Stephan Kreutzer
Algorithmic Applications of Sparse Classes of Graphs
Foundations of Software Technology and Theoretical Computer Science 13.12.2013
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text!Task: Choose a programme commi1ee You are the PC chair and want to put together a PC for a conference. You may choose 15 people for your PC.
Idea. Use DBLP and create a co-‐author graph. !!!!!!
�2
Choosing a Programme Committee
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text!Task: Choose a programme commi1ee You are the PC chair and want to put together a PC for a conference. You may choose 15 people for your PC.
Idea. Use DBLP and create a co-‐author graph. !!!!!!
�2
Choosing a Programme Committee
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text!!!!!!!!
�3
Choosing a Programme Committee
Domina'ng Set Input: Graph G, number k Problem: Find a set S ⊆ V(G) with |S| ≤ k such that for all v∈V(G)-‐S there is an u∈S with { u,v } ∈ E(G).
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text!!!!!!!!
�3
Choosing a Programme Committee
Domina'ng Set Input: Graph G, number k Problem: Find a set S ⊆ V(G) with |S| ≤ k such that for all v∈V(G)-‐S there is an u∈S with { u,v } ∈ E(G).
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text!!!!!!!!
�4
Choosing a Programme Committee
Distance d Domina'ng Set / Network Centres Input: Graph G, numbers k, d Problem: Find a set S ⊆ V(G) with |S| ≤ k such that for all v∈V(G) there is an u∈S with dist(u,v) ≤ d.
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text!Task: Judge the invited talk Now you have chosen an invited speaker and want feedback on his talk. You want to ask about 10 people from the audience.
Idea. Create audience graph. Edge: two people have similar taste/research area. !!!!!!
�5
Judging the Invited Talk
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text!Task: Judge the invited talk Now you have chosen an invited speaker and want feedback on his talk. You want to ask about 10 people from the audience.
Idea. Create audience graph. Edge: two people have similar taste/research area. !!!!!!
�5
Judging the Invited Talk
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text!!!!!!!
�6
Judging the Invited Talk
Independent Set Input: Graph G, number k Problem: Find a set S ⊆ V(G) with |S| ≥ k such that { u,v } ∉ E(G) for all v, u∊S with u ≠ v.
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text!!!!!!!
�7
Judging the Invited Talk
Distance d Independent Set (d-‐DIS) Input: Graph G, numbers k, d Problem: Find a set S ⊆ V(G) with |S| ≥ k such that dist(u,v) > d for all v, u∊V(G) with u ≠ v.
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text!!!!!!!
�8
Judging the Invited Talk
Coloured Distance d Independent Set Input: Graph G, numbers k, d, set B ⊆ V(G) Problem: Find a set S ⊆ B with |S| ≥ k such that dist(u,v) > d for all v, u∊S with u ≠ v.
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text!!!!!!!
�8
Judging the Invited Talk
Coloured Distance d Independent Set Input: Graph G, numbers k, d, set B ⊆ V(G) Problem: Find a set S ⊆ B with |S| ≥ k such that dist(u,v) > d for all v, u∊S with u ≠ v.
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�9
These are only two examples of many standard algorithmic problems on graphs. !Examples.
• Network Centres / Facility locaQon problems / dominaQng sets • Clique, Independent Set, Subgraph Containment • Network design problems: Steiner trees or networks • k-‐Colourability • k-‐disjoint paths, Hamiltonian paths
Complexity. The corresponding decision problems are all NP-‐complete in general. Hence, it is expected that no efficient algorithms solving them exist.
Complexity
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�9
These are only two examples of many standard algorithmic problems on graphs. !Examples.
• Network Centres / Facility locaQon problems / dominaQng sets • Clique, Independent Set, Subgraph Containment • Network design problems: Steiner trees or networks • k-‐Colourability • k-‐disjoint paths, Hamiltonian paths
Complexity. The corresponding decision problems are all NP-‐complete in general. Hence, it is expected that no efficient algorithms solving them exist.
Complexity
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�10
!!!Dealing with the complexity.
• Design heurisQcs
• Design exact algorithms opQmising the (exponenQal) running Qme.
• ApproximaQon algorithms
• IdenQfy special classes of admissible inputs on which the problemsbecome tractable.
Managing Complexity
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text!!!Restricted cases sufficient for applicaCons. We may have addiQonal informaQon about the structure of inputs.
Examples. road map: almost planar ! communicaQon networks: sparse (moderate number of edges) !!Restricted classes of inputs. Solve problems efficiently on restricted input classes relevant for applicaQons.
�11
Algorithmic Graph Structure Theory
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text!!!Restricted cases sufficient for applicaCons. We may have addiQonal informaQon about the structure of inputs.
Examples. road map: almost planar ! communicaQon networks: sparse (moderate number of edges) !!Restricted classes of inputs. Solve problems efficiently on restricted input classes relevant for applicaQons.
�11
Algorithmic Graph Structure Theory
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�12
Algorithmic Graph Structure Theory
all graph classes
bd expansion
planar
bd degree
bd local tree-‐w.
excluded minors
locally excl. minors
nowhere dense
trees
bd tree-‐width
apex minor freebd genus
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title TextModels of efficiency. Solvability in polynomial Qme. But DominaQng Set NP-‐hard on very restricted classes of graphs. !In our examples. The PC is rather small (10-‐15 people). So may be ok. Also you are not going to ask too many people how the talk was. !Parameterized Complexity. Restrict the exponenQal behaviour to a specific/small part of the input. For instance the size of the soluQon or some structural parameter.
Try to find algorithms running in Qme
!
�13
Efficiently?
2O(pk) · n
2O(pk) · n 2O(k) · n2 f(k) · ncoror
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title TextModels of efficiency. Solvability in polynomial Qme. But DominaQng Set NP-‐hard on very restricted classes of graphs. !In our examples. The PC is rather small (10-‐15 people). So may be ok. Also you are not going to ask too many people how the talk was. !Parameterized Complexity. Restrict the exponenQal behaviour to a specific/small part of the input. For instance the size of the soluQon or some structural parameter.
Try to find algorithms running in Qme
!
�13
Efficiently?
