form is a complete decomposition bounded tree-width.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 17/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

A High Level DescriptionEssentially, all algorithmic meta-theorems known to date are based onthe following simple idea.

Given a graph G and a formula ϕ, we recursively decompose the graphinto smaller, or simpler, subgraphs until we reach graphs of constant size.

The simplest form is a complete decomposition bounded tree-width.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 17/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

A High Level DescriptionEssentially, all algorithmic meta-theorems known to date are based onthe following simple idea.

Given a graph G and a formula ϕ, we recursively decompose the graphinto smaller, or simpler, subgraphs until we reach graphs of constant size.

The simplest form is a complete decomposition bounded tree-width.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 17/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

A High Level DescriptionGiven a graph G and a formula ϕ, we recursively decompose the graphinto smaller, or simpler, subgraphs until we reach graphs of constant size.

We can also cover the graph by sub-graphs in different forms.

These subgraphs should have a simpler structure than the original graph.

This leads to planar graphs, local tree-width and bounded expansion.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 18/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

A High Level DescriptionGiven a graph G and a formula ϕ, we recursively decompose the graphinto smaller, or simpler, subgraphs until we reach graphs of constant size.

We can also cover the graph by sub-graphs in different forms.

These subgraphs should have a simpler structure than the original graph.

This leads to planar graphs, local tree-width and bounded expansion.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 18/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

A High Level Description

For this idea to work we need concepts of graph decompositions and foreach of these a corresponding concept of composition of formulas.

decomposition of graphs composition of formulas

proper separation (bd. tree-width) Feferman-Vaught theorems

Automata + Transductions

neighbourhood cover (planar graphs) locality theorems

general cover (bd. expansion) quantifier elimination

Logical methods used in algorithmic meta-theorems.

1. Automata theoretic method

2. Transductions

3. Composition theorems (Feferman-Vaught style)

4. Locality arguments

5. Quantifier elimination procedures

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 19/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

A High Level Description

For this idea to work we need concepts of graph decompositions and foreach of these a corresponding concept of composition of formulas.

decomposition of graphs composition of formulas

proper separation (bd. tree-width) Feferman-Vaught theorems

Automata + Transductions

neighbourhood cover (planar graphs) locality theorems

general cover (bd. expansion) quantifier elimination

Logical methods used in algorithmic meta-theorems.

1. Automata theoretic method

2. Transductions

3. Composition theorems (Feferman-Vaught style)

4. Locality arguments

5. Quantifier elimination procedures

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

The Automata-Theoretic Method

Let C be a class of graphs and L be a logic.

Idea to solveMC(L, C).

1. Define a suitable automata model for C. Deciding whether anautomaton accepts G ∈ C should be in polynomial time.

2. Show that there is an effective translation of MSO-formulas on C intoautomata Aϕ.

3. Given G ∈ C and ϕ ∈ L, construct Aϕ and decide whether Aϕ

accepts G.

Example. MC(MSO, TREE).

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 20/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

The Transduction Method

Let C be a class of graphs and L be a logic.

Idea to solveMC(L, C).

1. Let D be a class of graphs on which MC(L,D) is tractable.2. Show that there is

• an effective translation of L-formulas on C to L-formulas ϕ∗ on D

• a polynomial-time translation of G ∈ C to G∗

∈ D

such that G |= ϕ iff G∗ |= ϕ∗.

3. Given G ∈ C and ϕ ∈ L, construct ϕ∗ and G∗ and decide G∗ |= ϕ∗.

Example. MC(MSO, C) is fpt for all classes C of bounded tree-width.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

First Proof of Courcelle’s TheoremTheorem: (Courcelle 1990)

For any class C of bounded tree-width

MC(MSO2, C)Input: Graph G ∈ C, ϕ ∈ MSO2

Parameter: |ϕ|Problem: Decide G |= ϕ

is fixed-parameter tractable (linear time for each fixed ϕ).

There is an effective transduction from classes of bounded tree-width intothe class of trees.

What about the parameter dependence?

