Introduction + Bidimensionality Theory

Ignasi Sau

CNRS, LIRMM, Montpellier (France)

Many thanks to Dimitrios M. Thilikos!!

14`emesJCALM. October 2013. UPC, Barcelona, Catalunya

1/57 Outline of the talk

1 Introduction, part II Treewidth Dynamic programming on tree decompositions Structure of H-minor-free graphs Some algorithmic issues A few words on other containment relations

2 Bidimensionality Some ingredients An illustrative example Meta-algorithms Further extensions

2/57 Next section is...

1 Introduction, part II Treewidth Dynamic programming on tree decompositions Structure of H-minor-free graphs Some algorithmic issues A few words on other containment relations

2 Bidimensionality Some ingredients An illustrative example Meta-algorithms Further extensions

3/57 Single-exponential FPT algorithm:2 O(k) · nO(1)

Subexponential FPT algorithm:2 o(k) · nO(1)

Parameterized complexity in one slide

Idea given an NP-hard problem, ﬁx one parameter of the input to see if the problem gets more “tractable”. Example: the size of a Vertex Cover.

Given a (NP-hard) problem with input of size n and a parameter k, a ﬁxed-parameter tractable( FPT) algorithm runs in

f (k) · nO(1), for some function f .

Examples: k-Vertex Cover, k-Longest Path.

4/57 Subexponential FPT algorithm:2 o(k) · nO(1)

Parameterized complexity in one slide

Idea given an NP-hard problem, ﬁx one parameter of the input to see if the problem gets more “tractable”. Example: the size of a Vertex Cover.

Given a (NP-hard) problem with input of size n and a parameter k, a ﬁxed-parameter tractable( FPT) algorithm runs in

f (k) · nO(1), for some function f .

Examples: k-Vertex Cover, k-Longest Path.

Single-exponential FPT algorithm:2 O(k) · nO(1)

4/57 Parameterized complexity in one slide

Idea given an NP-hard problem, ﬁx one parameter of the input to see if the problem gets more “tractable”. Example: the size of a Vertex Cover.

Given a (NP-hard) problem with input of size n and a parameter k, a ﬁxed-parameter tractable( FPT) algorithm runs in

f (k) · nO(1), for some function f .

Examples: k-Vertex Cover, k-Longest Path.

Single-exponential FPT algorithm:2 O(k) · nO(1)

Subexponential FPT algorithm:2 o(k) · nO(1)

4/57 Next subsection is...

1 Introduction, part II Treewidth Dynamic programming on tree decompositions Structure of H-minor-free graphs Some algorithmic issues A few words on other containment relations

2 Bidimensionality Some ingredients An illustrative example Meta-algorithms Further extensions

5/57 Any vertex v ∈ V (G) and the endpoints of any edge e ∈ E(G) belong to some node Xt of D; and

For any v ∈ V (G), the set {t ∈ V (T ) | v ∈ Xt } is a subtree of T .

The width of a tree decomposition is max{|Xt | | t ∈ V (T )} − 1.

The treewidth of a graph G, denoted tw(G), is the minimum width over all tree decompositions of G.

Invariant that measures the topological complexity of a graph.

Tree decompositions and treewidth

A tree decomposition of a graph G is a pair D = (T , X ) such that T is a tree and X = {Xt | t ∈ V (T )} is a collection of subsets of V (G) such that:

(each Xt ∈ X corresponds to a vertex t ∈ V (T ): we call Xt node of D)

6/57 The width of a tree decomposition is max{|Xt | | t ∈ V (T )} − 1.

The treewidth of a graph G, denoted tw(G), is the minimum width over all tree decompositions of G.

Invariant that measures the topological complexity of a graph.

Tree decompositions and treewidth

A tree decomposition of a graph G is a pair D = (T , X ) such that T is a tree and X = {Xt | t ∈ V (T )} is a collection of subsets of V (G) such that:

(each Xt ∈ X corresponds to a vertex t ∈ V (T ): we call Xt node of D) Any vertex v ∈ V (G) and the endpoints of any edge e ∈ E(G) belong to some node Xt of D; and

For any v ∈ V (G), the set {t ∈ V (T ) | v ∈ Xt } is a subtree of T .

6/57 The treewidth of a graph G, denoted tw(G), is the minimum width over all tree decompositions of G.

Invariant that measures the topological complexity of a graph.

Tree decompositions and treewidth

A tree decomposition of a graph G is a pair D = (T , X ) such that T is a tree and X = {Xt | t ∈ V (T )} is a collection of subsets of V (G) such that:

(each Xt ∈ X corresponds to a vertex t ∈ V (T ): we call Xt node of D) Any vertex v ∈ V (G) and the endpoints of any edge e ∈ E(G) belong to some node Xt of D; and

For any v ∈ V (G), the set {t ∈ V (T ) | v ∈ Xt } is a subtree of T .

The width of a tree decomposition is max{|Xt | | t ∈ V (T )} − 1.

6/57 Invariant that measures the topological complexity of a graph.

Tree decompositions and treewidth

A tree decomposition of a graph G is a pair D = (T , X ) such that T is a tree and X = {Xt | t ∈ V (T )} is a collection of subsets of V (G) such that:

(each Xt ∈ X corresponds to a vertex t ∈ V (T ): we call Xt node of D) Any vertex v ∈ V (G) and the endpoints of any edge e ∈ E(G) belong to some node Xt of D; and

For any v ∈ V (G), the set {t ∈ V (T ) | v ∈ Xt } is a subtree of T .

The width of a tree decomposition is max{|Xt | | t ∈ V (T )} − 1.

The treewidth of a graph G, denoted tw(G), is the minimum width over all tree decompositions of G.

6/57 Tree decompositions and treewidth

A tree decomposition of a graph G is a pair D = (T , X ) such that T is a tree and X = {Xt | t ∈ V (T )} is a collection of subsets of V (G) such that:

(each Xt ∈ X corresponds to a vertex t ∈ V (T ): we call Xt node of D) Any vertex v ∈ V (G) and the endpoints of any edge e ∈ E(G) belong to some node Xt of D; and

For any v ∈ V (G), the set {t ∈ V (T ) | v ∈ Xt } is a subtree of T .

The width of a tree decomposition is max{|Xt | | t ∈ V (T )} − 1.

The treewidth of a graph G, denoted tw(G), is the minimum width over all tree decompositions of G.

Invariant that measures the topological complexity of a graph.

6/57 Tree decompositions: example

7/57 Tree decompositions: example

8/57 Tree decompositions: example

9/57 Tree decompositions: example

10/57 Tree decompositions: example

11/57 2 Approximation algorithms on general graphs √ O(Opt · Opt)-approximation. [Feige, Hajiaghayi, Lee. 2008]

3 (Exact) FPT algorithms on general graphs (tw 6 k?) k+2 In time O(n ). [Arnborg, Corneil, Proskurowski. 1987] O(k3) In time k · n. [Bodlaender. 1996]

4 FPT approximation algorithms on general graphs O(k) 2 4.5-approximation in time2 · n . [Amir. 2010] O(k) 5-approx. in2 · n. [Bodlaender, Drange, Dregi, Fomin, Lokshtanov, Pilipczuk. 2013]

5 Approximation algorithms on sparse graphs

3/2-approximation on planar graphs. [Seymour and Thomas. 1994] cH -approximation on H-minor-free graphs. [Demaine and Hajiaghayi. 2008] Open Compute the treewidth of a planar graph.

About computing treewidth: a global picture

1 Computing treewidth is NP-hard [Arnborg, Corneil, Proskurowski. 1987]

12/57 3 (Exact) FPT algorithms on general graphs (tw 6 k?) k+2 In time O(n ). [Arnborg, Corneil, Proskurowski. 1987] O(k3) In time k · n. [Bodlaender. 1996]

4 FPT approximation algorithms on general graphs O(k) 2 4.5-approximation in time2 · n . [Amir. 2010] O(k) 5-approx. in2 · n. [Bodlaender, Drange, Dregi, Fomin, Lokshtanov, Pilipczuk. 2013]

5 Approximation algorithms on sparse graphs

3/2-approximation on planar graphs. [Seymour and Thomas. 1994] cH -approximation on H-minor-free graphs. [Demaine and Hajiaghayi. 2008] Open Compute the treewidth of a planar graph.

About computing treewidth: a global picture

1 Computing treewidth is NP-hard [Arnborg, Corneil, Proskurowski. 1987]

2 Approximation algorithms on general graphs √ O(Opt · Opt)-approximation. [Feige, Hajiaghayi, Lee. 2008]

12/57 k+2 In time O(n ). [Arnborg, Corneil, Proskurowski. 1987] O(k3) In time k · n. [Bodlaender. 1996]

4 FPT approximation algorithms on general graphs O(k) 2 4.5-approximation in time2 · n . [Amir. 2010] O(k) 5-approx. in2 · n. [Bodlaender, Drange, Dregi, Fomin, Lokshtanov, Pilipczuk. 2013]

5 Approximation algorithms on sparse graphs

3/2-approximation on planar graphs. [Seymour and Thomas. 1994] cH -approximation on H-minor-free graphs. [Demaine and Hajiaghayi. 2008] Open Compute the treewidth of a planar graph.

