Logic in Algorithmic Graph Structure Theory

Transcription

1 Logic in Algorithmic Graph Structure Theory Stephan Kreutzer Technical University Berlin Bruno s workshop June 18-20, 2012, LaBRI, Bordeaux

2 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 2/63 Introduction This talk is motivated by the rôle logic can play in algorithmic graph structure theory. We look at standard computational problems on graphs such as: Dominating Set Find a min. set of vertices which are neighbours to all others. 3-Colourability Colour the vertices by 3 colours without monochromatic edges. Hamiltonian path Find a path containing every vertex exactly once. Clearly, all these problems are NP-complete and hence we do not expect them to be solvable efficiently in general.

3 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 3/63 Algorithmic Graph Structure Theory Restricted classes of input instances. Study these problems on specific, more restricted 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 of graphs.

4 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 4/63 Overview of Graph Classes

5 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 4/63 Overview of Graph Classes

6 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 5/63 Algorithmic Graph Structure Theory The general goal of this area is to 1. explore the range and different types of problems that become tractable on any given class or type of graphs and 2. for certain types of problems such as domination problems explore how far the tractability barrier can be pushed.

7 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 6/63 Algorithmic Graph Structure Theory The general goal of this area is to 1. explore the range and different types of problems that become tractable on any given class or type of graphs and 2. for certain types of problems such as domination problems explore how far the tractability barrier can be pushed. Design of algorithms. Much research has gone into developing and improving algorithms for specific problems on certain classes of graphs. Meta-theorems. To explore general tractability barriers, results that establish tractability results for a whole range of problems for specific classes of graphs are very useful. All problems satisfying certain criteria are tractable on every class of graphs satisfying a property P. These results come in different flavours.

8 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 6/63 Algorithmic Graph Structure Theory The general goal of this area is to 1. explore the range and different types of problems that become tractable on any given class or type of graphs and 2. for certain types of problems such as domination problems explore how far the tractability barrier can be pushed. Design of algorithms. Much research has gone into developing and improving algorithms for specific problems on certain classes of graphs. Meta-theorems. To explore general tractability barriers, results that establish tractability results for a whole range of problems for specific classes of graphs are very useful. All problems satisfying certain criteria are tractable on every class of graphs satisfying a property P. These results come in different flavours.

9 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 7/63 Descriptive Approach to Meta-Theorems Descriptive approach. A different approach is to use logic. Many computational problems on graphs can be described elegantly in logics 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) ) x 1... x k y ( k i=1 y = x i E(x i, y) ) 3-Colourability. definable in Monadic Second-Order Logic (MSO) Colour the vertices by three colours without monochromatic edges. ( C 1 C 2 C 3 x 3 i=1 C i(x) x y(e(x, y) ) 3 i=1 (C i(x) C i (y))) Hamiltonian path. definable in Monadic Second-Order Logic (MSO 2 ) Find a path containing every vertex exactly once. P(Pis a path y y V(P))

10 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 7/63 Descriptive Approach to Meta-Theorems Descriptive approach. A different approach is to use logic. Many computational problems on graphs can be described elegantly in logics 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) ) x 1... x k y ( k i=1 y = x i E(x i, y) ) 3-Colourability. definable in Monadic Second-Order Logic (MSO) Colour the vertices by three colours without monochromatic edges. ( C 1 C 2 C 3 x 3 i=1 C i(x) x y(e(x, y) ) 3 i=1 (C i(x) C i (y))) Hamiltonian path. definable in Monadic Second-Order Logic (MSO 2 ) Find a path containing every vertex exactly once. P(Pis a path y y V(P))

11 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 7/63 Descriptive Approach to Meta-Theorems Descriptive approach. A different approach is to use logic. Many computational problems on graphs can be described elegantly in logics 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) ) x 1... x k y ( k i=1 y = x i E(x i, y) ) 3-Colourability. definable in Monadic Second-Order Logic (MSO) Colour the vertices by three colours without monochromatic edges. ( C 1 C 2 C 3 x 3 i=1 C i(x) x y(e(x, y) ) 3 i=1 (C i(x) C i (y))) Hamiltonian path. definable in Monadic Second-Order Logic (MSO 2 ) Find a path containing every vertex exactly once. P(Pis a path y y V(P))