2O(pk) · n
2O(pk) · n 2O(k) · n2 f(k) · ncoror
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title TextModels of efficiency. Solvability in polynomial Qme. But DominaQng Set NP-‐hard on very restricted classes of graphs. !In our examples. The PC is rather small (10-‐15 people). So may be ok. Also you are not going to ask too many people how the talk was. !Parameterized Complexity. Restrict the exponenQal behaviour to a specific/small part of the input. For instance the size of the soluQon or some structural parameter.
Try to find algorithms running in Qme
!
�13
Efficiently?
2O(pk) · n
2O(pk) · n 2O(k) · n2 f(k) · ncoror
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text!Parameterized Problem: Pair (P, k). !
!
!
DefiniCon. A parameterized problem (P, k) is called fixed-‐parameter tractable if it can be solved in Qme ! for some (computable) funcQon f and constant c. !Parameterized intractability: W[1]-‐hardness, W[2] ….. ! Problems such as Independent/DominaQng Set … W[1]-‐hard on general graphs
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Parameterized Complexity
f(k) · nc
Independent Set Input: Graph G, number k Parameter: k (or k+d or a struct. param.: tree-‐width…) Problem: Find S ⊆ V(G) with |S| ≥ k st{u,v}∉ E for all v ≠ u∊V(G).
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�15
Algorithmic Graph Structure Theory
all graph classes
bd expansion
planar
bd degree
bd local tree-‐w.
excluded minors
locally excl. minors
nowhere dense
trees
bd tree-‐width
apex minor free
Sub-‐Exp: DominaQng Set
k-‐Colourability (FPT in tree-‐w.)
FPT:Network Centres
PTAS:Independent Set
Meta-‐Kernels
Bidimensionality theory
topological methods
Kernels:Longest Path
Algorithmic MetaTheorems
dynamic programming
bd genus
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�16
Design of algorithms. Much research has gone into developing and improving algorithms for specific problems on certain classes of graphs. !In this talk. What are the largest graph classes on which we can solve certain types of problems.
Algorithmic Graph Structure Theory
Algorithmic(Applications(of(Sparse(GraphsStephan(Kreutzer
Title(Text
�14
Algorithmic(Graph(Structure(Theory
all)graph)classes
bd)expansion
planar
bd)degree
bd)local)tree9w.
excluded)minors
locally)excl.)minors
nowhere)dense
treesbd)tree9width
apex)minor)free
Sub?Exp:!DominaQng!Set!
k?Colourability!(FPT!in!tree?w.)
FPT:Network!Centres!
PTAS:Independent!Set
Meta?Kernels
Bidimensionality!theory
topological!methods
Kernels:Longest!Path
Algorithmic!MetaTheorems
dynamic!programming
bd)genus
Algorithmic Meta-‐Theorems
Algorithmic Applications of Sparse GraphsStephan Kreutzer �18
Aim. A simple way of verifying whether a problem is tractable on a specific class of graphs, e.g. of small tree-‐width. !Meta-‐theorems. To explore general tractability barriers, we want results that establish tractability for a whole range of problems for specific classes of graphs
All problems saKsfying certain criteria are tractable on every class of graphs saKsfying a property P.
Algorithmic Graph Structure Theory
Algorithmic(Applications(of(Sparse(GraphsStephan(Kreutzer
Title(Text
�14
Algorithmic(Graph(Structure(Theory
all)graph)classes
bd)expansion
planar
bd)degree
bd)local)tree9w.
excluded)minors
locally)excl.)minors
nowhere)dense
treesbd)tree9width
apex)minor)free
Sub?Exp:!DominaQng!Set!
k?Colourability!(FPT!in!tree?w.)
FPT:Network!Centres!
PTAS:Independent!Set
Meta?Kernels
Bidimensionality!theory
topological!methods
Kernels:Longest!Path
Algorithmic!MetaTheorems
dynamic!programming
bd)genus
Algorithmic Applications of Sparse GraphsStephan Kreutzer �18
Aim. A simple way of verifying whether a problem is tractable on a specific class of graphs, e.g. of small tree-‐width. !Meta-‐theorems. To explore general tractability barriers, we want results that establish tractability for a whole range of problems for specific classes of graphs
All problems saKsfying certain criteria are tractable on every class of graphs saKsfying a property P.
Algorithmic Graph Structure Theory
Algorithmic(Applications(of(Sparse(GraphsStephan(Kreutzer
Title(Text
�14
Algorithmic(Graph(Structure(Theory
all)graph)classes
bd)expansion
planar
bd)degree
bd)local)tree9w.
excluded)minors
locally)excl.)minors
nowhere)dense
treesbd)tree9width
apex)minor)free
Sub?Exp:!DominaQng!Set!
k?Colourability!(FPT!in!tree?w.)
FPT:Network!Centres!
PTAS:Independent!Set
Meta?Kernels
Bidimensionality!theory
topological!methods
Kernels:Longest!Path
Algorithmic!MetaTheorems
dynamic!programming
bd)genus
Algorithmic Applications of Sparse GraphsStephan Kreutzer �19
In parQcular, it would be nice to read tractability of a problem directly off its mathemaQcal descripQon.
!!Theorem. Every graph property that can be described using only
• there is a/for all sets of edges/verQces • there is a/for all verQces/edges • Boolean combinaQons • there is an edge or path between u and v …
can be solved in linear Qme on any class of graphs of bounded tree-‐width. !Example. 3-‐Colourability !A graph G is 3-‐Colourable if there are 3 sets of verQces such that every vertex of G belongs to a set and for all edges both endpoints have different colours.
Algorithmic Meta-‐Theorems
Algorithmic Applications of Sparse GraphsStephan Kreutzer �19
In parQcular, it would be nice to read tractability of a problem directly off its mathemaQcal descripQon.
!!Theorem. Every graph property that can be described using only
• there is a/for all sets of edges/verQces • there is a/for all verQces/edges • Boolean combinaQons • there is an edge or path between u and v …
can be solved in linear Qme on any class of graphs of bounded tree-‐width. !Example. 3-‐Colourability !A graph G is 3-‐Colourable if there are 3 sets of verQces such that every vertex of G belongs to a set and for all edges both endpoints have different colours.
Algorithmic Meta-‐Theorems
Algorithmic Applications of Sparse GraphsStephan Kreutzer �20
!Theorem. (Courcelle 90) !