About computing treewidth: a global picture

1 Computing treewidth is NP-hard [Arnborg, Corneil, Proskurowski. 1987]

2 Approximation algorithms on general graphs √ O(Opt · Opt)-approximation. [Feige, Hajiaghayi, Lee. 2008]

3 (Exact) FPT algorithms on general graphs (tw 6 k?)

12/57 O(k3) In time k · n. [Bodlaender. 1996]

4 FPT approximation algorithms on general graphs O(k) 2 4.5-approximation in time2 · n . [Amir. 2010] O(k) 5-approx. in2 · n. [Bodlaender, Drange, Dregi, Fomin, Lokshtanov, Pilipczuk. 2013]

5 Approximation algorithms on sparse graphs

3/2-approximation on planar graphs. [Seymour and Thomas. 1994] cH -approximation on H-minor-free graphs. [Demaine and Hajiaghayi. 2008] Open Compute the treewidth of a planar graph.

About computing treewidth: a global picture

1 Computing treewidth is NP-hard [Arnborg, Corneil, Proskurowski. 1987]

2 Approximation algorithms on general graphs √ O(Opt · Opt)-approximation. [Feige, Hajiaghayi, Lee. 2008]

3 (Exact) FPT algorithms on general graphs (tw 6 k?) k+2 In time O(n ). [Arnborg, Corneil, Proskurowski. 1987]

12/57 4 FPT approximation algorithms on general graphs O(k) 2 4.5-approximation in time2 · n . [Amir. 2010] O(k) 5-approx. in2 · n. [Bodlaender, Drange, Dregi, Fomin, Lokshtanov, Pilipczuk. 2013]

5 Approximation algorithms on sparse graphs

3/2-approximation on planar graphs. [Seymour and Thomas. 1994] cH -approximation on H-minor-free graphs. [Demaine and Hajiaghayi. 2008] Open Compute the treewidth of a planar graph.

About computing treewidth: a global picture

1 Computing treewidth is NP-hard [Arnborg, Corneil, Proskurowski. 1987]

2 Approximation algorithms on general graphs √ O(Opt · Opt)-approximation. [Feige, Hajiaghayi, Lee. 2008]

3 (Exact) FPT algorithms on general graphs (tw 6 k?) k+2 In time O(n ). [Arnborg, Corneil, Proskurowski. 1987] O(k3) In time k · n. [Bodlaender. 1996]

12/57 O(k) 5-approx. in2 · n. [Bodlaender, Drange, Dregi, Fomin, Lokshtanov, Pilipczuk. 2013]

5 Approximation algorithms on sparse graphs

3/2-approximation on planar graphs. [Seymour and Thomas. 1994] cH -approximation on H-minor-free graphs. [Demaine and Hajiaghayi. 2008] Open Compute the treewidth of a planar graph.

About computing treewidth: a global picture

1 Computing treewidth is NP-hard [Arnborg, Corneil, Proskurowski. 1987]

2 Approximation algorithms on general graphs √ O(Opt · Opt)-approximation. [Feige, Hajiaghayi, Lee. 2008]

3 (Exact) FPT algorithms on general graphs (tw 6 k?) k+2 In time O(n ). [Arnborg, Corneil, Proskurowski. 1987] O(k3) In time k · n. [Bodlaender. 1996]

4 FPT approximation algorithms on general graphs O(k) 2 4.5-approximation in time2 · n . [Amir. 2010]

12/57 5 Approximation algorithms on sparse graphs

3/2-approximation on planar graphs. [Seymour and Thomas. 1994] cH -approximation on H-minor-free graphs. [Demaine and Hajiaghayi. 2008] Open Compute the treewidth of a planar graph.

About computing treewidth: a global picture

1 Computing treewidth is NP-hard [Arnborg, Corneil, Proskurowski. 1987]

2 Approximation algorithms on general graphs √ O(Opt · Opt)-approximation. [Feige, Hajiaghayi, Lee. 2008]

3 (Exact) FPT algorithms on general graphs (tw 6 k?) k+2 In time O(n ). [Arnborg, Corneil, Proskurowski. 1987] O(k3) In time k · n. [Bodlaender. 1996]

4 FPT approximation algorithms on general graphs O(k) 2 4.5-approximation in time2 · n . [Amir. 2010] O(k) 5-approx. in2 · n. [Bodlaender, Drange, Dregi, Fomin, Lokshtanov, Pilipczuk. 2013]

12/57 cH -approximation on H-minor-free graphs. [Demaine and Hajiaghayi. 2008] Open Compute the treewidth of a planar graph.

About computing treewidth: a global picture

1 Computing treewidth is NP-hard [Arnborg, Corneil, Proskurowski. 1987]

2 Approximation algorithms on general graphs √ O(Opt · Opt)-approximation. [Feige, Hajiaghayi, Lee. 2008]

3 (Exact) FPT algorithms on general graphs (tw 6 k?) k+2 In time O(n ). [Arnborg, Corneil, Proskurowski. 1987] O(k3) In time k · n. [Bodlaender. 1996]

4 FPT approximation algorithms on general graphs O(k) 2 4.5-approximation in time2 · n . [Amir. 2010] O(k) 5-approx. in2 · n. [Bodlaender, Drange, Dregi, Fomin, Lokshtanov, Pilipczuk. 2013]

5 Approximation algorithms on sparse graphs

3/2-approximation on planar graphs. [Seymour and Thomas. 1994]

12/57 Open Compute the treewidth of a planar graph.

About computing treewidth: a global picture

1 Computing treewidth is NP-hard [Arnborg, Corneil, Proskurowski. 1987]

2 Approximation algorithms on general graphs √ O(Opt · Opt)-approximation. [Feige, Hajiaghayi, Lee. 2008]

3 (Exact) FPT algorithms on general graphs (tw 6 k?) k+2 In time O(n ). [Arnborg, Corneil, Proskurowski. 1987] O(k3) In time k · n. [Bodlaender. 1996]

4 FPT approximation algorithms on general graphs O(k) 2 4.5-approximation in time2 · n . [Amir. 2010] O(k) 5-approx. in2 · n. [Bodlaender, Drange, Dregi, Fomin, Lokshtanov, Pilipczuk. 2013]

5 Approximation algorithms on sparse graphs

3/2-approximation on planar graphs. [Seymour and Thomas. 1994] cH -approximation on H-minor-free graphs. [Demaine and Hajiaghayi. 2008]

12/57 About computing treewidth: a global picture

1 Computing treewidth is NP-hard [Arnborg, Corneil, Proskurowski. 1987]

2 Approximation algorithms on general graphs √ O(Opt · Opt)-approximation. [Feige, Hajiaghayi, Lee. 2008]

3 (Exact) FPT algorithms on general graphs (tw 6 k?) k+2 In time O(n ). [Arnborg, Corneil, Proskurowski. 1987] O(k3) In time k · n. [Bodlaender. 1996]

4 FPT approximation algorithms on general graphs O(k) 2 4.5-approximation in time2 · n . [Amir. 2010] O(k) 5-approx. in2 · n. [Bodlaender, Drange, Dregi, Fomin, Lokshtanov, Pilipczuk. 2013]

5 Approximation algorithms on sparse graphs

3/2-approximation on planar graphs. [Seymour and Thomas. 1994] cH -approximation on H-minor-free graphs. [Demaine and Hajiaghayi. 2008] Open Compute the treewidth of a planar graph.

12/57 Next subsection is...

1 Introduction, part II Treewidth Dynamic programming on tree decompositions Structure of H-minor-free graphs Some algorithmic issues A few words on other containment relations

2 Bidimensionality Some ingredients An illustrative example Meta-algorithms Further extensions

13/57 Tables of dynamic programming (DP) at each node Xi :( Si , ai ), where Si is an independent set in Gi , and ai is the maximum size of 0 an independent set in Gi which coincides with Si in Xi .

The tables (Si , ai ) at node Xi can be computed from the tables of its children as follows. For each Si independent set of Gi :

If Xj is a child of Xi , an independent set Sj of Gj is feasible for Si if

Xj ∩ Si = Xi ∩ Sj .

For each children Xj of Xi , let Sj be feasible for Si s.t. (Sj , aj ) is deﬁned and aj is maximized. P We set ai := |Si | + j (aj − |Si ∩ Sj |).

2|X | Running time for each node Xi : O(2 i ). Overall running time:2 O(tw(G)) · n.

Example: Max. Independent Set

For each node Xi , let Gi = G[Xi ], and let 0 Gi be the subgraph of G induced by node Xi and its descendants.

14/57 The tables (Si , ai ) at node Xi can be computed from the tables of its children as follows. For each Si independent set of Gi :

If Xj is a child of Xi , an independent set Sj of Gj is feasible for Si if

Xj ∩ Si = Xi ∩ Sj .

For each children Xj of Xi , let Sj be feasible for Si s.t. (Sj , aj ) is deﬁned and aj is maximized. P We set ai := |Si | + j (aj − |Si ∩ Sj |).

2|X | Running time for each node Xi : O(2 i ). Overall running time:2 O(tw(G)) · n.

Example: Max. Independent Set

For each node Xi , let Gi = G[Xi ], and let 0 Gi be the subgraph of G induced by node Xi and its descendants.

Tables of dynamic programming (DP) at each node Xi :( Si , ai ), where Si is an independent set in Gi , and ai is the maximum size of 0 an independent set in Gi which coincides with Si in Xi .