Theorem. (Frick, Grohe, 01)

1. Unless P=NP, there is no fpt-algorithm for MSO model checking ontrees with elementary parameter dependence.

2. Unless FPT=W[1], there is no fpt-algorithm for FO model checkingon trees with elementary parameter dependence.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

First Proof of Courcelle’s TheoremTheorem: (Courcelle 1990)

For any class C of bounded tree-width

MC(MSO2, C)Input: Graph G ∈ C, ϕ ∈ MSO2

Parameter: |ϕ|Problem: Decide G |= ϕ

is fixed-parameter tractable (linear time for each fixed ϕ).

There is an effective transduction from classes of bounded tree-width intothe class of trees.

What about the parameter dependence?

Theorem. (Frick, Grohe, 01)

1. Unless P=NP, there is no fpt-algorithm for MSO model checking ontrees with elementary parameter dependence.

2. Unless FPT=W[1], there is no fpt-algorithm for FO model checkingon trees with elementary parameter dependence.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Extensions of Courcelle’s TheoremTheorem. (Courcelle, Makowsky, Rotics 2001)

MC(MSO1, C) is fixed-parameter tractable for any class C of boundedclique-width.

Analogous results can be obtained for various extensions of MSO:

• Counting MSO: (see The Book)MSO + Quantifiers |X | = p mod q

• Order-Invariant MSO (MSO<-inv): (Engelmann, K., Siebertz ’12)

• Formulas ϕ can use an order relation < but truth must be independenton the specific order used.

• This allows to express even cardinality of a set of elements.• Bruno yesterday had more examples of problems that can be defined

order-invariant but not known to be definable without order.

MC(MSO<-inv2 , C) is fpt on any class of graphs of bounded tree-width.

MC(MSO<-inv1 , C) is fpt on any class of graphs of bounded

clique-width.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Extensions of Courcelle’s TheoremTheorem. (Courcelle, Makowsky, Rotics 2001)

MC(MSO1, C) is fixed-parameter tractable for any class C of boundedclique-width.

Analogous results can be obtained for various extensions of MSO:

• Counting MSO: (see The Book)MSO + Quantifiers |X | = p mod q

• Order-Invariant MSO (MSO<-inv): (Engelmann, K., Siebertz ’12)

• Formulas ϕ can use an order relation < but truth must be independenton the specific order used.

• This allows to express even cardinality of a set of elements.• Bruno yesterday had more examples of problems that can be defined

order-invariant but not known to be definable without order.

MC(MSO<-inv2 , C) is fpt on any class of graphs of bounded tree-width.

MC(MSO<-inv1 , C) is fpt on any class of graphs of bounded

clique-width.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

The Composition Method

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 24/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Feferman-Vaught Style Theorems

Notation:

G: graph v : tuple of vertices

tpMSO(G, v): full MSO-type of v in G (all MSO-formulae true at v )

tpMSOq (G, v): class of MSO-formulae of quantifier-rank ≤ q true at v

analogously for tpFO and tpFOq

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Feferman-Vaught Style Theorems

Theorem. (see Makowsky 04)Let G,H be graphs

v ∈ V (G) w ∈ V (H)

u ∈ V (G) such that u = V (G) ∩ V (H)

For all q ≥ 0,

tpq(G ∪ H,uvw) is determined by tpq(G,uv) and tpq(uw)

Furthermore, there is an algorithm that computes tpq(G ∪ H,uvw) fromtpq(G,uv) and tpq(uw).

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Feferman-Vaught Style TheoremsTheorem.Let G,H be graphs, v ∈ V (G) w ∈ V (H)

u ∈ V (G) such that u = V (G) ∩ V (H)

For all q ≥ 0, tpq(G ∪ H,uvw) is determined by tpq(G,uv) and tpq(uw)

Furthermore, there is an algorithm that computes tpq(G ∪ H,uvw) fromtpq(G,uv) and tpq(uw).

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Feferman-Vaught Style TheoremsTheorem.Let G,H be graphs, v ∈ V (G) w ∈ V (H)

u ∈ V (G) such that u = V (G) ∩ V (H)

For all q ≥ 0, tpq(G ∪ H,uvw) is determined by tpq(G,uv) and tpq(uw)

Furthermore, there is an algorithm that computes tpq(G ∪ H,uvw) fromtpq(G,uv) and tpq(uw).

This suggests a model-checking algorithm on graphs which ca nrecursively be decomposed into sub-graphs of constant size .