Every graph property definable in Monadic Second-‐Order Logic can be decided in linear Qme on any class of graphs of bounded tree-‐width. !
General form of an algorithmic meta-‐theorem. Every problem definable in a given logic L is tractable on any class of graphs saKsfying a certain property. !
Rephrased in parameterized complexity. Let C be a class of graphs. Then the following problem is fixed-‐parameter tractable !!!!!
Algorithmic Meta-‐Theorems
MC(L, C) Input: Graph G∈C, Formula 𝜑∈L Parameter: |𝜑| (or |𝜑| + tw(G)) Problem: Decide G ⊧ 𝜑?
Algorithmic Applications of Sparse GraphsStephan Kreutzer �20
!Theorem. (Courcelle 90) !
Every graph property definable in Monadic Second-‐Order Logic can be decided in linear Qme on any class of graphs of bounded tree-‐width. !
General form of an algorithmic meta-‐theorem. Every problem definable in a given logic L is tractable on any class of graphs saKsfying a certain property. !
Rephrased in parameterized complexity. Let C be a class of graphs. Then the following problem is fixed-‐parameter tractable !!!!!
Algorithmic Meta-‐Theorems
MC(L, C) Input: Graph G∈C, Formula 𝜑∈L Parameter: |𝜑| (or |𝜑| + tw(G)) Problem: Decide G ⊧ 𝜑?
Algorithmic Applications of Sparse GraphsStephan Kreutzer �21
Theorem. (Courcelle 90) !Every graph property definable in Monadic Second-‐Order Logic can be decided in linear Qme on any class of graphs of bounded tree-‐width.
!ApplicaCons of Courcelle’s theorem/Algorithmic Meta-‐Theorems. • Simple way of verifying whether a problem is tractable on a graph class • Basis of theories such as meta-‐kernalisaQon • Used as an essenQal part of some poly-‐Qme algorithms (e.g.structural decomposiQons)
ImplementaCons. (Langer, Rossmanith…)
Efficient implementaQon of the theorem available. !
Tested on real world examples: covering Hanover subway system by wifi transmijers
Beats ILP solvers (CPLEX) on some problems
Algorithmic Meta-‐Theorems
Algorithmic Applications of Sparse GraphsStephan Kreutzer �21
Theorem. (Courcelle 90) !Every graph property definable in Monadic Second-‐Order Logic can be decided in linear Qme on any class of graphs of bounded tree-‐width.
!ApplicaCons of Courcelle’s theorem/Algorithmic Meta-‐Theorems. • Simple way of verifying whether a problem is tractable on a graph class • Basis of theories such as meta-‐kernalisaQon • Used as an essenQal part of some poly-‐Qme algorithms (e.g.structural decomposiQons)
ImplementaCons. (Langer, Rossmanith…)
Efficient implementaQon of the theorem available. !
Tested on real world examples: covering Hanover subway system by wifi transmijers
Beats ILP solvers (CPLEX) on some problems
Algorithmic Meta-‐Theorems
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
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!
!Theorem. (K., Tazari 10)
Let C be a class of graphs closed under sub-‐graphs. If the tree-‐width of C is not poly-‐logarithmically, or log28 n, bounded then MC(MSO, C) is not FPT unless SAT can be solved in sub-‐exponenQal Qme. (fpt: with parameter |φ|) !
Theorem. (Seese 96) !Every graph property definable in First-‐Order Logic can be decided in linear Qme on any class of graphs of bounded maximum degree.
Algorithmic Meta-‐Theorems
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�22
!
!Theorem. (K., Tazari 10)
Let C be a class of graphs closed under sub-‐graphs. If the tree-‐width of C is not poly-‐logarithmically, or log28 n, bounded then MC(MSO, C) is not FPT unless SAT can be solved in sub-‐exponenQal Qme. (fpt: with parameter |φ|) !
Theorem. (Seese 96) !Every graph property definable in First-‐Order Logic can be decided in linear Qme on any class of graphs of bounded maximum degree.
Algorithmic Meta-‐Theorems
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�23
Algorithmic Graph Structure Theory
all graph classes
bd expansion
planar
bd degree
bd local tree-‐w.
excluded minors
locally excl. minors
nowhere dense
trees
bd tree-‐width
apex minor freebd genus
MSO (Courcelle 90)
FO FPT (Seese 96)
FO FPT (Frick, Grohe 01)
FO FPT+PTAS (Flum, Frick, Grohe 01)(Dawar, Gr, K., Schw 06)
FO FPT (Dawar, Grohe, K. 07)
FO FPT + Enum (Dvorak, Kral, Thomas 11) (Kazana, Segoufin 12)
FO???
FO FPT (Frick, Grohe 01)
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
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!!
Main result. (Grohe, K., Siebertz 13+) Every problem definable in first-‐order logic can be decided in KmeO(n1+𝜀) , for every 𝜀>0, on any class of graphs that is nowhere dense.In other words, FO-‐model checking is FPT on nowhere dense classes. !
Examples. • The (Coloured/Distance-‐d-‐) DominaQng Set problem is fixed-‐parameter tractable on nowhere dense classes.
• CompuQng Steiner trees is FPT with parameter the soluQon size on nowhere dense classes.
! Theorem. (K. 09, Dvorak, Kral, Thomas ’11) If a class C closed under subgraphs is not nowhere dense, then FO-‐model-‐checking is not fixed-‐parameter tractable (unless AW[∗] = FPT).
Algorithmic Meta-‐Theorems
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�24
!!
Main result. (Grohe, K., Siebertz 13+) Every problem definable in first-‐order logic can be decided in KmeO(n1+𝜀) , for every 𝜀>0, on any class of graphs that is nowhere dense.In other words, FO-‐model checking is FPT on nowhere dense classes. !
Examples. • The (Coloured/Distance-‐d-‐) DominaQng Set problem is fixed-‐parameter tractable on nowhere dense classes.
• CompuQng Steiner trees is FPT with parameter the soluQon size on nowhere dense classes.
! Theorem. (K. 09, Dvorak, Kral, Thomas ’11) If a class C closed under subgraphs is not nowhere dense, then FO-‐model-‐checking is not fixed-‐parameter tractable (unless AW[∗] = FPT).