12 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 7/63 Descriptive Approach to Meta-Theorems Descriptive approach. A different approach is to use logic. Many computational problems on graphs can be described elegantly in logics 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) ) x 1... x k y ( k i=1 y = x i E(x i, y) ) 3-Colourability. definable in Monadic Second-Order Logic (MSO) Colour the vertices by three colours without monochromatic edges. ( C 1 C 2 C 3 x 3 i=1 C i(x) x y(e(x, y) ) 3 i=1 (C i(x) C i (y))) Hamiltonian path. definable in Monadic Second-Order Logic (MSO 2 ) Find a path containing every vertex exactly once. P(Pis a path y y V(P))

13 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 8/63 Algorithmic Meta-Theorems Use logics to specify algorithmic problems and show that all problems definable 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 (MSO 2 ) can be decided in linear time on any class of structures of bounded tree-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 case quickly. 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)

14 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 8/63 Algorithmic Meta-Theorems Use logics to specify algorithmic problems and show that all problems definable 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 (MSO 2 ) can be decided in linear time on any class of structures of bounded tree-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 case quickly. 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)

15 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 8/63 Algorithmic Meta-Theorems Use logics to specify algorithmic problems and show that all problems definable 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 (MSO 2 ) can be decided in linear time on any class of structures of bounded tree-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 case quickly. 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)

16 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 9/63 Algorithmic Meta-Theorems

17 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 10/63 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 of graphs 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 are tractable on a class of graphs. 2. Establish general algorithmic techniques for solving them. 3. Corresponding intractability results for logics exhibit natural boundaries beyond which these techniques fail. In this talk. Present some of the result obtained in this area focussing on the logical tools available.

18 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 10/63 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 of graphs 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 are tractable on a class of graphs. 2. Establish general algorithmic techniques for solving them. 3. Corresponding intractability results for logics exhibit natural boundaries beyond which these techniques fail. In this talk. Present some of the result obtained in this area focussing on the logical tools available.

19 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 11/63 Rephrasing everything in terms of logic Let L be a logic and C be a class of structures. The Model-Checking Problem MC(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 of graphs 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.

20 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 11/63 Rephrasing everything in terms of logic Let L be a logic and C be a class of structures. The Model-Checking Problem MC(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 of graphs 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.

21 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 12/63 Parametrized Complexity The complexity theoretical framework we use is the framework of parameterized complexity introduced by Downey and Fellows. Fixed-Parameter tractability. A model-checking problem is fixed-parameter tractable (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 tractable on a class C of graphs if, on input G C and k, it can be decided in time f(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],...

22 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 12/63 Parametrized Complexity The complexity theoretical framework we use is the framework of parameterized complexity introduced by Downey and Fellows. Fixed-Parameter tractability. A model-checking problem is fixed-parameter tractable (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 tractable on a class C of graphs if, on input G C and k, it can be decided in time f(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],...

23 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 12/63 Parametrized Complexity The complexity theoretical framework we use is the framework of parameterized complexity introduced by Downey and Fellows. Fixed-Parameter tractability. A model-checking problem is fixed-parameter tractable (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 tractable on a class C of graphs if, on input G C and k, it can be decided in time f(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],...

24 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 13/63 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 general classes 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 whether MSO-model-checking is tractable on some class.

25 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 13/63 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 general classes 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 whether MSO-model-checking is tractable on some class.

26 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 13/63 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 general classes 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 complexity theoretical assumptions. We may not always get an exact characterisation, there may be gaps. But such a characterisation would give an easy tool to assess whether MSO-model-checking is tractable on some class.

27 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 13/63 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 general classes 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 complexity theoretical assumptions. We may not always get an exact characterisation, there may be gaps. But such a characterisation would give an easy tool to assess whether MSO-model-checking is tractable on some class.

28 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 14/63 Structural Characterisation of Model-Checking Problems To achieve such a characterisation we need upper bounds: tractability of model-checking on specific classes of graphs. Such results are known as algorithmic meta-theorems lower bounds: results establishing intractability of model-checking problems if certain structural parameters are not given.