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Feferman-Vaught Style TheoremsTheorem.Let G,H be graphs, v ∈ V (G) w ∈ V (H)

u ∈ V (G) such that u = V (G) ∩ V (H)

For all q ≥ 0, tpq(G ∪ H,uvw) is determined by tpq(G,uv) and tpq(uw)

Furthermore, there is an algorithm that computes tpq(G ∪ H,uvw) fromtpq(G,uv) and tpq(uw).

This suggests a model-checking algorithm on graphs which ca nrecursively be decomposed into sub-graphs of constant size .

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Feferman-Vaught Style TheoremsTheorem.Let G,H be graphs, v ∈ V (G) w ∈ V (H)

u ∈ V (G) such that u = V (G) ∩ V (H)

For all q ≥ 0, tpq(G ∪ H,uvw) is determined by tpq(G,uv) and tpq(uw)

Furthermore, there is an algorithm that computes tpq(G ∪ H,uvw) fromtpq(G,uv) and tpq(uw).

This suggests a model-checking algorithm on graphs which ca nrecursively be decomposed into sub-graphs of constant size .

graphs of bounded tree-width

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

An Overview of Graph Parameters

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

The Locality Method for First-Order Logic

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Locality of First-Order Logic

Notation: Let G be a graph e.g. the Gaifman graph of a structure

distG(u, v) : length of the shortest path between u and v

NGr (v) := {u ∈ V (G) : distG(u, v) ≤ r}

NGr (v): r -neighbourhood of v in G.

Definition:A formula ϕ(x) ∈ FO is r -local if for all graphs G and all v ∈ V (G)

G |= ϕ(v) ⇐⇒ G[Nr (v)

]|= ϕ(v).

Hence, truth at v only depends on the vertices around v .

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Gaifman’s Theorem

Theorem. (Gaifman, 1982)Every first-order sentence ϕ ∈ FO is equivalent to a Boolean combinationof basic local sentences.

Basic local sentence:

ϕ := ∃x1 . . . ∃xm

∧

i 6=j

dist(xi , xj)> 2r ∧k∧

i=1

ψ(xi).

where ψ is r -local.

Remark.Gaifman’s proof is constructive.

Theorem. (Dawar, Grohe, K., Schweikardt, 07)

For each k ≥ 1 there is ϕk ∈ FO[{E}] of length O(k4) such that everyequivalent sentence in Gaifman-NF has length at least tower(k).

(similar lower bounds for Feferman-Vaught and preservation thms)

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Gaifman’s Theorem

Theorem. (Gaifman, 1982)Every first-order sentence ϕ ∈ FO is equivalent to a Boolean combinationof basic local sentences.

Basic local sentence:

ϕ := ∃x1 . . . ∃xm

∧

i 6=j

dist(xi , xj)> 2r ∧k∧

i=1

ψ(xi).

where ψ is r -local.

Remark.Gaifman’s proof is constructive.

Theorem. (Dawar, Grohe, K., Schweikardt, 07)

For each k ≥ 1 there is ϕk ∈ FO[{E}] of length O(k4) such that everyequivalent sentence in Gaifman-NF has length at least tower(k).

(similar lower bounds for Feferman-Vaught and preservation thms)

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

The Locality MethodTheorem. (Follows from Frick, Grohe 01)

Fix d ≥ 0. For every r ≥ 0 let Dr be a class of graphs such thatMC(FO,Dr ) can be solved in time f (|ϕ|+ r) · |G|d .

Let C be a class of graphs such that for all G ∈ C, v ∈ V (G) and r ≥ 0

G[NGr (v)] ∈ Dr .

Then MC(FO, C) is fixed-parameter tractable.

Examples.• Graph classes of maximum degree k .

Take Dr to be graphs of size at most d r .

• Graph classes of bounded local tree-widthTake Dr to be graphs of tree-width at most g(r).

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

First-Order Logic on Bounded Degree Graphs

Theorem. (Follows from Frick, Grohe 01)

Fix d ≥ 0. For every r ≥ 0 let Dr be a class of graphs such thatMC(FO,Dr ) can be solved in time f (|ϕ|+ r) · |G|d .