Algorithmic Meta-‐Theorems
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�25
Algorithmic Graph Structure Theory
all graph classes
bd expansion
planar
bd degree
bd local tree-‐w.
excluded minors
locally excl. minors
nowhere dense
trees
bd tree-‐width
apex minor freebd genus
MSO (Courcelle 90)
FO FPT (Seese 96)
FO FPT (Frick, Grohe 01)
FO FPT+PTAS (Flum, Frick, Grohe 01)(Dawar, Gr, K., Schw 06)
FO (Dawar, Grohe, K. 07)
FO FPT + Enum (Dvorak, Kral, Thomas 11) (Kazana, Segoufin 12)
FO FPT (Grohe, K., Seibertz 13+)
FO FPT (Frick, Grohe 01)
Nowhere Dense Classes of Graphs
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�27
ObservaCon. The classes of graphs studied so far exhibit very different properQes. But they are all relaQvely sparse, i.e. a low number of edges compared to the number of verQces. !!!!
Nowhere Dense Classes of Graphs
Algorithmic(Applications(of(Sparse(GraphsStephan(Kreutzer
Title(Text
�15
Algorithmic(Graph(Structure(Theory
all)graph)classes
bd)expansion
planar
bd)degree
bd)local)tree9w.
excluded)minors
locally)excl.)minors
nowhere)dense
treesbd)tree9width
apex)minor)freebd)genus
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
QuesCon. What are sparse graphs or sparse classes of graphs? !A1empt 1. Bounded average degree Study classes of graphs G where for some constant d. !!!!!!!Property 1. A sparse class of graphs should be preserved by taking subgraphs.
�28
What are Sparse Casses of Graphs?
|E(G)||V (G)| d
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
QuesCon. What are sparse graphs or sparse classes of graphs? !A1empt 1. Bounded average degree Study classes of graphs G where for some constant d. !!!!!!!Property 1. A sparse class of graphs should be preserved by taking subgraphs.
�28
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
Property 1. A sparse class of graphs should be preserved by taking subgraphs. !A1empt 2. Bounded degeneracy A graph is d-‐degenerate if every subgraph H⊆G contains a vertex of degree ≤ d. !!!!!!!Property 2. A sparse class of graphs should be preserved by “undoing” subdivisions of bounded length. !Nowhere dense classes of graphs. Exactly the classes with Property 1 and 2.
�29
What are Sparse Casses of Graphs?
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
Property 1. A sparse class of graphs should be preserved by taking subgraphs. !A1empt 2. Bounded degeneracy A graph is d-‐degenerate if every subgraph H⊆G contains a vertex of degree ≤ d. !!!!!!!Property 2. A sparse class of graphs should be preserved by “undoing” subdivisions of bounded length. !Nowhere dense classes of graphs. Exactly the classes with Property 1 and 2.
�29
What are Sparse Casses of Graphs?
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
Property 1. A sparse class of graphs should be preserved by taking subgraphs. !A1empt 2. Bounded degeneracy A graph is d-‐degenerate if every subgraph H⊆G contains a vertex of degree ≤ d. !!!!!!!Property 2. A sparse class of graphs should be preserved by “undoing” subdivisions of bounded length. !Nowhere dense classes of graphs. Exactly the classes with Property 1 and 2.
�29
What are Sparse Casses of Graphs?
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�30
DefiniCon. Let H be a graph. 1. A subdivision of H is a graph obtained from H by replacing edges by
pairwise vertex disjoint paths. 2. H is a topological minor of G, , if a subdivision of H is isomorphic
to a subgraph of G. !!! An r-‐subdivision of H is a graph obtained from H by replacing edges by pairwise vertex disjoint paths of length at most r. ! H is an r-‐shallow topological minor, , of G if an r-‐subdivision of H is isomorphic to a subgraph of G. !
Topological Minors
H � G
H �r G
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�30
DefiniCon. Let H be a graph. 1. A subdivision of H is a graph obtained from H by replacing edges by
pairwise vertex disjoint paths. 2. H is a topological minor of G, , if a subdivision of H is isomorphic
to a subgraph of G. !!! An r-‐subdivision of H is a graph obtained from H by replacing edges by pairwise vertex disjoint paths of length at most r. ! H is an r-‐shallow topological minor, , of G if an r-‐subdivision of H is isomorphic to a subgraph of G. !
Topological Minors
H � G
H �r G
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�31
Nowhere Dense Classes of GraphsIntroduction Algorithmic Meta-Theorems Nowhere Dense Classes of Graphs Conclusion
Introduction Definition FO Model-Checking
Nowhere Dense Classes of Graphs
Definition. (Nešetril, Ossona de Mendez)
A class C of graphs is nowhere dense if for every r ≥ 1 there is a numberf (r) such that Kf (r) ≺r G for all G ∈ C.
If the function f : r → f (r) is computable then we call C effectivelynowhere dense.
Examples.
• Graph classes excluding a fixed minor such as planar graphs orgraph classes of bounded tree-width.
• Graph classes of bounded local tree-width or locally excluding aminor such as graph classes of bounded degree.
• Classes of bounded expansion.
Non-Examples. The class of graphs of average degree ≤ 2 is not nowheredense.
Stephan Kreutzer Nohere Dense Classes of Graphs 55/69
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�31
Nowhere Dense Classes of GraphsIntroduction Algorithmic Meta-Theorems Nowhere Dense Classes of Graphs Conclusion
Introduction Definition FO Model-Checking
Nowhere Dense Classes of Graphs
Definition. (Nešetril, Ossona de Mendez)
A class C of graphs is nowhere dense if for every r ≥ 1 there is a numberf (r) such that Kf (r) ≺r G for all G ∈ C.
If the function f : r → f (r) is computable then we call C effectivelynowhere dense.
Examples.
• Graph classes excluding a fixed minor such as planar graphs orgraph classes of bounded tree-width.
• Graph classes of bounded local tree-width or locally excluding aminor such as graph classes of bounded degree.
• Classes of bounded expansion.
Non-Examples. The class of graphs of average degree ≤ 2 is not nowheredense.