14/57 2|X | Running time for each node Xi : O(2 i ). Overall running time:2 O(tw(G)) · n.

Example: Max. Independent Set

For each node Xi , let Gi = G[Xi ], and let 0 Gi be the subgraph of G induced by node Xi and its descendants.

Tables of dynamic programming (DP) at each node Xi :( Si , ai ), where Si is an independent set in Gi , and ai is the maximum size of 0 an independent set in Gi which coincides with Si in Xi .

The tables (Si , ai ) at node Xi can be computed from the tables of its children as follows. For each Si independent set of Gi :

If Xj is a child of Xi , an independent set Sj of Gj is feasible for Si if

Xj ∩ Si = Xi ∩ Sj .

For each children Xj of Xi , let Sj be feasible for Si s.t. (Sj , aj ) is deﬁned and aj is maximized. P We set ai := |Si | + j (aj − |Si ∩ Sj |).

14/57 Example: Max. Independent Set

For each node Xi , let Gi = G[Xi ], and let 0 Gi be the subgraph of G induced by node Xi and its descendants.

Tables of dynamic programming (DP) at each node Xi :( Si , ai ), where Si is an independent set in Gi , and ai is the maximum size of 0 an independent set in Gi which coincides with Si in Xi .

The tables (Si , ai ) at node Xi can be computed from the tables of its children as follows. For each Si independent set of Gi :

If Xj is a child of Xi , an independent set Sj of Gj is feasible for Si if

Xj ∩ Si = Xi ∩ Sj .

For each children Xj of Xi , let Sj be feasible for Si s.t. (Sj , aj ) is deﬁned and aj is maximized. P We set ai := |Si | + j (aj − |Si ∩ Sj |).

2|X | Running time for each node Xi : O(2 i ). Overall running time:2 O(tw(G)) · n. 14/57 In other words, all these problems are ﬁxed -parameter tractable (FPT) when parameterized by the treewidth of their input graphs.

Running time (tight): k 2...2 22 · nO(1).

Courcelle’s Theorem - algorithmic importance of treewidth

What we have seen with Max. Independent Set can be generalized to a wide family of problems:

Theorem (Courcelle. 1988) Graph problems expressible in Monadic Second Order Logic (MSOL) can O(1) be solved in timef (k) · n in graphs withn vertices and tw 6 k.

15/57 Running time (tight): k 2...2 22 · nO(1).

Courcelle’s Theorem - algorithmic importance of treewidth

What we have seen with Max. Independent Set can be generalized to a wide family of problems:

Theorem (Courcelle. 1988) Graph problems expressible in Monadic Second Order Logic (MSOL) can O(1) be solved in timef (k) · n in graphs withn vertices and tw 6 k.

In other words, all these problems are ﬁxed -parameter tractable (FPT) when parameterized by the treewidth of their input graphs.

15/57 Courcelle’s Theorem - algorithmic importance of treewidth

What we have seen with Max. Independent Set can be generalized to a wide family of problems:

Theorem (Courcelle. 1988) Graph problems expressible in Monadic Second Order Logic (MSOL) can O(1) be solved in timef (k) · n in graphs withn vertices and tw 6 k.

In other words, all these problems are ﬁxed -parameter tractable (FPT) when parameterized by the treewidth of their input graphs.

Running time (tight): k 2...2 22 · nO(1).

15/57 Next subsection is...

1 Introduction, part II Treewidth Dynamic programming on tree decompositions Structure of H-minor-free graphs Some algorithmic issues A few words on other containment relations

2 Bidimensionality Some ingredients An illustrative example Meta-algorithms Further extensions

16/57 Apex in an embedded graph: add a vertex with any neighbors in the embedded graph. Vortex of depth h in an embedded graph: paste a graph of pathwidth at most h in a face of the embedding.

Structure Theorem [Robertson and Seymour]: Fix a graph H. There exists a constant h = f (|V (H)|) such that any H-minor-free graph G can be decomposed (in a tree-like way) into h-clique-sums from h-almost-embeddable graphs: obtained from graphs of genus at most h by adding at most h apices and at most h vortices of depth at most h.

Structure of H-minor-free graphs: roughly “bidimensional”

Some (simpliﬁed) preliminaries:

h-clique-sum of two graphs G1 and G2: choose cliques K1 ⊆ G1 and K2 ⊆ G2 with |V (K1)| = |V (K2)| = h, identify them, and possibly remove some edges of that clique.

17/57 Vortex of depth h in an embedded graph: paste a graph of pathwidth at most h in a face of the embedding.

Structure of H-minor-free graphs: roughly “bidimensional”

Some (simpliﬁed) preliminaries:

h-clique-sum of two graphs G1 and G2: choose cliques K1 ⊆ G1 and K2 ⊆ G2 with |V (K1)| = |V (K2)| = h, identify them, and possibly remove some edges of that clique. Apex in an embedded graph: add a vertex with any neighbors in the embedded graph.

17/57 Structure Theorem [Robertson and Seymour]: Fix a graph H. There exists a constant h = f (|V (H)|) such that any H-minor-free graph G can be decomposed (in a tree-like way) into h-clique-sums from h-almost-embeddable graphs: obtained from graphs of genus at most h by adding at most h apices and at most h vortices of depth at most h.

Structure of H-minor-free graphs: roughly “bidimensional”

Some (simpliﬁed) preliminaries:

17/57 There exists a constant h = f (|V (H)|) such that any H-minor-free graph G can be decomposed (in a tree-like way) into h-clique-sums from h-almost-embeddable graphs: obtained from graphs of genus at most h by adding at most h apices and at most h vortices of depth at most h.

Structure of H-minor-free graphs: roughly “bidimensional”

Some (simpliﬁed) preliminaries:

Structure Theorem [Robertson and Seymour]: Fix a graph H.

17/57 obtained from graphs of genus at most h by adding at most h apices and at most h vortices of depth at most h.

Structure of H-minor-free graphs: roughly “bidimensional”

Some (simpliﬁed) preliminaries:

17/57 Structure of H-minor-free graphs: roughly “bidimensional”

Some (simpliﬁed) preliminaries:

17/57 Next subsection is...

1 Introduction, part II Treewidth Dynamic programming on tree decompositions Structure of H-minor-free graphs Some algorithmic issues A few words on other containment relations

2 Bidimensionality Some ingredients An illustrative example Meta-algorithms Further extensions

18/57 2 Improved algorithm running in time O(n ). [Kawarabayashi, Kobayashi, Reed. 2012] Even faster algorithms have been claimed... [Reed ??]

F Membership in a minor-closed graph class Theorem (Robertson and Seymour) Let G be a minor-closed graph class. Given ann-vertex graphG, one can test whetherG ∈ G in timeO (n2).

Proof. By the Graph Minors Theorem, there existsa ﬁnite list of minimal

excluded minors {F1,..., F`} for the class G. Then we can do minor testing 2 and check whether Fi m G in time O(n ) for each 1 6 i 6 `.

Some algorithmic issues of graph minors

F Minor testing Theorem (Robertson and Seymour)

Fix a graphH. Given ann-vertex graphG, one can test whetherH m G in timeO (n3).

19/57 Even faster algorithms have been claimed... [Reed ??]

F Membership in a minor-closed graph class Theorem (Robertson and Seymour) Let G be a minor-closed graph class. Given ann-vertex graphG, one can test whetherG ∈ G in timeO (n2).

Proof. By the Graph Minors Theorem, there existsa ﬁnite list of minimal

excluded minors {F1,..., F`} for the class G. Then we can do minor testing 2 and check whether Fi m G in time O(n ) for each 1 6 i 6 `.

Some algorithmic issues of graph minors

F Minor testing Theorem (Robertson and Seymour)

Fix a graphH. Given ann-vertex graphG, one can test whetherH m G in timeO (n3).

2 Improved algorithm running in time O(n ). [Kawarabayashi, Kobayashi, Reed. 2012]

19/57 F Membership in a minor-closed graph class Theorem (Robertson and Seymour) Let G be a minor-closed graph class. Given ann-vertex graphG, one can test whetherG ∈ G in timeO (n2).

Proof. By the Graph Minors Theorem, there existsa ﬁnite list of minimal

excluded minors {F1,..., F`} for the class G. Then we can do minor testing 2 and check whether Fi m G in time O(n ) for each 1 6 i 6 `.

Some algorithmic issues of graph minors

F Minor testing Theorem (Robertson and Seymour)

Fix a graphH. Given ann-vertex graphG, one can test whetherH m G in timeO (n3).

2 Improved algorithm running in time O(n ). [Kawarabayashi, Kobayashi, Reed. 2012] Even faster algorithms have been claimed... [Reed ??]

19/57 Proof. By the Graph Minors Theorem, there existsa ﬁnite list of minimal

excluded minors {F1,..., F`} for the class G. Then we can do minor testing 2 and check whether Fi m G in time O(n ) for each 1 6 i 6 `.

Some algorithmic issues of graph minors

F Minor testing Theorem (Robertson and Seymour)

Fix a graphH. Given ann-vertex graphG, one can test whetherH m G in timeO (n3).

2 Improved algorithm running in time O(n ). [Kawarabayashi, Kobayashi, Reed. 2012] Even faster algorithms have been claimed... [Reed ??]

F Membership in a minor-closed graph class Theorem (Robertson and Seymour) Let G be a minor-closed graph class. Given ann-vertex graphG, one can test whetherG ∈ G in timeO (n2).