Let C be a class of graphs such that for all G ∈ C, v ∈ V (G) and r ≥ 0

G[NGr (v)] ∈ Dr .

Then MC(FO, C) is fixed-parameter tractable.

Proof. By Gaifman’s theorem it suffices to consider formulae of the form

∃x1 . . . ∃xm

∧

1≤i<j≤m

dist(xi , xj) > 2r ∧k∧

i=1

ψ(xi)

for some r -local formula ψ(x).

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

First-Order Logic on Bounded Degree Graphs

Theorem. (Follows from Frick, Grohe 01)

Fix d ≥ 0. For every r ≥ 0 let Dr be a class of graphs such thatMC(FO,Dr ) can be solved in time f (|ϕ|+ r) · |G|d .

Let C be a class of graphs such that for all G ∈ C, v ∈ V (G) and r ≥ 0

G[NGr (v)] ∈ Dr .

Then MC(FO, C) is fixed-parameter tractable.

Proof. By Gaifman’s theorem it suffices to consider formulae of the form

∃x1 . . . ∃xm

∧

1≤i<j≤m

dist(xi , xj) > 2r ∧k∧

i=1

ψ(xi)

for some r -local formula ψ(x).

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Proof of Theorem

Suppose

ϕ := ∃x1 . . . ∃xm

∧

1≤i<j≤m

dist(xi , xj) > 2r ∧m∧

i=1

ψ(xi)

for some r -local formula ψ(x).

Let G ∈ C.

Find m vertices of distance > 2r whose r -neighbourhoods satisfy ψ.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Proof of Theorem

Suppose

ϕ := ∃x1 . . . ∃xm

∧

1≤i<j≤m

dist(xi , xj) > 2r ∧m∧

i=1

ψ(xi)

for some r -local formula ψ(x).

Let G ∈ C.

Find m vertices of distance > 2r whose r -neighbourhoods satisfy ψ.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Localisation of Graph Invariants

Theorem. First-order model checking is fixed-parameter tractable on

• graph classes of bounded degree (Seese 96)

• planar graphs (Frick, Grohe 01)

• graph classes of locally bounded tree-width (Frick, Grohe 01)

Theorem. (Flum, Grohe 01)First-order model-checking is fixed-parameter tractable on graph classesexcluding a minor.

Theorem. (Dawar, Grohe, K. 07)First-order model-checking is fixed-parameter tractable on graph classeslocally excluding a minor.

Theorem: (Dvorák, Kral, Thomas 10, Dawar, K. 10; Grohe, K. 11)

First-order model-checking is fixed-parameter tractable on graph classesof locally bounded expansion.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Logical Tools in Algorithmic Meta-Theorems

Logical methods used in algorithmic meta-theorems.

1. Automata theoretic method

2. Transductions

3. Composition theorems (Feferman-Vaught style)

4. Locality arguments

5. Quantifier elimination procedures

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

First-Order Logic and Excluded Minors

Theorem: (Flum, Grohe 2001)

Let C be a class of graphs excluding a minor.

The problem

MC(FO, C)Input: Graph G ∈ C, ϕ ∈ MSO

Parameter: |ϕ|Problem: G |= ϕ?

is fixed-parameter tractable.

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FO Model Checking: Decomposing a Graph

Given: C class of graphs excluding a minor HInput: Graph G such that H 6� G and ϕ ∈ FOProblem: G |= ϕ

G excludes H Decomp. theorem, Robertson, SeymourLogic: Composition Theorems

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

FO Model Checking: Decomposing a Graph

Given: C class of graphs excluding a minor HInput: Graph G such that H 6� G and ϕ ∈ FOProblem: G |= ϕ

G excludes H Decomp. theorem, Robertson, SeymourLogic: Composition Theorems

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

FO Model Checking: Decomposing a Graph

In a block: Local tree-width (almost) bounded by a function λ

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

FO Model Checking: Decomposing a Graph

In a block: Local tree-width (almost) bounded by a function λ

Logic: Transduction to get rid of the extra vertices

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

FO Model Checking: Decomposing a Graph

In a block: Local tree-width (almost) bounded by a function λ

Logic: Transduction to get rid of the extra verticesLogic: Locality Method

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

FO Model Checking: Decomposing a Graph

In a block: Local tree-width (almost) bounded by a function λ

Logic: Transduction to get rid of the extra verticesLogic: Locality MethodFollowed by Transduction into Trees

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

FO Model Checking: Decomposing a Graph

In a block: Local tree-width (almost) bounded by a function λ

Logic: Transduction to get rid of the extra verticesLogic: Locality MethodFollowed by Transduction into TreesFollowed by Automata Method

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

First-Order Logic and Excluded Minors

Theorem. (Flum, Grohe 01)First-order model-checking is fixed-parameter tractable on graph classesexcluding a minor.