Stephan Kreutzer Nohere Dense Classes of Graphs 55/69
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�32
Equivalent DefinitionIntroduction Algorithmic Meta-Theorems Nowhere Dense Classes of Graphs Conclusion
Introduction Definition FO Model-Checking
Nowhere Dense Classes of Graphs
Nowhere dense classes of graphs have many equivalentcharacterisations, making it a very robust and natural concept.
Equivalent characterisation. (Nešetril, Ossona de Mendez)
A class C is nowhere dense if, and only if,
limr ,n→∞
max
!
log |E(H)|
log |V (H)|: G ∈ C, |G| = n and H ≼r G
"
≤ 1.
Theorem. (Nešetril, Ossona de Mendez)
For every class C
limr ,n→∞
max
!
log |E(H)|
log |V (H)|: G ∈ C, |G| = n and H ≼r G
"
∈ {0,1,2}
Stephan Kreutzer Nohere Dense Classes of Graphs 56/69
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�32
Equivalent DefinitionIntroduction Algorithmic Meta-Theorems Nowhere Dense Classes of Graphs Conclusion
Introduction Definition FO Model-Checking
Nowhere Dense Classes of Graphs
Nowhere dense classes of graphs have many equivalentcharacterisations, making it a very robust and natural concept.
Equivalent characterisation. (Nešetril, Ossona de Mendez)
A class C is nowhere dense if, and only if,
limr ,n→∞
max
!
log |E(H)|
log |V (H)|: G ∈ C, |G| = n and H ≼r G
"
≤ 1.
Theorem. (Nešetril, Ossona de Mendez)
For every class C
limr ,n→∞
max
!
log |E(H)|
log |V (H)|: G ∈ C, |G| = n and H ≼r G
"
∈ {0,1,2}
Stephan Kreutzer Nohere Dense Classes of Graphs 56/69
A Game Characterisation of Sparseness
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�34
(l, m, r)-‐Splijer Game on G Graph G, parameters l,m,r>0 Players Connector and Splijer !Ini'alisa'on:
Round (i+1): 1. C chooses 2. S chooses
of size at most m !We set !S wins if . Otherwise the game conQnues.
If S has not won ater l rounds, then C wins.
The Splitter Game
Gi+1 := Gi[NGir (vi+1) \Wi+1]
Gi+1 = ;
G0 := G
vi+1 2 V (Gi)
Wi+1 ✓ NGir (vi+1)
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�34
(l, m, r)-‐Splijer Game on G Graph G, parameters l,m,r>0 Players Connector and Splijer !Ini'alisa'on:
Round (i+1): 1. C chooses 2. S chooses
of size at most m !We set !S wins if . Otherwise the game conQnues.
If S has not won ater l rounds, then C wins.
The Splitter Game
Gi+1 := Gi[NGir (vi+1) \Wi+1]
Gi+1 = ;
G0 := G
vi+1 2 V (Gi)
Wi+1 ✓ NGir (vi+1)
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�34
(l, m, r)-‐Splijer Game on G Graph G, parameters l,m,r>0 Players Connector and Splijer !Ini'alisa'on:
Round (i+1): 1. C chooses 2. S chooses
of size at most m !We set !S wins if . Otherwise the game conQnues.
If S has not won ater l rounds, then C wins.
The Splitter Game
Gi+1 := Gi[NGir (vi+1) \Wi+1]
Gi+1 = ;
G0 := G
vi+1 2 V (Gi)
Wi+1 ✓ NGir (vi+1)
v1
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�34
(l, m, r)-‐Splijer Game on G Graph G, parameters l,m,r>0 Players Connector and Splijer !Ini'alisa'on:
Round (i+1): 1. C chooses 2. S chooses
of size at most m !We set !S wins if . Otherwise the game conQnues.
If S has not won ater l rounds, then C wins.
The Splitter Game
Gi+1 := Gi[NGir (vi+1) \Wi+1]
Gi+1 = ;
G0 := G
vi+1 2 V (Gi)
Wi+1 ✓ NGir (vi+1)
v1
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�34
(l, m, r)-‐Splijer Game on G Graph G, parameters l,m,r>0 Players Connector and Splijer !Ini'alisa'on:
Round (i+1): 1. C chooses 2. S chooses
of size at most m !We set !S wins if . Otherwise the game conQnues.
If S has not won ater l rounds, then C wins.
The Splitter Game
Gi+1 := Gi[NGir (vi+1) \Wi+1]
Gi+1 = ;
G0 := G
vi+1 2 V (Gi)
Wi+1 ✓ NGir (vi+1)
v1
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�34
(l, m, r)-‐Splijer Game on G Graph G, parameters l,m,r>0 Players Connector and Splijer !Ini'alisa'on:
Round (i+1): 1. C chooses 2. S chooses
of size at most m !We set !S wins if . Otherwise the game conQnues.
If S has not won ater l rounds, then C wins.
The Splitter Game
Gi+1 := Gi[NGir (vi+1) \Wi+1]
Gi+1 = ;
G0 := G
vi+1 2 V (Gi)
Wi+1 ✓ NGir (vi+1)
v1
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�34
(l, m, r)-‐Splijer Game on G Graph G, parameters l,m,r>0 Players Connector and Splijer !Ini'alisa'on:
Round (i+1): 1. C chooses 2. S chooses
of size at most m !We set !S wins if . Otherwise the game conQnues.
If S has not won ater l rounds, then C wins.
The Splitter Game
Gi+1 := Gi[NGir (vi+1) \Wi+1]
Gi+1 = ;
G0 := G
vi+1 2 V (Gi)
Wi+1 ✓ NGir (vi+1)
v1
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�34
(l, m, r)-‐Splijer Game on G Graph G, parameters l,m,r>0 Players Connector and Splijer !Ini'alisa'on:
Round (i+1): 1. C chooses 2. S chooses
of size at most m !We set !S wins if . Otherwise the game conQnues.
If S has not won ater l rounds, then C wins.