29 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 14/63 Structural Characterisation of Model-Checking Problems To achieve such a characterisation we need upper bounds: tractability of model-checking on specific classes of graphs. Such results are known as algorithmic meta-theorems Part I of this talk lower bounds: results establishing intractability of model-checking problems if certain structural parameters are not given. Part II of this talk

30 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 15/63 Upper Bounds on the Complexity of Model-Checking Problems

31 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 16/63 Overview of Algorithmic Meta-Theorems

32 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 17/63 A High Level Description Essentially, all algorithmic meta-theorems known to date are based on the following simple idea. Given a graph G and a formula ϕ, we recursively decompose the graph into smaller, or simpler, subgraphs until we reach graphs of constant size. The simplest form is a complete decomposition bounded tree-width.

33 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 17/63 A High Level Description Essentially, all algorithmic meta-theorems known to date are based on the following simple idea. Given a graph G and a formula ϕ, we recursively decompose the graph into smaller, or simpler, subgraphs until we reach graphs of constant size. The simplest form is a complete decomposition bounded tree-width.

34 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 17/63 A High Level Description Essentially, all algorithmic meta-theorems known to date are based on the following simple idea. Given a graph G and a formula ϕ, we recursively decompose the graph into smaller, or simpler, subgraphs until we reach graphs of constant size. The simplest form is a complete decomposition bounded tree-width.

35 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 18/63 A High Level Description Given a graph G and a formula ϕ, we recursively decompose the graph into 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.

36 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 18/63 A High Level Description Given a graph G and a formula ϕ, we recursively decompose the graph into 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.

37 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 19/63 A High Level Description For this idea to work we need concepts of graph decompositions and for each of these a corresponding concept of composition of formulas. decomposition of graphs proper separation (bd. tree-width) neighbourhood cover (planar graphs) general cover (bd. expansion) composition of formulas Feferman-Vaught theorems Automata + Transductions locality theorems 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

38 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 19/63 A High Level Description For this idea to work we need concepts of graph decompositions and for each of these a corresponding concept of composition of formulas. decomposition of graphs proper separation (bd. tree-width) neighbourhood cover (planar graphs) general cover (bd. expansion) composition of formulas Feferman-Vaught theorems Automata + Transductions locality theorems 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

39 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 20/63 The Automata-Theoretic Method Let C be a class of graphs and L be a logic. Idea to solve MC(L,C). 1. Define a suitable automata model for C. Deciding whether an automaton accepts G C should be in polynomial time. 2. Show that there is an effective translation of MSO-formulas on C into automata A ϕ. 3. Given G C and ϕ L, construct A ϕ and decide whether A ϕ accepts G. Example. MC(MSO, TREE).

40 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 21/63 The Transduction Method Let C be a class of graphs and L be a logic. Idea to solve MC(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.

41 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 22/63 Theorem: First Proof of Courcelle s Theorem For any class C of bounded tree-width MC(MSO 2, C) Input: Graph G C, ϕ MSO 2 Parameter: ϕ Problem: Decide G = ϕ is fixed-parameter tractable (linear time for each fixed ϕ). (Courcelle 1990) There is an effective transduction from classes of bounded tree-width into the class of trees. Theorem. What about the parameter dependence? (Frick, Grohe, 01) 1. Unless P=NP, there is no fpt-algorithm for MSO model checking on trees with elementary parameter dependence. 2. Unless FPT=W[1], there is no fpt-algorithm for FO model checking on trees with elementary parameter dependence.

42 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 22/63 Theorem: First Proof of Courcelle s Theorem For any class C of bounded tree-width MC(MSO 2, C) Input: Graph G C, ϕ MSO 2 Parameter: ϕ Problem: Decide G = ϕ is fixed-parameter tractable (linear time for each fixed ϕ). (Courcelle 1990) There is an effective transduction from classes of bounded tree-width into the class of trees. Theorem. What about the parameter dependence? (Frick, Grohe, 01) 1. Unless P=NP, there is no fpt-algorithm for MSO model checking on trees with elementary parameter dependence. 2. Unless FPT=W[1], there is no fpt-algorithm for FO model checking on trees with elementary parameter dependence.