Theorem: (Dawar, Grohe, K. 2007)

Let C be a class of graphs excluding a minor H.

The problem

MC(FO, C)Input: Graph G ∈ C, ϕ ∈ MSO

Parameter: |ϕ|+ |H|Problem: G |= ϕ?

is fixed-parameter tractable.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

First-Order Logic and Excluded Minors

Theorem. (Flum, Grohe 01)First-order model-checking is fixed-parameter tractable on graph classesexcluding a minor.

Theorem: (Dawar, Grohe, K. 2007)

Let C be a class of graphs excluding a minor H.

The problem

MC(FO, C)Input: Graph G ∈ C, ϕ ∈ MSO

Parameter: |ϕ|+ |H|Problem: G |= ϕ?

is fixed-parameter tractable.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Localisation of Graph Invariants

Theorem. First-order model checking is fixed-parameter tractable on

• graph classes of bounded degree (Seese 96)

• planar graphs (Frick, Grohe 01)

• graph classes of locally bounded tree-width (Frick, Grohe 01)

Theorem. (Flum, Grohe 01)First-order model-checking is fixed-parameter tractable on graph classesexcluding a minor.

Theorem. (Dawar, Grohe, K. 07)First-order model-checking is fixed-parameter tractable on graph classeslocally excluding a minor.

Theorem: (Dvorák, Kral, Thomas 10, Dawar, K. 10; Grohe, K. 11)

First-order model-checking is fixed-parameter tractable on graph classesof locally bounded expansion.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Localisation of Graph Invariants

Theorem. First-order model checking is fixed-parameter tractable on

• graph classes of bounded degree (Seese 96)

• planar graphs (Frick, Grohe 01)

• graph classes of locally bounded tree-width (Frick, Grohe 01)

Theorem. (Flum, Grohe 01)First-order model-checking is fixed-parameter tractable on graph classesexcluding a minor.

Theorem. (Dawar, Grohe, K. 07)First-order model-checking is fixed-parameter tractable on graph classeslocally excluding a minor.

Theorem: (Dvorák, Kral, Thomas 10, Dawar, K. 10; Grohe, K. 11)

First-order model-checking is fixed-parameter tractable on graph classesof locally bounded expansion.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

The Quantifier-Elimination Method

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

The Quantifier-Elimination Method

We will show the following result, proved using quantifier-elimination.

Theorem. (Dvorák, Kral, Thomas 10; Dawar, K. 10; Grohe, K. 11)

First-Order Model-Checking is fixed-parameter tractable on any class ofgraphs of bounded expansion.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Examples of Classes of Bounded Expansion

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Tree-Depth

Definition. (Nešetril and Ossona de Mendez 06)

1. The tree-depth td(G) is the minimum height of a tree T s.t.G ⊆ clos(T ).

2. A class C of graphs has bounded tree-depth if there is k ≥ 0 s.t.td(G) ≤ k for all G ∈ C.

Example.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Tree-DepthDefinition. (Nešetril and Ossona de Mendez 06)

1. The tree-depth td(G) is the minimum height of a tree T s.t.G ⊆ clos(T ).

2. A class C of graphs has bounded tree-depth if there is k ≥ 0 s.t.td(G) ≤ k for all G ∈ C.

Example.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Tree-DepthDefinition. (Nešetril and Ossona de Mendez 06)

1. The tree-depth td(G) is the minimum height of a tree T s.t.G ⊆ clos(T ).

2. A class C of graphs has bounded tree-depth if there is k ≥ 0 s.t.td(G) ≤ k for all G ∈ C.

Example.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Tree-DepthDefinition. (Nešetril and Ossona de Mendez 06)

1. The tree-depth td(G) is the minimum height of a tree T s.t.G ⊆ clos(T ).

2. A class C of graphs has bounded tree-depth if there is k ≥ 0 s.t.td(G) ≤ k for all G ∈ C.

Example.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Tree-DepthDefinition. (Nešetril and Ossona de Mendez 06)

1. The tree-depth td(G) is the minimum height of a tree T s.t.G ⊆ clos(T ).

2. A class C of graphs has bounded tree-depth if there is k ≥ 0 s.t.td(G) ≤ k for all G ∈ C.

Example.