The Splitter Game
Gi+1 := Gi[NGir (vi+1) \Wi+1]
Gi+1 = ;
G0 := G
vi+1 2 V (Gi)
Wi+1 ✓ NGir (vi+1)
v1
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�35
An ExampleLet G be the following graph and let r = 1 m = 2 l = 3
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�36
An ExampleLet G be the following graph and let r = 1 m = 2 l = 3
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�37
An ExampleLet G be the following graph and let r = 1 m = 2 l = 3
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�38
An ExampleLet G be the following graph and let r = 1 m = 2 l = 3
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�39
An ExampleLet G be the following graph and let r = 1 m = 2 l = 3
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�40
An ExampleLet G be the following graph and let r = 1 m = 2 l = 3
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�41
An ExampleLet G be the following graph and let r = 1 m = 2 l = 3
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�42
An ExampleLet G be the following graph and let r = 1 m = 2 l = 3
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�43
An ExampleLet G be the following graph and let r = 1 m = 2 l = 3
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�44
An ExampleLet G be the following graph and let r = 1 m = 2 l = 3
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�45
An ExampleLet G be the following graph and let r = 1 m = 2 l = 3
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�46
An ExampleLet G be the following graph and let r = 1 m = 2 l = 3
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�47
!!Theorem. (Grohe, K., Siebertz 2013+) A class C is nowhere dense if, and only if, for all r there are l, m such that Splijer wins the (l, m, r)-‐splijer game on every graph in C. !!Theorem. (Grohe, K., Siebertz 2013+) Let C be a nowhere dense class of graphs. The Coloured Distance d Independent Set can be solved on C in Qme f(k+d)n1+𝜀, for every 𝜀>0.
A Game Characterisation
Coloured Distance d Independent Set Input: Graph G∈C numbers k, d, set R ⊆ V(G) Problem: Find a set S ⊆ R with |S| ≥ k st dist(u,v) > d for all v≠u∊V(G).
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�47
!!Theorem. (Grohe, K., Siebertz 2013+) A class C is nowhere dense if, and only if, for all r there are l, m such that Splijer wins the (l, m, r)-‐splijer game on every graph in C. !!Theorem. (Grohe, K., Siebertz 2013+) Let C be a nowhere dense class of graphs. The Coloured Distance d Independent Set can be solved on C in Qme f(k+d)n1+𝜀, for every 𝜀>0.
A Game Characterisation
Coloured Distance d Independent Set Input: Graph G∈C numbers k, d, set R ⊆ V(G) Problem: Find a set S ⊆ R with |S| ≥ k st dist(u,v) > d for all v≠u∊V(G).
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�48
Coloured Distance d Independent Set
Coloured Distance d Independent Set Input: Graph G∈C numbers k, d, set R ⊆ V(G) Problem: Find a set S ⊆ R with |S| ≥ k such that dist(u,v) > d for all v ≠ u∈V(G).
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�48
Coloured Distance d Independent Set
Coloured Distance d Independent Set Input: Graph G∈C numbers k, d, set R ⊆ V(G) Problem: Find a set S ⊆ R with |S| ≥ k such that dist(u,v) > d for all v ≠ u∈V(G).
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�48
Coloured Distance d Independent Set
Coloured Distance d Independent Set Input: Graph G∈C numbers k, d, set R ⊆ V(G) Problem: Find a set S ⊆ R with |S| ≥ k such that dist(u,v) > d for all v ≠ u∈V(G).
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�48
Coloured Distance d Independent Set
Coloured Distance d Independent Set Input: Graph G∈C numbers k, d, set R ⊆ V(G) Problem: Find a set S ⊆ R with |S| ≥ k such that dist(u,v) > d for all v ≠ u∈V(G).
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
Coloured Distance d Independent Set Input: Graph G∈C numbers k, d, set R ⊆ V(G) Problem: Find a set S ⊆ R with |S| ≥ k such that dist(u,v) > d for all v ≠ u∈V(G).
�49
Coloured Distance d Independent Set
ObservaCon. This idenQfies a set of size k or all red nodes are in the d neighbourhood of S.
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
Coloured Distance d Independent Set Input: Graph G∈C numbers k, d, set R ⊆ V(G) Problem: Find a set S ⊆ R with |S| ≥ k such that dist(u,v) > d for all v ≠ u∈V(G).
�49
Coloured Distance d Independent Set
ObservaCon. This idenQfies a set of size k or all red nodes are in the d neighbourhood of S.
So we need some way of controlling neighbourhoods.
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�50
Coloured Distance d Independent Set
For simplicity, assume the neighbourhoods touch, i.e. is connected. Then there is such that . Take this as Connector’s move in the Splijer game. Let W ⊆ V(G) be Splijer’s response.
v1 2 V (G)
G[NGd (S)]
G[NGd (S)] ✓ NG
k·d(v1)
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�50
Coloured Distance d Independent Set
For simplicity, assume the neighbourhoods touch, i.e. is connected. Then there is such that . Take this as Connector’s move in the Splijer game. Let W ⊆ V(G) be Splijer’s response.
v1 2 V (G)
G[NGd (S)]
G[NGd (S)] ✓ NG
k·d(v1)
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�50
Coloured Distance d Independent Set
For simplicity, assume the neighbourhoods touch, i.e. is connected. Then there is such that . Take this as Connector’s move in the Splijer game. Let W ⊆ V(G) be Splijer’s response.
v1 2 V (G)
G[NGd (S)]
G[NGd (S)] ✓ NG
k·d(v1)
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�51
Coloured Distance d Independent Set
For simplicity, assume the neighbourhoods touch, i.e. is connected. Then there is such that . Take this as Connector’s move in the Splijer game. Let W ⊆ V(G) be Splijer’s response.
We conQnue recursively in the new graph . This process stops ater l(k·∙d) steps.
G[NGr (S)]
G[NGr (S)] ✓ NG
k·d(v1)v1 2 V (G)
G1 := NGk·d(v) \W
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�51
Coloured Distance d Independent Set
For simplicity, assume the neighbourhoods touch, i.e. is connected. Then there is such that . Take this as Connector’s move in the Splijer game. Let W ⊆ V(G) be Splijer’s response.
We conQnue recursively in the new graph . This process stops ater l(k·∙d) steps.
G[NGr (S)]
G[NGr (S)] ✓ NG
k·d(v1)v1 2 V (G)
G1 := NGk·d(v) \W
Theorem. (Grohe, K., Siebertz 2013+) Let C be a nowhere dense class of graphs. The Coloured d-‐DIS can be solved on C in Qme f(k+d)n1+𝜀, for every 𝜀>0.