43 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 23/63 Extensions of Courcelle s Theorem Theorem. (Courcelle, Makowsky, Rotics 2001) MC(MSO 1, C) is fixed-parameter tractable for any class C of bounded clique-width. Analogous results can be obtained for various extensions of MSO: Counting MSO: MSO + Quantifiers X = p mod q (see The Book) Order-Invariant MSO (MSO <-inv ): (Engelmann, K., Siebertz 12) Formulas ϕ can use an order relation < but truth must be independent on 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 <-inv 2,C) is fpt on any class of graphs of bounded tree-width. MC(MSO <-inv 1,C) is fpt on any class of graphs of bounded clique-width.

44 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 23/63 Extensions of Courcelle s Theorem Theorem. (Courcelle, Makowsky, Rotics 2001) MC(MSO 1, C) is fixed-parameter tractable for any class C of bounded clique-width. Analogous results can be obtained for various extensions of MSO: Counting MSO: MSO + Quantifiers X = p mod q (see The Book) Order-Invariant MSO (MSO <-inv ): (Engelmann, K., Siebertz 12) Formulas ϕ can use an order relation < but truth must be independent on 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 <-inv 2,C) is fpt on any class of graphs of bounded tree-width. MC(MSO <-inv 1,C) is fpt on any class of graphs of bounded clique-width.

45 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 24/63 The Composition Method

46 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 25/63 Feferman-Vaught Style Theorems Notation: G: graph v: tuple of vertices tp MSO (G, v): full MSO-type of v in G (all MSO-formulae true at v) tp MSO q (G, v): class of MSO-formulae of quantifier-rank q true at v analogously for tp FO and tp FO q

47 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 26/63 Feferman-Vaught Style Theorems Theorem. Let G, H be graphs v V(G) w V(H) u V(G) such that u = V(G) V(H) (see Makowsky 04) For all q 0, tp q (G H, uvw) is determined by tp q (G, uv) and tp q (uw) Furthermore, there is an algorithm that computes tp q (G H, uvw) from tp q (G, uv) and tp q (uw).

48 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 27/63 Feferman-Vaught Style Theorems Theorem. 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, tp q (G H, uvw) is determined by tp q (G, uv) and tp q (uw) Furthermore, there is an algorithm that computes tp q (G H, uvw) from tp q (G, uv) and tp q (uw).

49 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 27/63 Feferman-Vaught Style Theorems Theorem. 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, tp q (G H, uvw) is determined by tp q (G, uv) and tp q (uw) Furthermore, there is an algorithm that computes tp q (G H, uvw) from tp q (G, uv) and tp q (uw). This suggests a model-checking algorithm on graphs which can recursively be decomposed into sub-graphs of constant size.

50 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 27/63 Feferman-Vaught Style Theorems Theorem. 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, tp q (G H, uvw) is determined by tp q (G, uv) and tp q (uw) Furthermore, there is an algorithm that computes tp q (G H, uvw) from tp q (G, uv) and tp q (uw). This suggests a model-checking algorithm on graphs which can recursively be decomposed into sub-graphs of constant size.

51 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 27/63 Feferman-Vaught Style Theorems Theorem. 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, tp q (G H, uvw) is determined by tp q (G, uv) and tp q (uw) Furthermore, there is an algorithm that computes tp q (G H, uvw) from tp q (G, uv) and tp q (uw). This suggests a model-checking algorithm on graphs which can recursively be decomposed into sub-graphs of constant size. graphs of bounded tree-width

52 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 28/63 An Overview of Graph Parameters

53 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 29/63 The Locality Method for First-Order Logic

54 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 30/63 Locality of First-Order Logic Notation: Let G be a graph dist G (u, v) : e.g. the Gaifman graph of a structure length of the shortest path between u and v N G r (v) := {u V(G) : dist G (u, v) r} N G r (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 [ N r (v) ] = ϕ(v). Hence, truth at v only depends on the vertices around v.