Observation.• Classes with bounded tree-depth also have bounded tree-width.• Bounded tree-depth is equivalent to having tree-decompositions

where the tree has bounded height (see Achim’s talk)Theorem. MSO2 model-checking is fixed-parameter tractable on any class C

of graphs of bounded tree-depth.STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 45/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Tree-DepthObservation. Let T be a tree of height k .

Suppose (a1, . . . ,al), (b1, . . . ,bl) ∈ V (T )l are such that

1. tpTq (ai) = tpT

q (bi) for all 1 ≤ i ≤ l and

2. the “relative position” of the ai is the same as for the bi

Then for all ϕ(x1, . . . , xl) ∈ FO of qantifier-rank ≤ q:

T |= ϕ[a] ⇐⇒ T |= ϕ[b].

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Tree-Depth

Observation. Let T be a tree of height k .Suppose (a1, . . . ,al), (b1, . . . ,bl) ∈ V (T )l are such that

1. tpTq (ai) = tpT

q (bi) for all 1 ≤ i ≤ l and

2. the “relative position” of the ai is the same as for the bi

Then for all ϕ(x1, . . . , xl) ∈ FO of qantifier-rank ≤ q:

T |= ϕ[a] ⇐⇒ T |= ϕ[b].

Lemma. Fix q, k ≥ 0.

Let Ck be the class of trees of height ≤ k and let Dk be the class of treesobtained from T ∈ Ck by colouring every vertex v of T by tpT

q (v).

Then there is q′ ≥ 0 s.t. on Dk every ϕ(x) ∈ FOq is equivalent to anexistential formula ϕ∗(x) ∈ FOq′ .

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Bounded Expansion

Definition. (Nešetril and Ossona de Mendez 06)

A class C of graphs has bounded expansion if

for all k ≥ 0 there is a number N(k) such that

for all G ∈ C there is a vertex colouring by N(k) colours such that

the union of ≤ k colour classes has tree-depth at most k .

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Bounded Expansion

Definition. (Nešetril and Ossona de Mendez 06)

A class C of graphs has bounded expansion if

for all k ≥ 0 there is a number N(k) such that

for all G ∈ C there is a vertex colouring by N(k) colours such that

the union of ≤ k colour classes has tree-depth at most k .

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Bounded Expansion

Definition. (Nešetril and Ossona de Mendez 06)

A class C of graphs has bounded expansion if

for all k ≥ 0 there is a number N(k) such that

for all G ∈ C there is a vertex colouring by N(k) colours such that

the union of ≤ k colour classes has tree-depth at most k .

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

First-Order Model-Checking on Bd. Expansion

Theorem. (Dvorák, Kral, Thomas 10; Dawar, K. 10; Grohe, K. 11)First-Order Model-Checking is fixed-parameter tractable on any class ofgraphs of bounded expansion.

Proof. We first show the theorem for existential formulas.

Let G ∈ C and ϕ ∈ FO be given, where

ϕ := ∃x1 . . . xqψ(x) with ψ quantifier-free.