19/57 Some algorithmic issues of graph minors

F Minor testing Theorem (Robertson and Seymour)

Fix a graphH. Given ann-vertex graphG, one can test whetherH m G in timeO (n3).

2 Improved algorithm running in time O(n ). [Kawarabayashi, Kobayashi, Reed. 2012] Even faster algorithms have been claimed... [Reed ??]

F Membership in a minor-closed graph class Theorem (Robertson and Seymour) Let G be a minor-closed graph class. Given ann-vertex graphG, one can test whetherG ∈ G in timeO (n2).

Proof. By the Graph Minors Theorem, there existsa ﬁnite list of minimal

excluded minors {F1,..., F`} for the class G. Then we can do minor testing 2 and check whether Fi m G in time O(n ) for each 1 6 i 6 `. 19/57 Using this tree-like structure, a number of hard optimization problems can be solved eﬃciently in H-minor-free graphs usingDP.

But... what about the constant h = f (|V (H)|) ??

Roughly the same constant appears in the Minor Testing algorithms. The proofs say “there exists a constant h such that...”

F Very recently, this constant has been made explicit and “reasonable”! [Geelen, Huynh, and Richter. 2013] [Mazoit. 2013]

Some algorithmic issues of graph minors (II)

Let us recall the Structure Theorem of H-minor-free graphs: Fix a graph H. There exists a constant h = f (|V (H)|) such that any H-minor-free graph G can be decomposed (in a tree-like way) into h-clique-sums from h-almost-embeddable graphs.

20/57 But... what about the constant h = f (|V (H)|) ??

Roughly the same constant appears in the Minor Testing algorithms. The proofs say “there exists a constant h such that...”

F Very recently, this constant has been made explicit and “reasonable”! [Geelen, Huynh, and Richter. 2013] [Mazoit. 2013]

Some algorithmic issues of graph minors (II)

Using this tree-like structure, a number of hard optimization problems can be solved eﬃciently in H-minor-free graphs usingDP.

20/57 Roughly the same constant appears in the Minor Testing algorithms. The proofs say “there exists a constant h such that...”

F Very recently, this constant has been made explicit and “reasonable”! [Geelen, Huynh, and Richter. 2013] [Mazoit. 2013]

Some algorithmic issues of graph minors (II)

Using this tree-like structure, a number of hard optimization problems can be solved eﬃciently in H-minor-free graphs usingDP.

But... what about the constant h = f (|V (H)|) ??

20/57 F Very recently, this constant has been made explicit and “reasonable”! [Geelen, Huynh, and Richter. 2013] [Mazoit. 2013]

Some algorithmic issues of graph minors (II)

Using this tree-like structure, a number of hard optimization problems can be solved eﬃciently in H-minor-free graphs usingDP.

But... what about the constant h = f (|V (H)|) ??

Roughly the same constant appears in the Minor Testing algorithms. The proofs say “there exists a constant h such that...”

20/57 Some algorithmic issues of graph minors (II)

Using this tree-like structure, a number of hard optimization problems can be solved eﬃciently in H-minor-free graphs usingDP.

But... what about the constant h = f (|V (H)|) ??

Roughly the same constant appears in the Minor Testing algorithms. The proofs say “there exists a constant h such that...”

F Very recently, this constant has been made explicit and “reasonable”! [Geelen, Huynh, and Richter. 2013] [Mazoit. 2013]

20/57 Next subsection is...

1 Introduction, part II Treewidth Dynamic programming on tree decompositions Structure of H-minor-free graphs Some algorithmic issues A few words on other containment relations

2 Bidimensionality Some ingredients An illustrative example Meta-algorithms Further extensions

21/57 1. Graphs are WQO w.r.t. the minor relation. 2. Minor Testing is FPT when parameterized by |V (H)|. 3. H-minor-free graphs have a nice structure.

Contraction minor: H cm G if H can be obtained from G by F contracting edges. 1. Graphs are WQO w.r.t. the contraction minor relation? NO! 2. Contraction Minor Testing is FPT when param. by |V (H)|? NO! NP-hard already for |V (H)| 6 4. [Brouwer and Veldman. 1987] 3. Nice structure? Not really: They contain cliques, chordal graphs...

Topological minor: H tp G if H can be obtained from a subgraph F of G by contracting edges with at least one endpoint of degree 6 2. 1. Graphs are WQO w.r.t. the topological minor relation? NO! 2. Topological Minor Testing is FPT when param. by |V (H)|? YES! [Grohe, Kawarabayashi, Marx, Wollan. 2011] 3. Nice structure? YES! [Grohe and Marx. 2012]

Minor: H m G if H can be obtained from a subgraph of G by F contracting edges.

22/57 Contraction minor: H cm G if H can be obtained from G by F contracting edges. 1. Graphs are WQO w.r.t. the contraction minor relation? NO! 2. Contraction Minor Testing is FPT when param. by |V (H)|? NO! NP-hard already for |V (H)| 6 4. [Brouwer and Veldman. 1987] 3. Nice structure? Not really: They contain cliques, chordal graphs...

Minor: H m G if H can be obtained from a subgraph of G by F contracting edges. 1. Graphs are WQO w.r.t. the minor relation. 2. Minor Testing is FPT when parameterized by |V (H)|. 3. H-minor-free graphs have a nice structure.

22/57 1. Graphs are WQO w.r.t. the contraction minor relation? NO! 2. Contraction Minor Testing is FPT when param. by |V (H)|? NO! NP-hard already for |V (H)| 6 4. [Brouwer and Veldman. 1987] 3. Nice structure? Not really: They contain cliques, chordal graphs...

Contraction minor: H cm G if H can be obtained from G by F contracting edges.

22/57 NO! 2. Contraction Minor Testing is FPT when param. by |V (H)|? NO! NP-hard already for |V (H)| 6 4. [Brouwer and Veldman. 1987] 3. Nice structure? Not really: They contain cliques, chordal graphs...

Contraction minor: H cm G if H can be obtained from G by F contracting edges. 1. Graphs are WQO w.r.t. the contraction minor relation?

22/57 2. Contraction Minor Testing is FPT when param. by |V (H)|? NO! NP-hard already for |V (H)| 6 4. [Brouwer and Veldman. 1987] 3. Nice structure? Not really: They contain cliques, chordal graphs...

Contraction minor: H cm G if H can be obtained from G by F contracting edges. 1. Graphs are WQO w.r.t. the contraction minor relation? NO!

22/57 NO! NP-hard already for |V (H)| 6 4. [Brouwer and Veldman. 1987] 3. Nice structure? Not really: They contain cliques, chordal graphs...

Contraction minor: H cm G if H can be obtained from G by F contracting edges. 1. Graphs are WQO w.r.t. the contraction minor relation? NO! 2. Contraction Minor Testing is FPT when param. by |V (H)|?

22/57 3. Nice structure? Not really: They contain cliques, chordal graphs...

22/57 Not really: They contain cliques, chordal graphs...

22/57 Topological minor: H tp G if H can be obtained from a subgraph F of G by contracting edges with at least one endpoint of degree 6 2. 1. Graphs are WQO w.r.t. the topological minor relation? NO! 2. Topological Minor Testing is FPT when param. by |V (H)|? YES! [Grohe, Kawarabayashi, Marx, Wollan. 2011] 3. Nice structure? YES! [Grohe and Marx. 2012]

22/57 1. Graphs are WQO w.r.t. the topological minor relation? NO! 2. Topological Minor Testing is FPT when param. by |V (H)|? YES! [Grohe, Kawarabayashi, Marx, Wollan. 2011] 3. Nice structure? YES! [Grohe and Marx. 2012]

Topological minor: H tp G if H can be obtained from a subgraph F of G by contracting edges with at least one endpoint of degree 6 2.

22/57 NO! 2. Topological Minor Testing is FPT when param. by |V (H)|? YES! [Grohe, Kawarabayashi, Marx, Wollan. 2011] 3. Nice structure? YES! [Grohe and Marx. 2012]

Topological minor: H tp G if H can be obtained from a subgraph F of G by contracting edges with at least one endpoint of degree 6 2. 1. Graphs are WQO w.r.t. the topological minor relation?

22/57 2. Topological Minor Testing is FPT when param. by |V (H)|? YES! [Grohe, Kawarabayashi, Marx, Wollan. 2011] 3. Nice structure? YES! [Grohe and Marx. 2012]

Topological minor: H tp G if H can be obtained from a subgraph F of G by contracting edges with at least one endpoint of degree 6 2. 1. Graphs are WQO w.r.t. the topological minor relation? NO!

22/57 YES! [Grohe, Kawarabayashi, Marx, Wollan. 2011] 3. Nice structure? YES! [Grohe and Marx. 2012]

22/57 3. Nice structure? YES! [Grohe and Marx. 2012]

22/57 YES! [Grohe and Marx. 2012]

22/57 Minor: H m G if H can be obtained from a subgraph of G by F contracting edges. 1. Graphs are WQO w.r.t. the minor relation. 2. Minor Testing is FPT when parameterized by |V (H)|. 3. H-minor-free graphs have a nice structure.

22/57 Structure of sparse graphs - a nice picture by Felix Reidl

23/57 Next section is...