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�52
A Bounded Depth Search Tree
G
G1 := Nr(v1)
G2 := Nr(v2)
G3 := Nr(v3)
Gn := Nr(vn)
NG2\W(w1) NG2\W(w2) NG2\W(w3)
The enQre search tree has size l·∙nO(l)
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�52
A Bounded Depth Search Tree
G
G1 := Nr(v1)
G2 := Nr(v2)
G3 := Nr(v3)
Gn := Nr(vn)
NG2\W(w1) NG2\W(w2) NG2\W(w3)
} l(r)
The enQre search tree has size l·∙nO(l)
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�52
A Bounded Depth Search Tree
G
G1 := Nr(v1)
G2 := Nr(v2)
G3 := Nr(v3)
Gn := Nr(vn)
NG2\W(w1) NG2\W(w2) NG2\W(w3)
} l(r)
The enQre search tree has size l·∙nO(l)
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�53
Neighbourhood Covers
Defini&on. Let r ∈ ℕ. An r-‐neighbourhood cover 𝒩 of a graph G is a set of connected subgraphs of G called clusters such that for every v ∈ V(G) there is some N ∈ 𝒩 with Nr ⊆ N. !The radius of 𝒩 is the maximum radius of any of its cluster. !The degree of 𝒩 is max
v2V (G)
��{N 2 N : v 2 N}��
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�53
Neighbourhood Covers
Defini&on. Let r ∈ ℕ. An r-‐neighbourhood cover 𝒩 of a graph G is a set of connected subgraphs of G called clusters such that for every v ∈ V(G) there is some N ∈ 𝒩 with Nr ⊆ N. !The radius of 𝒩 is the maximum radius of any of its cluster. !The degree of 𝒩 is max
v2V (G)
��{N 2 N : v 2 N}��
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�53
Neighbourhood Covers
Defini&on. Let r ∈ ℕ. An r-‐neighbourhood cover 𝒩 of a graph G is a set of connected subgraphs of G called clusters such that for every v ∈ V(G) there is some N ∈ 𝒩 with Nr ⊆ N. !The radius of 𝒩 is the maximum radius of any of its cluster. !The degree of 𝒩 is max
v2V (G)
��{N 2 N : v 2 N}��
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�54
Neighbourhood Covers
Neighbourhood covers are studied for instance for • distributed algorithms • graph spanners
Goal. Minimise the radius of the cover. Minimise the degree.
Defini&on. Let r ∈ ℕ. An r-‐neighbourhood cover 𝒩 of a graph G is a set of connected subgraphs of G called clusters such that for every v ∈ V(G) there is some N ∈ 𝒩 with Nr ⊆ N. !The radius of 𝒩 is the maximum radius of any of its cluster. !The degree of 𝒩 is max
v2V (G)
��{N 2 N : v 2 N}��
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�55
Neighbourhood Covers
Theorem. (Grohe, K., Siebertz 13+) Let C be a nowhere dense class of graphs. For every radius r>0 and 𝜀>0 there is an n0 such that for every G ∈ C with |V(G)| = n > n0 we can compute an
r-‐neighbourhood cover 𝒩 of G with radius 2r and degree n𝜀 in Qme O(n1+𝜀). !The radius of 𝒩 is the maximum radius of any of its cluster. !The degree of 𝒩 is !
Note. For classes C excluding a fixed minor or of bounded expansion we get constant degree.
max
v2V (G)
��{N 2 N : v 2 N}��
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�56
A Bounded Size Search Tree
G
N(v1)G2 := N(v2)
N(v3) N(vn)
NG2(w1) NG2(w2) NG2(w3)
} l(r)
If we replace neighbourhoods by clusters of a sparse neighbourhood cover, then the enQre search tree only has size l·∙n1+𝜀
Further Results on Nowhere Dense Graphs
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�58
Nowhere dense classes of graphs have many equivalent characterisaQons, making it a very robust and natural concept. !A class C is nowhere dense if, and only if, 1. for every r there is a graph not contained as r-‐shallow topological minor
2. the edge density of every r-‐shallow minor is bounded by no(1)
3. Splijer wins the Splijer-‐Game
4. for every k, every graph G ∈ C can be coloured by colours no(1) so that
every k colour classes induce a subgraph of tree-‐width ≤ k.
5. C is uniformly quasi-‐wide
6. the weak colouring number is bounded.
Nowhere Dense Classes of Graphs
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�59
Theorem. (Nesetril, Ossona de Mendez)
A class C of graphs is nowhere dense if, and only if, for every k and every 𝜀>0, every (large enough) graph G ∈ C can be coloured by n𝜀 colours so that every k colour classes induce a subgraph of tree-‐width ≤ k. These colourings can be computed in Qme O(n1+𝜀).
!
!
!
!Corollary. Fix a graph H. Let C be a nowhere dense class of graphs. Then we can decide H ⊆ G for all G∈C in Qme O(n1+𝜀).
Low Tree-‐Width Colourings
G
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�59
Theorem. (Nesetril, Ossona de Mendez)
A class C of graphs is nowhere dense if, and only if, for every k and every 𝜀>0, every (large enough) graph G ∈ C can be coloured by n𝜀 colours so that every k colour classes induce a subgraph of tree-‐width ≤ k. These colourings can be computed in Qme O(n1+𝜀).
!
!
!
!Corollary. Fix a graph H. Let C be a nowhere dense class of graphs. Then we can decide H ⊆ G for all G∈C in Qme O(n1+𝜀).
Low Tree-‐Width Colourings
G
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�59
Theorem. (Nesetril, Ossona de Mendez)
A class C of graphs is nowhere dense if, and only if, for every k and every 𝜀>0, every (large enough) graph G ∈ C can be coloured by n𝜀 colours so that every k colour classes induce a subgraph of tree-‐width ≤ k. These colourings can be computed in Qme O(n1+𝜀).
!
!
!
!Corollary. Fix a graph H. Let C be a nowhere dense class of graphs. Then we can decide H ⊆ G for all G∈C in Qme O(n1+𝜀).
Low Tree-‐Width Colourings
G
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�59
Theorem. (Nesetril, Ossona de Mendez)
A class C of graphs is nowhere dense if, and only if, for every k and every 𝜀>0, every (large enough) graph G ∈ C can be coloured by n𝜀 colours so that every k colour classes induce a subgraph of tree-‐width ≤ k. These colourings can be computed in Qme O(n1+𝜀).