1. Colour G by N(q) colours γ s.t. any q colour classes together havetree-depth ≤ q.

Obviously, ϕ ≡∨

C1,...,Cq∈γ∃x1 ∈ C1 . . . ∃xq ∈ Cq ψ

2. For any q colour classes C1, . . . ,Cq decide G[C1 ∪ · · · ∪ Gq] |= ϕ.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

First-Order Model-Checking on Bd. Expansion

Now suppose ϕ(x1, . . . , xl) := ∀xl+1ψ(x1, . . . , xl+1) where ψ existential

Then ϕ ≡∧

C1,...,Cl+1

(l∧

i=1

xi ∈ Ci

)

→ ∀xl+1 ∈ Cl+1ψ(x)

︸ ︷︷ ︸

equiv. to existential formula

Hence, if we expand G by

• relations for the colour classes and

• the types tpG[C1∪···∪Cq ]q (v) for all v ∈ G[C1 ∪ · · · ∪ Cq],

then on the expansion G′, ϕ(x1, . . . , xl) is equivalent to an exist. formula

By iterating this proceedure, given G ∈ C and ϕ ∈ FO, we can compute

• an expansion G′ of G and

• an existential formula ϕ′ ∈ FO

such that G′ still has small (bounded) expansion. We can test G′ |= ϕ′.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

The Quantifier Elimination Method

We have shown the following result using quantifier-elimination.

Theorem. (Dvorák, Kral, Thomas 10; Dawar, K. 10; Grohe, K. 11)

First-Order Model-Checking is fixed-parameter tractable on any class ofgraphs of bounded expansion.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Part II: Lower Bounds

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Lower Bounds

Research programme.For each of the natural logics L such as FO or MSO,identify a structural property P of classes C of graphs such that MC(L, C)is tractable if, and only if, C has the property P under suitable complexitytheoretical assumptions.

So far, we have focussed on establishing upper bounds, i.e. tractabilityresults.

Surprisingly, much less is known about corresponding lower bounds.

In this (short) second part of the talk we will briefly look at lower boundsfor Courcelle’s theorem.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Courcelle’s Theorem

Theorem. (Courcelle 1990)

For any class C of bounded tree-width

MC(MSO2, C)Input: Graph G ∈ C, ϕ ∈ MSO2

Parameter: |ϕ|Problem: Decide G |= ϕ

is fixed-parameter tractable (linear time for each fixed ϕ).

MSO2: MSO with edge-set quantification.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 54/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Lower Bounds for Monadic Second-Order Logic

We would like to show.If a class C of graphs has unbounded tree-width thenMC(MSO2, C) is not fixed-parameter tractable.

Sadly, in this generality this is not true.

Theorem. (Makowsky, Mariño 04)

There are classes C of graphs of unbounded tree-width on whichMC(MSO2, C) is tractable.

But something similar is true.

Unbounded Tree-Width.We first need to classify the unboundedness of tree-width.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Classes of Unbounded Tree-Width

Definition. Let f : N → N be a non-decreasing function.A class C of graphs has f -bounded tree-width if tw(G) ≤ f (|G|) for allG ∈ C.

Examples.

• In Courcelle’s theorem, f (n) := c is constant.

• f (n) := n is the maximal function that makes sense here.

• We will look at f (n) := logc n for a small constant c > 0.

Theorem by Makowsky, Mariño.

There are classes C of graphs of logarithmic tree-width on whichMC(MSO2, C) is tractable.

What we would like to show.If the tree-width of C is not bounded by logc n, forsmall constant c, then MC(MSO, C) is not FPT.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Classes of Unbounded Tree-Width

Definition. Let f : N → N be a non-decreasing function.A class C of graphs has f -bounded tree-width if tw(G) ≤ f (|G|) for allG ∈ C.

Examples.

• In Courcelle’s theorem, f (n) := c is constant.

• f (n) := n is the maximal function that makes sense here.

• We will look at f (n) := logc n for a small constant c > 0.

Theorem by Makowsky, Mariño.

There are classes C of graphs of logarithmic tree-width on whichMC(MSO2, C) is tractable.

What we would like to show.If the tree-width of C is not bounded by logc n, forsmall constant c, then MC(MSO, C) is not FPT.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Classes of Unbounded Tree-Width

Definition. Let f : N → N be a non-decreasing function.A class C of graphs has f -bounded tree-width if tw(G) ≤ f (|G|) for allG ∈ C.

Examples.

• In Courcelle’s theorem, f (n) := c is constant.

• f (n) := n is the maximal function that makes sense here.

• We will look at f (n) := logc n for a small constant c > 0.

Theorem by Makowsky, Mariño.

There are classes C of graphs of logarithmic tree-width on whichMC(MSO2, C) is tractable.