55 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 31/63 Gaifman s Theorem Theorem. (Gaifman, 1982) Every first-order sentence ϕ FO is equivalent to a Boolean combination of basic local sentences. Basic local sentence: where ψ is r-local. k ϕ := x 1... x m dist(x i, x j )> 2r ψ(x i ). i j Remark. Gaifman s proof is constructive. Theorem. (Dawar, Grohe, K., Schweikardt, 07) For each k 1 there is ϕ k FO[{E}] of length O(k 4 ) such that every equivalent sentence in Gaifman-NF has length at least tower(k). (similar lower bounds for Feferman-Vaught and preservation thms) i=1

56 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 31/63 Gaifman s Theorem Theorem. (Gaifman, 1982) Every first-order sentence ϕ FO is equivalent to a Boolean combination of basic local sentences. Basic local sentence: where ψ is r-local. k ϕ := x 1... x m dist(x i, x j )> 2r ψ(x i ). i j Remark. Gaifman s proof is constructive. Theorem. (Dawar, Grohe, K., Schweikardt, 07) For each k 1 there is ϕ k FO[{E}] of length O(k 4 ) such that every equivalent sentence in Gaifman-NF has length at least tower(k). (similar lower bounds for Feferman-Vaught and preservation thms) i=1

57 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 32/63 The Locality Method Theorem. (Follows from Frick, Grohe 01) Fix d 0. For every r 0 let D r be a class of graphs such that MC(FO,D r ) 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[N G r (v)] D r. Then MC(FO, C) is fixed-parameter tractable. Examples. Graph classes of maximum degree k. Take D r to be graphs of size at most d r. Graph classes of bounded local tree-width Take D r to be graphs of tree-width at most g(r).

58 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 33/63 First-Order Logic on Bounded Degree Graphs Theorem. (Follows from Frick, Grohe 01) Fix d 0. For every r 0 let D r be a class of graphs such that MC(FO,D r ) 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[N G r (v)] D r. Then MC(FO, C) is fixed-parameter tractable. Proof. By Gaifman s theorem it suffices to consider formulae of the form x 1... x m 1 i<j m for some r-local formula ψ(x). dist(x i, x j ) > 2r k ψ(x i ) i=1

59 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 33/63 First-Order Logic on Bounded Degree Graphs Theorem. (Follows from Frick, Grohe 01) Fix d 0. For every r 0 let D r be a class of graphs such that MC(FO,D r ) 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[N G r (v)] D r. Then MC(FO, C) is fixed-parameter tractable. Proof. By Gaifman s theorem it suffices to consider formulae of the form x 1... x m 1 i<j m for some r-local formula ψ(x). dist(x i, x j ) > 2r k ψ(x i ) i=1

60 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 34/63 Proof of Theorem Suppose m ϕ := x 1... x m dist(x i, x j ) > 2r ψ(x i ) for some r-local formula ψ(x). Let G C. 1 i<j m i=1 Find m vertices of distance > 2r whose r-neighbourhoods satisfy ψ.

61 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 34/63 Proof of Theorem Suppose m ϕ := x 1... x m dist(x i, x j ) > 2r ψ(x i ) for some r-local formula ψ(x). Let G C. 1 i<j m i=1 Find m vertices of distance > 2r whose r-neighbourhoods satisfy ψ.

62 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 35/63 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 classes excluding a minor. Theorem. (Dawar, Grohe, K. 07) First-order model-checking is fixed-parameter tractable on graph classes locally excluding a minor. Theorem: (Dvořák, Kral, Thomas 10, Dawar, K. 10; Grohe, K. 11) First-order model-checking is fixed-parameter tractable on graph classes of locally bounded expansion.

63 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 36/63 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

64 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 37/63 First-Order Logic and Excluded Minors Theorem: Let C be a class of graphs excluding a minor. The problem (Flum, Grohe 2001) MC(FO, C) Input: Graph G C, ϕ MSO Parameter: ϕ Problem: G = ϕ? is fixed-parameter tractable.

65 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 38/63 FO Model Checking: Decomposing a Graph Given: Input: Problem: C class of graphs excluding a minor H Graph G such that H G and ϕ FO G = ϕ G excludes H Decomp. theorem, Robertson, Seymour Logic: Composition Theorems

66 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 38/63 FO Model Checking: Decomposing a Graph Given: Input: Problem: C class of graphs excluding a minor H Graph G such that H G and ϕ FO G = ϕ G excludes H Decomp. theorem, Robertson, Seymour Logic: Composition Theorems

67 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 39/63 FO Model Checking: Decomposing a Graph In a block: Local tree-width (almost) bounded by a function λ

68 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 39/63 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