1 Introduction, part II Treewidth Dynamic programming on tree decompositions Structure of H-minor-free graphs Some algorithmic issues A few words on other containment relations

2 Bidimensionality Some ingredients An illustrative example Meta-algorithms Further extensions

24/57 Next subsection is...

1 Introduction, part II Treewidth Dynamic programming on tree decompositions Structure of H-minor-free graphs Some algorithmic issues A few words on other containment relations

2 Bidimensionality Some ingredients An illustrative example Meta-algorithms Further extensions

25/57 Long Path Input: A graph G = (V , E) and a positive integers k. Parameter: k. Question: Does there exist a path P in G of length at least k?

A few representative problems

Vertex Cover Input: A graph G = (V , E) and a positive integers k. Parameter: k. Question: Does there exist a subset C ⊆ V of size at most k such that G[V \ C] is an independent set?

26/57 A few representative problems

Vertex Cover Input: A graph G = (V , E) and a positive integers k. Parameter: k. Question: Does there exist a subset C ⊆ V of size at most k such that G[V \ C] is an independent set?

Long Path Input: A graph G = (V , E) and a positive integers k. Parameter: k. Question: Does there exist a path P in G of length at least k?

26/57 Dominating Set Input: A graph G = (V , E) and a positive integers k. Parameter: k. Question: Does there exist a subset D ⊆ V of size at most k such that for all v ∈ V , N[v] ∩ D =6 ∅?

A few representative problems (II)

Feedback Vertex Set Input: A graph G = (V , E) and a positive integers k. Parameter: k. Question: Does there exist a subset F ⊆ V of size at most k such that for G[V \ F ] is a forest?

27/57 A few representative problems (II)

Feedback Vertex Set Input: A graph G = (V , E) and a positive integers k. Parameter: k. Question: Does there exist a subset F ⊆ V of size at most k such that for G[V \ F ] is a forest?

Dominating Set Input: A graph G = (V , E) and a positive integers k. Parameter: k. Question: Does there exist a subset D ⊆ V of size at most k such that for all v ∈ V , N[v] ∩ D =6 ∅?

27/57 A parameter P is any function mapping graphs to nonnegative integers.

The parameterized problem associated withP asks, for some ﬁxed k, whether for a given graph G, P(G) 6 k (for minimization) or P(G) > k (for maximization problem).

We say that a parameter P is closed under taking of minors/contractions (or, brieﬂy, minor/contraction closed) if for every graph H, H m G/H cm G implies that P(H) 6 P(G).

Minor closed parameters

A graph class G is minor (contraction) closed if any minor (contraction) of a graph in G is also in G.

28/57 The parameterized problem associated withP asks, for some ﬁxed k, whether for a given graph G, P(G) 6 k (for minimization) or P(G) > k (for maximization problem).

We say that a parameter P is closed under taking of minors/contractions (or, brieﬂy, minor/contraction closed) if for every graph H, H m G/H cm G implies that P(H) 6 P(G).

Minor closed parameters

A graph class G is minor (contraction) closed if any minor (contraction) of a graph in G is also in G.

A parameter P is any function mapping graphs to nonnegative integers.

28/57 We say that a parameter P is closed under taking of minors/contractions (or, brieﬂy, minor/contraction closed) if for every graph H, H m G/H cm G implies that P(H) 6 P(G).

Minor closed parameters

A graph class G is minor (contraction) closed if any minor (contraction) of a graph in G is also in G.

A parameter P is any function mapping graphs to nonnegative integers.

The parameterized problem associated withP asks, for some ﬁxed k, whether for a given graph G, P(G) 6 k (for minimization) or P(G) > k (for maximization problem).

28/57 Minor closed parameters

A graph class G is minor (contraction) closed if any minor (contraction) of a graph in G is also in G.

A parameter P is any function mapping graphs to nonnegative integers.

The parameterized problem associated withP asks, for some ﬁxed k, whether for a given graph G, P(G) 6 k (for minimization) or P(G) > k (for maximization problem).

We say that a parameter P is closed under taking of minors/contractions (or, brieﬂy, minor/contraction closed) if for every graph H, H m G/H cm G implies that P(H) 6 P(G).

28/57 Contraction closed parameters: Dominating Set, Connected Vertex Cover, r-Dominating Set, ...

Examples of minor/contraction closed parameters

Minor closed parameters: Vertex Cover, Feedback Vertex Set, Long Path, Treewidth, ...

29/57 Examples of minor/contraction closed parameters

Minor closed parameters: Vertex Cover, Feedback Vertex Set, Long Path, Treewidth, ...

Contraction closed parameters: Dominating Set, Connected Vertex Cover, r-Dominating Set, ...

29/57 We have tw (H`,`) = `.

As Treewidth is minor closed, if ` m G, then tw(G) > tw(H`,`) = `. Does the reverse implication hold?

Theorem (Robertson and Seymour. 1986) For every integer ` > 0, there is an integerc (`) such that every graph of

treewidth > c(`) contains ` as a minor.

2 2`5 Smallest possible function c(`)? Ω(` log `) 6 c(`) 6 20 O(` log `) Some improvement: c(`) = 2 . [Leaf and Seymour. 2012]

Recent breakthrough: c(`) = poly(`). [Chekuri and Chuzhoy. 2013]

Important message grid-minors are the certiﬁcate of large treewidth.

Grid Exclusion Theorem

Let H`,` be the( ` × `)-grid: `

30/57 As Treewidth is minor closed, if ` m G, then tw(G) > tw(H`,`) = `. Does the reverse implication hold?

Theorem (Robertson and Seymour. 1986) For every integer ` > 0, there is an integerc (`) such that every graph of

treewidth > c(`) contains ` as a minor.

2 2`5 Smallest possible function c(`)? Ω(` log `) 6 c(`) 6 20 O(` log `) Some improvement: c(`) = 2 . [Leaf and Seymour. 2012]

Recent breakthrough: c(`) = poly(`). [Chekuri and Chuzhoy. 2013]

Important message grid-minors are the certiﬁcate of large treewidth.

Grid Exclusion Theorem

Let H`,` be the( ` × `)-grid: ` We have tw (H`,`) = `.

30/57 Does the reverse implication hold?

Theorem (Robertson and Seymour. 1986) For every integer ` > 0, there is an integerc (`) such that every graph of

treewidth > c(`) contains ` as a minor.

2 2`5 Smallest possible function c(`)? Ω(` log `) 6 c(`) 6 20 O(` log `) Some improvement: c(`) = 2 . [Leaf and Seymour. 2012]

Recent breakthrough: c(`) = poly(`). [Chekuri and Chuzhoy. 2013]

Important message grid-minors are the certiﬁcate of large treewidth.

Grid Exclusion Theorem

Let H`,` be the( ` × `)-grid: ` We have tw (H`,`) = `.

As Treewidth is minor closed, if ` m G, then tw(G) > tw(H`,`) = `.

30/57 Theorem (Robertson and Seymour. 1986) For every integer ` > 0, there is an integerc (`) such that every graph of

treewidth > c(`) contains ` as a minor.

2 2`5 Smallest possible function c(`)? Ω(` log `) 6 c(`) 6 20 O(` log `) Some improvement: c(`) = 2 . [Leaf and Seymour. 2012]

Recent breakthrough: c(`) = poly(`). [Chekuri and Chuzhoy. 2013]

Important message grid-minors are the certiﬁcate of large treewidth.

Grid Exclusion Theorem

Let H`,` be the( ` × `)-grid: ` We have tw (H`,`) = `.

As Treewidth is minor closed, if ` m G, then tw(G) > tw(H`,`) = `. Does the reverse implication hold?

30/57 2 2`5 Smallest possible function c(`)? Ω(` log `) 6 c(`) 6 20 O(` log `) Some improvement: c(`) = 2 . [Leaf and Seymour. 2012]

Recent breakthrough: c(`) = poly(`). [Chekuri and Chuzhoy. 2013]

Important message grid-minors are the certiﬁcate of large treewidth.

Grid Exclusion Theorem

Let H`,` be the( ` × `)-grid: ` We have tw (H`,`) = `.

As Treewidth is minor closed, if ` m G, then tw(G) > tw(H`,`) = `. Does the reverse implication hold?

Theorem (Robertson and Seymour. 1986) For every integer ` > 0, there is an integerc (`) such that every graph of

treewidth > c(`) contains ` as a minor.

30/57 2 2`5 Ω(` log `) 6 c(`) 6 20 O(` log `) Some improvement: c(`) = 2 . [Leaf and Seymour. 2012]

Recent breakthrough: c(`) = poly(`). [Chekuri and Chuzhoy. 2013]

Important message grid-minors are the certiﬁcate of large treewidth.

Grid Exclusion Theorem

Let H`,` be the( ` × `)-grid: ` We have tw (H`,`) = `.

As Treewidth is minor closed, if ` m G, then tw(G) > tw(H`,`) = `. Does the reverse implication hold?

Theorem (Robertson and Seymour. 1986) For every integer ` > 0, there is an integerc (`) such that every graph of

treewidth > c(`) contains ` as a minor.

Smallest possible function c(`)?

30/57 O(` log `) Some improvement: c(`) = 2 . [Leaf and Seymour. 2012]

Recent breakthrough: c(`) = poly(`). [Chekuri and Chuzhoy. 2013]

Important message grid-minors are the certiﬁcate of large treewidth.