!
!
!
!Corollary. Fix a graph H. Let C be a nowhere dense class of graphs. Then we can decide H ⊆ G for all G∈C in Qme O(n1+𝜀).
Low Tree-‐Width Colourings
G
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�60
Theorem. (Nesetril, Ossona de Mendez) A class C of graphs is nowhere dense if, and only if, for every radius r>0 there is an s=s(r) such that for every m>0 there is a N=N(r,m)>0 such that for every G∈C and W⊆V(G) with |W|>N there is a set S⊆V(G) of size |S|≤s and a set A⊆W of size |A|=m such that dist(G-‐S)(u,v) > r for all u≠v∈A.
Uniformly Quasi-‐Wideness
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�60
Theorem. (Nesetril, Ossona de Mendez) A class C of graphs is nowhere dense if, and only if, for every radius r>0 there is an s=s(r) such that for every m>0 there is a N=N(r,m)>0 such that for every G∈C and W⊆V(G) with |W|>N there is a set S⊆V(G) of size |S|≤s and a set A⊆W of size |A|=m such that dist(G-‐S)(u,v) > r for all u≠v∈A.
Uniformly Quasi-‐Wideness
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�60
Theorem. (Nesetril, Ossona de Mendez) A class C of graphs is nowhere dense if, and only if, for every radius r>0 there is an s=s(r) such that for every m>0 there is a N=N(r,m)>0 such that for every G∈C and W⊆V(G) with |W|>N there is a set S⊆V(G) of size |S|≤s and a set A⊆W of size |A|=m such that dist(G-‐S)(u,v) > r for all u≠v∈A.
Uniformly Quasi-‐Wideness
S
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�60
Theorem. (Nesetril, Ossona de Mendez) A class C of graphs is nowhere dense if, and only if, for every radius r>0 there is an s=s(r) such that for every m>0 there is a N=N(r,m)>0 such that for every G∈C and W⊆V(G) with |W|>N there is a set S⊆V(G) of size |S|≤s and a set A⊆W of size |A|=m such that dist(G-‐S)(u,v) > r for all u≠v∈A.
Uniformly Quasi-‐Wideness
S
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�61
Subgraph Isomorphism (-‐homomorphism). (Nesetril, Ossonda de Mendez 07) This is nearly linear Qme for any fixed template H. (This and many similar problems also follow from our result on MC(FO, C).) !Fixed-‐Parameter Algorithms. (Dawar, K. FSTTCS 09) DominaQng Set, Independent Set, … are FPT on nowhere dense classes of graphs. !Polynomial Kernels. (Gajarsky et al. 13) Several problems (Longest Path) have polynomial kernels. On bd. expansion classes they become linear. !Approxima&on. (Dvorak 13) DominaQon Set has a constant factor approximaQon algorithm on bd. expansion classes.
Further Algorithmic Results
Back to Algorithmic Meta-‐Theorems
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�63
!!
Main result. (Grohe, K., Siebertz 13+) Every problem definable in first-‐order logic can be decided in KmeO(n1+𝜀) , for every 𝜀>0, on any class of graphs that is nowhere dense. !
This result is opQmal in the following sense. ! Theorem. (K. 09, Dvorak, Kral, Thomas ’11) If a class C closed under subgraphs is not nowhere dense, then FO-‐model-‐ checking is not fpt (unless AW[∗] = FPT).
Algorithmic Meta-‐Theorems
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�64
Main result. (Grohe, K., Siebertz 13+) Every problem definable in first-‐order logic can be decided in KmeO(n1+𝜀) , for every 𝜀>0, on any class of graphs that is nowhere dense.
Proof. Given: a graph G and a first-‐order formula 𝜑. Decide whether 𝜑 is true in G. ! Show: this can be reduced to a combinaKon of Coloured Distance Independent Set Problem EvaluaKng first-‐order formulas in r-‐neighbourhoods of G.
Algorithmic Meta-‐Theorems
Does N(v) ⊧ 𝜓? If yes, mark red.
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�65
Main result. (Grohe, K., Siebertz 13+) Every problem definable in first-‐order logic can be decided in KmeO(n1+𝜀) , for every 𝜀>0, on any class of graphs that is nowhere dense.
Proof. Given: a graph G and a first-‐order formula 𝜑. Decide whether 𝜑 is true in G. ! Show: this can be reduced to a combinaKon of Coloured Distance Independent Set Problem EvaluaKng first-‐order formulas in r-‐neighbourhoods of G.
Algorithmic Meta-‐Theorems
Does N(v) ⊧ 𝜓? If yes, mark red.
Find an r-‐independ. set of size k.
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�66
Algorithmic Graph Structure Theory
all graph classes
bd expansion
planar
bd degree
bd local tree-‐w.
excluded minors
locally excl. minors
nowhere dense
trees
bd tree-‐width
apex minor freeMSO
(Courcelle 90)
FO (Seese 96)
FO (Frick, Grohe 01)
FO (Flum, Frick, Grohe 01)(Dawar, Gr, K., Schw 06)
FO (Dawar, Grohe, K. 07)
FO (Dvorak, Kral, Thomas 11) (Kazana, Segoufin 12)
FO is FPT
FO (Frick, Grohe 01)
Conclusion
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�68
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 dominaQon problems explore how far the tractability barrier can be pushed. !
Nowhere dense classes seem to be a natural limit for many techniques.
The study of nowhere dense classes has only just begun.
We need more tools and techniques for analysing these classes and designing good algorithms.
If you are interested: a (hopefully) good start might be the companion paper in the proceedings. There is a book: Sparsity by Nesetril and Ossona de Mendez.
Conclusion
Algorithmic Applications of Sparse GraphsStephan Kreutzer
Title Text
�68
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 dominaQon problems explore how far the tractability barrier can be pushed. !
Nowhere dense classes seem to be a natural limit for many techniques.
The study of nowhere dense classes has only just begun.
We need more tools and techniques for analysing these classes and designing good algorithms.
If you are interested: a (hopefully) good start might be the companion paper in the proceedings. There is a book: Sparsity by Nesetril and Ossona de Mendez.
Conclusion