69 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 39/63 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 Logic: Locality Method

70 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 39/63 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 Logic: Locality Method Followed by Transduction into Trees

71 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 39/63 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 Logic: Locality Method Followed by Transduction into Trees Followed by Automata Method

72 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 40/63 First-Order Logic and Excluded Minors Theorem. (Flum, Grohe 01) First-order model-checking is fixed-parameter tractable on graph classes excluding a minor. Theorem: Let C be a class of graphs excluding a minor H. The problem (Dawar, Grohe, K. 2007) MC(FO, C) Input: Graph G C, ϕ MSO Parameter: ϕ + H Problem: G = ϕ? is fixed-parameter tractable.

73 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 40/63 First-Order Logic and Excluded Minors Theorem. (Flum, Grohe 01) First-order model-checking is fixed-parameter tractable on graph classes excluding a minor. Theorem: Let C be a class of graphs excluding a minor H. The problem (Dawar, Grohe, K. 2007) MC(FO, C) Input: Graph G C, ϕ MSO Parameter: ϕ + H Problem: G = ϕ? is fixed-parameter tractable.

74 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 41/63 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 classes excluding a minor. Theorem. (Dawar, Grohe, K. 07) First-order model-checking is fixed-parameter tractable on graph classes locally excluding a minor. Theorem: (Dvořák, Kral, Thomas 10, Dawar, K. 10; Grohe, K. 11) First-order model-checking is fixed-parameter tractable on graph classes of locally bounded expansion.

75 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 41/63 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 classes excluding a minor. Theorem. (Dawar, Grohe, K. 07) First-order model-checking is fixed-parameter tractable on graph classes locally excluding a minor. Theorem: (Dvořák, Kral, Thomas 10, Dawar, K. 10; Grohe, K. 11) First-order model-checking is fixed-parameter tractable on graph classes of locally bounded expansion.

76 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 42/63 The Quantifier-Elimination Method

77 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 43/63 The Quantifier-Elimination Method We will show the following result, proved using quantifier-elimination. Theorem. (Dvořák, Kral, Thomas 10; Dawar, K. 10; Grohe, K. 11) First-Order Model-Checking is fixed-parameter tractable on any class of graphs of bounded expansion.

78 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 44/63 Examples of Classes of Bounded Expansion

79 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 45/63 Tree-Depth Definition. (Nešetřil and Ossona de Mendez 06) Example. 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.

80 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 45/63 Tree-Depth Definition. (Nešetřil 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.

81 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 45/63 Tree-Depth Definition. (Nešetřil 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.

82 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 45/63 Tree-Depth Definition. (Nešetřil 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.

83 Tree-Depth Definition. (Nešetřil 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. MSO 2 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

84 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 46/63 Tree-Depth Observation. Let T be a tree of height k. Suppose (a 1,...,a l ),(b 1,...,b l ) V(T) l are such that 1. tp T q (a i) = tp T q (b i) for all 1 i l and 2. the relative position of the a i is the same as for the b i Then for all ϕ(x 1,...,x l ) FO of qantifier-rank q: T = ϕ[ā] T = ϕ[ b].

85 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 47/63 Tree-Depth Observation. Let T be a tree of height k. Suppose (a 1,...,a l ),(b 1,...,b l ) V(T) l are such that 1. tp T q (a i) = tp T q (b i) for all 1 i l and 2. the relative position of the a i is the same as for the b i Then for all ϕ(x 1,...,x l ) FO of qantifier-rank q: T = ϕ[ā] T = ϕ[ b]. Lemma. Fix q, k 0. Let C k be the class of trees of height k and let D k be the class of trees obtained from T C k by colouring every vertex v of T by tp T q (v). Then there is q 0 s.t. on D k every ϕ( x) FO q is equivalent to an existential formula ϕ ( x) FO q.

86 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 48/63 Bounded Expansion Definition. (Nešetřil 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.

87 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 48/63 Bounded Expansion Definition. (Nešetřil 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.

88 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 48/63 Bounded Expansion Definition. (Nešetřil 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.