Grid Exclusion Theorem

Let H`,` be the( ` × `)-grid: ` We have tw (H`,`) = `.

As Treewidth is minor closed, if ` m G, then tw(G) > tw(H`,`) = `. Does the reverse implication hold?

Theorem (Robertson and Seymour. 1986) For every integer ` > 0, there is an integerc (`) such that every graph of

treewidth > c(`) contains ` as a minor.

2 2`5 Smallest possible function c(`)? Ω(` log `) 6 c(`) 6 20

30/57 Recent breakthrough: c(`) = poly(`). [Chekuri and Chuzhoy. 2013]

Important message grid-minors are the certiﬁcate of large treewidth.

Grid Exclusion Theorem

Let H`,` be the( ` × `)-grid: ` We have tw (H`,`) = `.

As Treewidth is minor closed, if ` m G, then tw(G) > tw(H`,`) = `. Does the reverse implication hold?

Theorem (Robertson and Seymour. 1986) For every integer ` > 0, there is an integerc (`) such that every graph of

treewidth > c(`) contains ` as a minor.

2 2`5 Smallest possible function c(`)? Ω(` log `) 6 c(`) 6 20 O(` log `) Some improvement: c(`) = 2 . [Leaf and Seymour. 2012]

30/57 Important message grid-minors are the certiﬁcate of large treewidth.

Grid Exclusion Theorem

Let H`,` be the( ` × `)-grid: ` We have tw (H`,`) = `.

As Treewidth is minor closed, if ` m G, then tw(G) > tw(H`,`) = `. Does the reverse implication hold?

Theorem (Robertson and Seymour. 1986) For every integer ` > 0, there is an integerc (`) such that every graph of

treewidth > c(`) contains ` as a minor.

2 2`5 Smallest possible function c(`)? Ω(` log `) 6 c(`) 6 20 O(` log `) Some improvement: c(`) = 2 . [Leaf and Seymour. 2012]

Recent breakthrough: c(`) = poly(`). [Chekuri and Chuzhoy. 2013]

30/57 Grid Exclusion Theorem

Let H`,` be the( ` × `)-grid: ` We have tw (H`,`) = `.

As Treewidth is minor closed, if ` m G, then tw(G) > tw(H`,`) = `. Does the reverse implication hold?

Theorem (Robertson and Seymour. 1986) For every integer ` > 0, there is an integerc (`) such that every graph of

treewidth > c(`) contains ` as a minor.

2 2`5 Smallest possible function c(`)? Ω(` log `) 6 c(`) 6 20 O(` log `) Some improvement: c(`) = 2 . [Leaf and Seymour. 2012]

Recent breakthrough: c(`) = poly(`). [Chekuri and Chuzhoy. 2013]

Important message grid-minors are the certiﬁcate of large treewidth. 30/57 Theorem (Demaine, Fomin, Hajiaghayi, Thilikos. 2005)

For every ﬁxed g, there is a constantc g such that every graph of genus g

and of treewidth > cg · ` contains ` as a minor.

Theorem (Demaine and Hajiaghayi. 2008)

For every ﬁxed graph H, there is a constantc H such that every

H-minor-free graph of treewidth > cH · ` contains ` as a minor.

Best constant in the above theorem is by [Kawarabayashi and Kobayashi. 2012]

In sparse graphs: linear dependency between treewidth and grid-minors

Grid Exclusion Theorems on sparse graphs

Theorem (Robertson, Seymour, Thomas. 1994)

Every planar graph of treewidth > 6 · ` contains ` as a minor.

31/57 Theorem (Demaine and Hajiaghayi. 2008)

For every ﬁxed graph H, there is a constantc H such that every

H-minor-free graph of treewidth > cH · ` contains ` as a minor.

Best constant in the above theorem is by [Kawarabayashi and Kobayashi. 2012]

In sparse graphs: linear dependency between treewidth and grid-minors

Grid Exclusion Theorems on sparse graphs

Theorem (Robertson, Seymour, Thomas. 1994)

Every planar graph of treewidth > 6 · ` contains ` as a minor.

Theorem (Demaine, Fomin, Hajiaghayi, Thilikos. 2005)

For every ﬁxed g, there is a constantc g such that every graph of genus g

and of treewidth > cg · ` contains ` as a minor.

31/57 In sparse graphs: linear dependency between treewidth and grid-minors

Grid Exclusion Theorems on sparse graphs

Theorem (Robertson, Seymour, Thomas. 1994)

Every planar graph of treewidth > 6 · ` contains ` as a minor.

Theorem (Demaine, Fomin, Hajiaghayi, Thilikos. 2005)

For every ﬁxed g, there is a constantc g such that every graph of genus g

and of treewidth > cg · ` contains ` as a minor.

Theorem (Demaine and Hajiaghayi. 2008)

For every ﬁxed graph H, there is a constantc H such that every

H-minor-free graph of treewidth > cH · ` contains ` as a minor.

Best constant in the above theorem is by [Kawarabayashi and Kobayashi. 2012]

31/57 Grid Exclusion Theorems on sparse graphs

Theorem (Robertson, Seymour, Thomas. 1994)

Every planar graph of treewidth > 6 · ` contains ` as a minor.

Theorem (Demaine, Fomin, Hajiaghayi, Thilikos. 2005)

For every ﬁxed g, there is a constantc g such that every graph of genus g

and of treewidth > cg · ` contains ` as a minor.

Theorem (Demaine and Hajiaghayi. 2008)

For every ﬁxed graph H, there is a constantc H such that every

H-minor-free graph of treewidth > cH · ` contains ` as a minor.

Best constant in the above theorem is by [Kawarabayashi and Kobayashi. 2012]

In sparse graphs: linear dependency between treewidth and grid-minors

31/57 How to use Grid Theorems algorithmically?

32/57 Next subsection is...

1 Introduction, part II Treewidth Dynamic programming on tree decompositions Structure of H-minor-free graphs Some algorithmic issues A few words on other containment relations

2 Bidimensionality Some ingredients An illustrative example Meta-algorithms Further extensions

33/57 Example: FPT algorithm for Planar Vertex Cover

A vertex cover C of a graph G, vc(G), is a set of vertices such that every edge of G has at least one endpoint in C.

34/57 Example: FPT algorithm for Planar Vertex Cover

INPUT: Planar graph G on n vertices, and an integer k.

OUTPUT: Either a vertex cover of G of size 6 k, or a proof that G has no such a vertex cover. √ RUNNING TIME:2 O( k) · nO(1).

Objective subexponential FPT algorithm for Planar Vertex Cover.

35/57 Example: FPT algorithm for Planar Vertex Cover

`2 vc(H`,`) > 2

36/57 G × =⇒ contains the (` `)-grid H`,` as a minor

2 The size of any vertex cover of H`,` is at least ` /2.

Recall that Vertex Cover is a minor closed parameter.

2 Since H`,` m G, it holds that vc(G) > vc(H`,`) > ` /2.

Example: FPT algorithm for Planar Vertex Cover

Let G be a planar graph of treewidth > 6 · `

37/57 2 The size of any vertex cover of H`,` is at least ` /2.

Recall that Vertex Cover is a minor closed parameter.

2 Since H`,` m G, it holds that vc(G) > vc(H`,`) > ` /2.

Example: FPT algorithm for Planar Vertex Cover

Let G be a planar graph of G × =⇒ contains the (` `)-grid treewidth > 6 · ` H`,` as a minor

37/57 Example: FPT algorithm for Planar Vertex Cover

Let G be a planar graph of G × =⇒ contains the (` `)-grid treewidth > 6 · ` H`,` as a minor

2 The size of any vertex cover of H`,` is at least ` /2.

Recall that Vertex Cover is a minor closed parameter.

2 Since H`,` m G, it holds that vc(G) > vc(H`,`) > ` /2.

37/57 WIN/WIN approach:

If k <` 2/2, we can safely answer “NO”. √ 2 If k > ` /2, then tw(G) = O(`) = O( k), and we can√ solve the problem by standard DP in time 2O(tw(G)) · nO(1) = 2O( k) · nO(1).

This gives a subexponential FPT algorithm!

We are already very close to an algorithm...

Recall: k is the parameter of the problem. We have that tw(G) = 6 · ` and ` is the size of a grid-minor of G. 2 Therefore, vc(G) > ` /2.

38/57 √ 2 If k > ` /2, then tw(G) = O(`) = O( k), and we can√ solve the problem by standard DP in time 2O(tw(G)) · nO(1) = 2O( k) · nO(1).

This gives a subexponential FPT algorithm!

We are already very close to an algorithm...

Recall: k is the parameter of the problem. We have that tw(G) = 6 · ` and ` is the size of a grid-minor of G. 2 Therefore, vc(G) > ` /2.

WIN/WIN approach:

If k <` 2/2, we can safely answer “NO”.

38/57 and we can√ solve the problem by standard DP in time 2O(tw(G)) · nO(1) = 2O( k) · nO(1).

This gives a subexponential FPT algorithm!

We are already very close to an algorithm...

Recall: k is the parameter of the problem. We have that tw(G) = 6 · ` and ` is the size of a grid-minor of G. 2 Therefore, vc(G) > ` /2.

WIN/WIN approach:

If k <` 2/2, we can safely answer “NO”. √ 2 If k > ` /2, then tw(G) = O(`) = O( k),

38/57 √ = 2O( k) · nO(1).

