Algorithmic(Applications(...

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. !!!!!!

Choosing a Programme Committee

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. !!!!!!

Title Text!!!!!!!!

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).

Title Text!!!!!!!!

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).

Title Text!!!!!!!!

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.

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. !!!!!!

Judging the Invited Talk

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. !!!!!!

Title Text!!!!!!!

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.

Title Text!!!!!!!

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.

Title Text!!!!!!!

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.

Title Text!!!!!!!

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.

Title Text

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

Title Text

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

Title Text

!!!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

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.

Algorithmic Graph Structure Theory

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.

Title Text

all graph classes

bd expansion

planar

bd degree

bd local tree-‐w.

excluded minors

locally excl. minors

nowhere dense

bd tree-‐width

apex minor freebd genus

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

Efficiently?

2O(pk) · n

2O(pk) · n 2O(k) · n2 f(k) · ncoror

Efficiently?

2O(pk) · n

Efficiently?

2O(pk) · n

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

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).

Title Text

all graph classes

bd expansion

planar

bd degree

bd local tree-‐w.

excluded minors

nowhere dense

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

Title Text

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(Applications(of(Sparse(GraphsStephan(Kreutzer

Title(Text

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.

Title(Text

all)graph)classes

bd)expansion

planar

bd)degree

bd)local)tree9w.

excluded)minors

nowhere)dense

treesbd)tree9width

apex)minor)free

Meta?Kernels

topological!methods

dynamic!programming

bd)genus

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.

Title(Text

all)graph)classes

bd)expansion

planar

bd)degree

bd)local)tree9w.

excluded)minors

nowhere)dense

treesbd)tree9width

apex)minor)free

Meta?Kernels

topological!methods

dynamic!programming

bd)genus

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.

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.

!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 !!!!!

MC(L, C) Input: Graph G∈C, Formula 𝜑∈L Parameter: |𝜑| (or |𝜑| + tw(G)) Problem: Decide G ⊧ 𝜑?

!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 !!!!!

MC(L, C) Input: Graph G∈C, Formula 𝜑∈L Parameter: |𝜑| (or |𝜑| + tw(G)) Problem: Decide G ⊧ 𝜑?

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

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

Title Text

!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.

Title Text

!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.

Title Text

all graph classes

bd expansion

planar

bd degree

bd local tree-‐w.

excluded minors

nowhere dense

bd tree-‐width

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)

Title Text

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).

Title Text

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).

Title Text

all graph classes

bd expansion

planar

bd degree

bd local tree-‐w.

excluded minors

nowhere dense

bd tree-‐width

MSO (Courcelle 90)

FO FPT (Seese 96)

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+)

Nowhere Dense Classes of Graphs

Title Text

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. !!!!

Title(Text

all)graph)classes

bd)expansion

planar

bd)degree

bd)local)tree9w.

excluded)minors

nowhere)dense

treesbd)tree9width

apex)minor)freebd)genus

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.

What are Sparse Casses of Graphs?

|E(G)||V (G)| d

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.

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.

Title Text

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

Title Text

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

Title Text

Nowhere Dense Classes of GraphsIntroduction Algorithmic Meta-Theorems Nowhere Dense Classes of Graphs Conclusion

Introduction Definition FO Model-Checking

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

Title Text

Nowhere Dense Classes of GraphsIntroduction Algorithmic Meta-Theorems Nowhere Dense Classes of Graphs Conclusion

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.

Title Text

Equivalent DefinitionIntroduction Algorithmic Meta-Theorems Nowhere Dense Classes of Graphs Conclusion

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→∞

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→∞

log |E(H)|

∈ {0,1,2}

Title Text

Equivalent DefinitionIntroduction Algorithmic Meta-Theorems Nowhere Dense Classes of Graphs Conclusion

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→∞

log |E(H)|

≤ 1.

Theorem. (Nešetril, Ossona de Mendez)

For every class C

limr ,n→∞

log |E(H)|

∈ {0,1,2}

A Game Characterisation of Sparseness

Title Text

(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)

Title Text

The Splitter Game

Gi+1 = ;

G0 := G

vi+1 2 V (Gi)

Title Text

The Splitter Game

Gi+1 = ;

G0 := G

vi+1 2 V (Gi)

Title Text

The Splitter Game

Gi+1 = ;

G0 := G

vi+1 2 V (Gi)

Title Text

The Splitter Game

Gi+1 = ;

G0 := G

vi+1 2 V (Gi)

Title Text

The Splitter Game

Gi+1 = ;

G0 := G

vi+1 2 V (Gi)

Title Text

The Splitter Game

Gi+1 = ;

G0 := G

vi+1 2 V (Gi)

Title Text

The Splitter Game

Gi+1 = ;

G0 := G

vi+1 2 V (Gi)

Title Text

An ExampleLet G be the following graph and let r = 1 m = 2 l = 3

Title Text

!!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).

Title Text

!!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).

Title Text

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).

Title Text

ObservaCon. This idenQfies a set of size k or all red nodes are in the d neighbourhood of S.

Title Text

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.

Title Text

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)

Title Text

v1 2 V (G)

G[NGd (S)]

G[NGd (S)] ✓ NG

k·d(v1)

Title Text

v1 2 V (G)

G[NGd (S)]

G[NGd (S)] ✓ NG

k·d(v1)

Title Text

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

Title Text

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.

Title Text

A Bounded Depth Search Tree

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)

Title Text

G1 := Nr(v1)

G2 := Nr(v2)

G3 := Nr(v3)

Gn := Nr(vn)

} l(r)

Title Text

G1 := Nr(v1)

G2 := Nr(v2)

G3 := Nr(v3)

Gn := Nr(vn)

} l(r)

Title Text

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}��

Title Text

v2V (G)

��{N 2 N : v 2 N}��

Title Text

v2V (G)

��{N 2 N : v 2 N}��

Title Text

Neighbourhood covers are studied for instance for • distributed algorithms • graph spanners

Goal. Minimise the radius of the cover. Minimise the degree.

v2V (G)

��{N 2 N : v 2 N}��

Title Text

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.

v2V (G)

��{N 2 N : v 2 N}��

Title Text

A Bounded Size Search Tree

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

Title Text

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.

Title Text

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

Title Text

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

Title Text

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

Title Text

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).

Title Text

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.

Does N(v) ⊧ 𝜓? If yes, mark red.

Title Text

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.

Does N(v) ⊧ 𝜓? If yes, mark red.

Find an r-‐independ. set of size k.

Title Text

all graph classes

bd expansion

planar

bd degree

bd local tree-‐w.

excluded minors

nowhere dense

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

Title Text

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

Title Text

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(...

Documents

Transcript of Algorithmic(Applications(...