89 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 49/63 First-Order Model-Checking on Bd. Expansion Theorem. (Dvořák, Kral, Thomas 10; Dawar, K. 10; Grohe, K. 11) First-Order Model-Checking is fixed-parameter tractable on any class of graphs of bounded expansion. Proof. We first show the theorem for existential formulas. Let G C and ϕ FO be given, where ϕ := x 1...x q ψ( x) with ψ quantifier-free. 1. Colour G by N(q) colours γ s.t. any q colour classes together have tree-depth q. Obviously, ϕ C 1,...,C q γ x 1 C 1... x q C q ψ 2. For any q colour classes C 1,...,C q decide G[C 1 G q ] = ϕ.

90 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 50/63 First-Order Model-Checking on Bd. Expansion Now suppose ϕ(x 1,...,x l ) := x l+1 ψ(x 1,...,x l+1 ) where ψ existential Then ϕ ( l ) C 1,...,C l+1 x i C i x l+1 C l+1 ψ( x) i=1 }{{} equiv. to existential formula Hence, if we expand G by relations for the colour classes and the types tp G[C 1 C q] q (v) for all v G[C 1 C q ], then on the expansion G, ϕ(x 1,...,x l ) 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.

91 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 51/63 The Quantifier Elimination Method We have shown the following result using quantifier-elimination. Theorem. (Dvořák, Kral, Thomas 10; Dawar, K. 10; Grohe, K. 11) First-Order Model-Checking is fixed-parameter tractable on any class of graphs of bounded expansion.

92 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 52/63 Part II: Lower Bounds

93 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 53/63 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 complexity theoretical assumptions. So far, we have focussed on establishing upper bounds, i.e. tractability results. Surprisingly, much less is known about corresponding lower bounds. In this (short) second part of the talk we will briefly look at lower bounds for Courcelle s theorem.

94 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 54/63 Courcelle s Theorem Theorem. For any class C of bounded tree-width (Courcelle 1990) MC(MSO 2, C) Input: Graph G C, ϕ MSO 2 Parameter: ϕ Problem: Decide G = ϕ is fixed-parameter tractable (linear time for each fixed ϕ). MSO 2 : MSO with edge-set quantification.

95 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 55/63 Lower Bounds for Monadic Second-Order Logic We would like to show. If a class C of graphs has unbounded tree-width then MC(MSO 2,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 which MC(MSO 2,C) is tractable. But something similar is true. Unbounded Tree-Width. We first need to classify the unboundedness of tree-width.

96 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 56/63 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 all G 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) := log c n for a small constant c > 0. Theorem by Makowsky, Mariño. There are classes C of graphs of logarithmic tree-width on which MC(MSO 2,C) is tractable. What we would like to show. If the tree-width of C is not bounded by log c n, for small constant c, then MC(MSO,C) is not FPT.

97 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 56/63 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 all G 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) := log c n for a small constant c > 0. Theorem by Makowsky, Mariño. There are classes C of graphs of logarithmic tree-width on which MC(MSO 2,C) is tractable. What we would like to show. If the tree-width of C is not bounded by log c n, for small constant c, then MC(MSO,C) is not FPT.

98 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 56/63 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 all G 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) := log c n for a small constant c > 0. Theorem by Makowsky, Mariño. There are classes C of graphs of logarithmic tree-width on which MC(MSO 2,C) is tractable. What we would like to show. If the tree-width of C is not bounded by log c n, for small constant c, then MC(MSO,C) is not FPT.

99 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 57/63 Lower Bounds for Courcelle s Theorem Easy observation. Let G be the class of subgraphs of grids. Then MSO 1 -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 graph of 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(MSO 2,C) is not fpt unless P=NP. We would like to replace closure under minors by closure under subgraphs.

100 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 57/63 Lower Bounds for Courcelle s Theorem Easy observation. Let G be the class of subgraphs of grids. Then MSO 1 -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 graph of 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(MSO 2,C) is not fpt unless P=NP. We would like to replace closure under minors by closure under subgraphs.

101 STEPHAN KREUTZER LOGIC IN ALGORITHMIC GRAPH STRUCTURE THEORY 57/63 Lower Bounds for Courcelle s Theorem Easy observation. Let G be the class of subgraphs of grids. Then MSO 1 -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 graph of 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(MSO 2,C) is not fpt unless P=NP. We would like to replace closure under minors by closure under subgraphs.