This gives a subexponential FPT algorithm!

We are already very close to an algorithm...

Recall: k is the parameter of the problem. We have that tw(G) = 6 · ` and ` is the size of a grid-minor of G. 2 Therefore, vc(G) > ` /2.

WIN/WIN approach:

If k <` 2/2, we can safely answer “NO”. √ 2 If k > ` /2, then tw(G) = O(`) = O( k), and we can solve the problem by standard DP in time 2O(tw(G)) · nO(1)

38/57 This gives a subexponential FPT algorithm!

We are already very close to an algorithm...

Recall: k is the parameter of the problem. We have that tw(G) = 6 · ` and ` is the size of a grid-minor of G. 2 Therefore, vc(G) > ` /2.

WIN/WIN approach:

If k <` 2/2, we can safely answer “NO”. √ 2 If k > ` /2, then tw(G) = O(`) = O( k), and we can√ solve the problem by standard DP in time 2O(tw(G)) · nO(1) = 2O( k) · nO(1).

38/57 We are already very close to an algorithm...

Recall: k is the parameter of the problem. We have that tw(G) = 6 · ` and ` is the size of a grid-minor of G. 2 Therefore, vc(G) > ` /2.

WIN/WIN approach:

If k <` 2/2, we can safely answer “NO”. √ 2 If k > ` /2, then tw(G) = O(`) = O( k), and we can√ solve the problem by standard DP in time 2O(tw(G)) · nO(1) = 2O( k) · nO(1).

This gives a subexponential FPT algorithm!

38/57 F Nothing special! It is just a minor bidimensional parameter:

2 minor-closed + vc( k ) = Ω(k ).

F Only the linear Grid Theorem! Arguments go through up to H-minor-free graphs.

Was Vertex Cover really just an example...?

What is so special in Vertex Cover?

Where did we use planarity?

39/57 F Only the linear Grid Theorem! Arguments go through up to H-minor-free graphs.

Was Vertex Cover really just an example...?

What is so special in Vertex Cover?

F Nothing special! It is just a minor bidimensional parameter:

2 minor-closed + vc( k ) = Ω(k ).

Where did we use planarity?

39/57 Was Vertex Cover really just an example...?

What is so special in Vertex Cover?

F Nothing special! It is just a minor bidimensional parameter:

2 minor-closed + vc( k ) = Ω(k ).

Where did we use planarity?

F Only the linear Grid Theorem! Arguments go through up to H-minor-free graphs.

39/57 Next subsection is...

1 Introduction, part II Treewidth Dynamic programming on tree decompositions Structure of H-minor-free graphs Some algorithmic issues A few words on other containment relations

2 Bidimensionality Some ingredients An illustrative example Meta-algorithms Further extensions

40/57 Minor Bidimensionality: [Demaine, Fomin, Hajiaghayi, Thilikos. 2005]

Deﬁnition A parameter p is minor bidimensional if

1 p is closed under taking of minors (minor closed), and ! 2 2 p k = Ω(k ).

41/57 Vertex Cover of a Grid

Hr,r for r = 10

42/57 Vertex Cover of a Grid

2 vc(Hr,r ) > r /2

43/57 Feedback Vertex Set of a Grid

44/57 Feedback Vertex Set of a Grid

2 fvs(Hr,r ) > r /4

45/57 √ Show√ that√ if the graph has large treewidth( > c k) then it has a ( k × k)-grid as a minor, and hence the answer to the problem is YES (or NO) immediately. √ Otherwise, the treewidth is bounded by c k, and hence we can use a dynamic programming (DP) algorithm on graphs of bounded treewidth.

t If we have a DP algorithm√ for bounded√ treewidth running in time c or tt , then it implies2 O( k) or 2O( k log k) algorithm.

How to obtain subexponential algorithms for BP?

First we must restrict ourselves to special graph classes, like planar or H-minor-free graphs.

46/57 √ Otherwise, the treewidth is bounded by c k, and hence we can use a dynamic programming (DP) algorithm on graphs of bounded treewidth.

t If we have a DP algorithm√ for bounded√ treewidth running in time c or tt , then it implies2 O( k) or 2O( k log k) algorithm.

How to obtain subexponential algorithms for BP?

46/57 t If we have a DP algorithm√ for bounded√ treewidth running in time c or tt , then it implies2 O( k) or 2O( k log k) algorithm.

How to obtain subexponential algorithms for BP?

First we must restrict ourselves to special graph classes, like planar or H-minor-free graphs. √ Show√ that√ if the graph has large treewidth( > c k) then it has a ( k × k)-grid as a minor, and hence the answer to the problem is YES (or NO) immediately. √ Otherwise, the treewidth is bounded by c k, and hence we can use a dynamic programming (DP) algorithm on graphs of bounded treewidth.

46/57 How to obtain subexponential algorithms for BP?

t If we have a DP algorithm√ for bounded√ treewidth running in time c or tt , then it implies2 O( k) or 2O( k log k) algorithm.

46/57 We can use a fast FPT algorithm or a constant-factor approx.

This follows because of the linear Grid Exclusion Theorems.

Doing DP in time2 O(tw(G)) · nO(1) is a whole area of research: Exploiting Catalan structures on sparse graphs. [Dorn et al. 2005-2008] [Ru´e,S., Thilikos. 2010] Randomized algorithms using Cut&Count. [Cygan et al. 2011] Deterministic algorithms based on rank of matrices. [Boadlaender et al. 2012] Deterministic algorithms based on matroids. [Fomin et al. 2013]

1 Compute (or approximate) tw(G). √ 2 If tw(G) = Ω( k), then answerNO. √ 3 Otherwise tw(G) = O( k), and we solve the problem by DP.

Piecing everything together

Theorem Let G be an H-minor-free graph, and let p be a minor bidimensional graph O(tw(G)) O(1) parameter computable in time 2 · n . √ Then deciding “ p(G) = k” can be done in time 2O( k) · nO(1).

47/57 We can use a fast FPT algorithm or a constant-factor approx.

This follows because of the linear Grid Exclusion Theorems.

Piecing everything together

1 Compute (or approximate) tw(G). √ 2 If tw(G) = Ω( k), then answerNO. √ 3 Otherwise tw(G) = O( k), and we solve the problem by DP.

47/57 This follows because of the linear Grid Exclusion Theorems.

Piecing everything together

1 Compute (or approximate) tw(G). We can use a fast FPT algorithm or a constant-factor approx. √ 2 If tw(G) = Ω( k), then answerNO. √ 3 Otherwise tw(G) = O( k), and we solve the problem by DP.

47/57 Doing DP in time2 O(tw(G)) · nO(1) is a whole area of research: Exploiting Catalan structures on sparse graphs. [Dorn et al. 2005-2008] [Ru´e,S., Thilikos. 2010] Randomized algorithms using Cut&Count. [Cygan et al. 2011] Deterministic algorithms based on rank of matrices. [Boadlaender et al. 2012] Deterministic algorithms based on matroids. [Fomin et al. 2013]

Piecing everything together

47/57 Exploiting Catalan structures on sparse graphs. [Dorn et al. 2005-2008] [Ru´e,S., Thilikos. 2010] Randomized algorithms using Cut&Count. [Cygan et al. 2011] Deterministic algorithms based on rank of matrices. [Boadlaender et al. 2012] Deterministic algorithms based on matroids. [Fomin et al. 2013]

Piecing everything together

47/57 Randomized algorithms using Cut&Count. [Cygan et al. 2011] Deterministic algorithms based on rank of matrices. [Boadlaender et al. 2012] Deterministic algorithms based on matroids. [Fomin et al. 2013]

Piecing everything together

47/57 Piecing everything together

1 Compute (or approximate) tw(G). We can use a fast FPT algorithm or a constant-factor approx. √ 2 If tw(G) = Ω( k), then answerNO. This follows because of the linear Grid Exclusion Theorems. √ 3 Otherwise tw(G) = O( k), and we solve the problem by DP. Doing DP in time2 O(tw(G)) · nO(1) is a whole area of research: Exploiting Catalan structures on sparse graphs. [Dorn et al. 2005-2008] [Ru´e,S., Thilikos. 2010] Randomized algorithms using Cut&Count. [Cygan et al. 2011] Deterministic algorithms based on rank of matrices. [Boadlaender et al. 2012] Deterministic algorithms based on matroids. [Fomin et al. 2013]

47/57 What about contraction closed parameters?? Dominating Set, Connected Vertex Cover, r-Dominating Set, ...

Minor Bidimensionality provides a meta-algorithm

This result applies to all minor closed parameters: Vertex Cover, Feedback Vertex Set, Long Path, Cycle Cover, ...

48/57 Minor Bidimensionality provides a meta-algorithm

This result applies to all minor closed parameters: Vertex Cover, Feedback Vertex Set, Long Path, Cycle Cover, ...

What about contraction closed parameters?? Dominating Set, Connected Vertex Cover, r-Dominating Set, ...

48/57 Next subsection is...

1 Introduction, part II Treewidth Dynamic programming on tree decompositions Structure of H-minor-free graphs Some algorithmic issues A few words on other containment relations

2 Bidimensionality Some ingredients An illustrative example Meta-algorithms Further extensions

49/57 But it is contraction closed...