What we would like to show.If the tree-width of C is not bounded by logc n, forsmall constant c, then MC(MSO, C) is not FPT.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Lower Bounds for Courcelle’s Theorem

Easy observation.Let G be the class of subgraphs of grids.

Then MSO1-model-checking is not fixed-par. tractable on G unless P=NP.

Recall the excluded grid theorem from Achim’s talk.

Theorem. There is a function f : N → N such that for all k ≥ 0, every graphof tree-width at least f (k) contains a k × k-grid as minor.

Combining the two results we obtain a first lower bound.

Theorem. (Makowsky, Mariño 04)

If C is a class of graphs of unbounded tree-width which is closed under(topological) minors, then MC(MSO2, C) is not fpt unless P=NP.

We would like to replace closure under minors by closure under subgraphs.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Lower Bounds for Courcelle’s Theorem

Easy observation.Let G be the class of subgraphs of grids.

Then MSO1-model-checking is not fixed-par. tractable on G unless P=NP.

Recall the excluded grid theorem from Achim’s talk.

Theorem. There is a function f : N → N such that for all k ≥ 0, every graphof tree-width at least f (k) contains a k × k-grid as minor.

Combining the two results we obtain a first lower bound.

Theorem. (Makowsky, Mariño 04)

If C is a class of graphs of unbounded tree-width which is closed under(topological) minors, then MC(MSO2, C) is not fpt unless P=NP.

We would like to replace closure under minors by closure under subgraphs.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 57/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Lower Bounds for Courcelle’s Theorem

Easy observation.Let G be the class of subgraphs of grids.

Then MSO1-model-checking is not fixed-par. tractable on G unless P=NP.

Recall the excluded grid theorem from Achim’s talk.

Theorem. There is a function f : N → N such that for all k ≥ 0, every graphof tree-width at least f (k) contains a k × k-grid as minor.

Combining the two results we obtain a first lower bound.

Theorem. (Makowsky, Mariño 04)

If C is a class of graphs of unbounded tree-width which is closed under(topological) minors, then MC(MSO2, C) is not fpt unless P=NP.

We would like to replace closure under minors by closure under subgraphs.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 57/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Lower Bounds for Courcelle’s Theorem

Theorem. (K., Tazari 10)For k ≥ 0 let Ck be a class of subgraphs of graphs G such thattw(G) ≤ logc(|G|).

Then for all k > 28, MC(MSO, Ck ) is not fpt (not even in XP) unless SATcan be solved in sub-exponential time.

(fpt: with parameter |ϕ|)

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Lower Bounds for First-Order Logic

Theorem. (Dvorák, Kral, Thomas 10)First-Order Model-Checking is fpt on any class of graphs of (locally)bounded expansion.

Theorem. (K. 09)If C is not nowhere dense, closed under sub-graphs and satisfies sometechnical condition, then MC(FO, C) is not fpt unless P=NP.

STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 59/63

INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Lower Bounds for First-Order Logic

Theorem. (Dvorák, Kral, Thomas 10)First-Order Model-Checking is fpt on any class of graphs of (locally)bounded expansion.

Theorem. (K. 09)If C is not nowhere dense, closed under sub-graphs and satisfies sometechnical condition, then MC(FO, C) is not fpt unless P=NP.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Overview of Algorithmic Meta-Theorems

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Conclusion

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Conclusion

Algorithmic Meta-Theorems.We have seen various tools for establishing algorithmic meta-theorems.

First Algorithmic Meta-Theorem. (Courcelle ’90)For any class C of bounded tree-width, MC(MSO2, C) is fixed-parametertractable (linear time for each fixed ϕ).

As Mike pointed out in his talk, these theorems are extremely useful inestablising tractability results on specific classes of graphs.

Research programme.For each of the natural logics L such as FO or MSO,identify a structural property P of classes C of graphs such that MC(L, C)is tractable if, and only if, C has the property P under suitable complexitytheoretical assumptions.

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INTRODUCTION UPPER BOUNDS COMPOSITION LOCALITY QUANT. ELIM.

Open Problems

Main Open Problems.

• The complexity of first-order logic on nowhere dense classes ofgraphs.

• Lower bounds for MSO1.

• To further advance the research programme, we should look at theclosure of tractable classes under transductions etc.

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