Contraction Bidimensionality: [Demaine, Fomin, Hajiaghayi, Thilikos. 2005]

Deﬁnition A parameter p is contraction bidimensional if

1 p is closed under taking of contractions (contraction closed), and

2 for a “(k × k)-grid-like graph” Γ, p(Γ) = Ω(k2).

What is a( k × k)-grid-like graph...?

Contraction bidimensionality

Dominating Set is NOT minor closed, so we cannot use Grid Exclusion Theorems!!

50/57 Contraction Bidimensionality: [Demaine, Fomin, Hajiaghayi, Thilikos. 2005]

Deﬁnition A parameter p is contraction bidimensional if

1 p is closed under taking of contractions (contraction closed), and

2 for a “(k × k)-grid-like graph” Γ, p(Γ) = Ω(k2).

What is a( k × k)-grid-like graph...?

Contraction bidimensionality

Dominating Set is NOT minor closed, so we cannot use Grid Exclusion Theorems!! But it is contraction closed...

50/57 What is a( k × k)-grid-like graph...?

Contraction bidimensionality

Dominating Set is NOT minor closed, so we cannot use Grid Exclusion Theorems!! But it is contraction closed...

Contraction Bidimensionality: [Demaine, Fomin, Hajiaghayi, Thilikos. 2005]

Deﬁnition A parameter p is contraction bidimensional if

1 p is closed under taking of contractions (contraction closed), and

2 for a “(k × k)-grid-like graph” Γ, p(Γ) = Ω(k2).

50/57 Contraction bidimensionality

Dominating Set is NOT minor closed, so we cannot use Grid Exclusion Theorems!! But it is contraction closed...

Contraction Bidimensionality: [Demaine, Fomin, Hajiaghayi, Thilikos. 2005]

Deﬁnition A parameter p is contraction bidimensional if

1 p is closed under taking of contractions (contraction closed), and

2 for a “(k × k)-grid-like graph” Γ, p(Γ) = Ω(k2).

What is a( k × k)-grid-like graph...?

50/57 F For planar graphs this is a partially triangulated (k × k)-grid. [Demaine, Fomin, Hajiaghayi, Thilikos. 2006]

F For graphs of Euler genus γ, this is a partially triangulated (k × k)-grid with up to γ additional handles.

[Demaine, Hajiaghayi, Thilikos. 2006]

F For apex-minor-free graphs, this is a (k × k)-augmented grid, i.e., partially triangulated grid augmented with additional edges such that each vertex is incident to O(1) edges to non-boundary vertices of the grid.

[Demaine, Fomin, Hajiaghayi, Thilikos. 2005]

H is an apex graph if ∃v ∈ V (H): H − v is planar

Contraction bidimensionality: old setting

A“(k × k)-grid-like graph” was diﬀerent for each graph class:

51/57 F For graphs of Euler genus γ, this is a partially triangulated (k × k)-grid with up to γ additional handles.

[Demaine, Hajiaghayi, Thilikos. 2006]

[Demaine, Fomin, Hajiaghayi, Thilikos. 2005]

H is an apex graph if ∃v ∈ V (H): H − v is planar

Contraction bidimensionality: old setting

A“(k × k)-grid-like graph” was diﬀerent for each graph class:

F For planar graphs this is a partially triangulated (k × k)-grid. [Demaine, Fomin, Hajiaghayi, Thilikos. 2006]

51/57 F For apex-minor-free graphs, this is a (k × k)-augmented grid, i.e., partially triangulated grid augmented with additional edges such that each vertex is incident to O(1) edges to non-boundary vertices of the grid.

[Demaine, Fomin, Hajiaghayi, Thilikos. 2005]

H is an apex graph if ∃v ∈ V (H): H − v is planar

Contraction bidimensionality: old setting

A“(k × k)-grid-like graph” was diﬀerent for each graph class:

F For planar graphs this is a partially triangulated (k × k)-grid. [Demaine, Fomin, Hajiaghayi, Thilikos. 2006]

F For graphs of Euler genus γ, this is a partially triangulated (k × k)-grid with up to γ additional handles.

[Demaine, Hajiaghayi, Thilikos. 2006]

51/57 Contraction bidimensionality: old setting

A“(k × k)-grid-like graph” was diﬀerent for each graph class:

F For planar graphs this is a partially triangulated (k × k)-grid. [Demaine, Fomin, Hajiaghayi, Thilikos. 2006]

F For graphs of Euler genus γ, this is a partially triangulated (k × k)-grid with up to γ additional handles.

[Demaine, Hajiaghayi, Thilikos. 2006]

[Demaine, Fomin, Hajiaghayi, Thilikos. 2005]

H is an apex graph if ∃v ∈ V (H): H − v is planar 51/57 Deﬁnition A parameter p is contraction bidimensional if the following hold: 1 p is contraction closed, and

2 2 p( k ) = Ω(k ).

Contraction bidimensionality: new deﬁnition

Finally, the right “(k × k)-grid-like graph” was found: [Fomin, Golovach, Thilikos. 2009]

52/57 Contraction bidimensionality: new deﬁnition

Finally, the right “(k × k)-grid-like graph” was found: [Fomin, Golovach, Thilikos. 2009]

Deﬁnition A parameter p is contraction bidimensional if the following hold: 1 p is contraction closed, and

2 2 p( k ) = Ω(k ). 52/57 As for minor bidimensionality, we need to prove that

I If tw(G) = Ω(k) then G contains k as a contraction.

Meta-algorithms for contraction bidimensional parameters

Theorem Let H be a ﬁxed apex graph, let G be an H-minor free graph, and let p O(tw(G)) O(1) be a contraction bidimensional parameter computable√ in 2 · n . Then deciding p(G) = k can be done in time 2O( k) · nO(1).

53/57 Meta-algorithms for contraction bidimensional parameters

As for minor bidimensionality, we need to prove that

I If tw(G) = Ω(k) then G contains k as a contraction.

53/57 Πk = Γk+ a new vertex vnew, connected to all the vertices in V (Γk).

TwoBidimensionality important for minors and contractions grid-like graphs The irrelevant vertex technique

Limits of bidimensionality

What about contraction-closed parameters? Two pattern graphs Γk and Πk : We deﬁne the following two pattern graphs Γk and Πk:

vnew

Γk = Πk =

Πk = Γk + a new universal vertex vnew. Dimitrios M. Thilikos ΕΚΠΑ-NKUA Algorithmic Graph Minor Theory Part 2 77

54/57 Theorem (Fomin, Golovach, Thilikos. 2009)

For every graph H, there isc H > 0 such that every connected H-minor-free 2 graph of treewidth at least cH · ` contains Γ` or Π` as a contraction.

Theorem (Fomin, Golovach, Thilikos. 2009)

For every apex graph H, there isc H > 0 such that every connected H-minor-free graph of treewidth at least cH · ` contains Γ` as a contraction.

The “contraction-certiﬁcates” for large treewidth

Theorem (Fomin, Golovach, Thilikos. 2009)

For any integer ` > 0, there isc ` such that every connected graph of treewidth at least c`, containsK `, Γ`, or Π` as a contraction.

55/57 Theorem (Fomin, Golovach, Thilikos. 2009)

For every apex graph H, there isc H > 0 such that every connected H-minor-free graph of treewidth at least cH · ` contains Γ` as a contraction.

The “contraction-certiﬁcates” for large treewidth

Theorem (Fomin, Golovach, Thilikos. 2009)

For any integer ` > 0, there isc ` such that every connected graph of treewidth at least c`, containsK `, Γ`, or Π` as a contraction.

Theorem (Fomin, Golovach, Thilikos. 2009)

For every graph H, there isc H > 0 such that every connected H-minor-free 2 graph of treewidth at least cH · ` contains Γ` or Π` as a contraction.

55/57 The “contraction-certiﬁcates” for large treewidth

Theorem (Fomin, Golovach, Thilikos. 2009)

For any integer ` > 0, there isc ` such that every connected graph of treewidth at least c`, containsK `, Γ`, or Π` as a contraction.

Theorem (Fomin, Golovach, Thilikos. 2009)

For every graph H, there isc H > 0 such that every connected H-minor-free 2 graph of treewidth at least cH · ` contains Γ` or Π` as a contraction.

Theorem (Fomin, Golovach, Thilikos. 2009)

For every apex graph H, there isc H > 0 such that every connected H-minor-free graph of treewidth at least cH · ` contains Γ` as a contraction.

55/57 2 Bidimensionality+ separation properties ⇒ (E)PTAS

[Demaine and Hajiaghayi. 2005] [Fomin, Lokshtanov, Raman, Saurabh. 2011]

3 Bidimensionality+ separation properties ⇒ Kernelization [Fomin, Lokshtanov, Saurabh, Thilikos. 2009-2010]

4 Bidimensionality+ new Grid Theorems ⇒ Geometric graphs [Fomin, Lokshtanov, Saurabh. 2012] [Grigoriev, Koutsonas, Thilikos. 2013]

Further applications of Bidimensionality

1 Bidimensionality+DP ⇒ Subexponential FPT algorithms [Demaine, Fomin, Hajiaghayi, Thilikos. 2004-2005] [Fomin, Golovach, Thilikos. 2009]

56/57 3 Bidimensionality+ separation properties ⇒ Kernelization [Fomin, Lokshtanov, Saurabh, Thilikos. 2009-2010]

4 Bidimensionality+ new Grid Theorems ⇒ Geometric graphs [Fomin, Lokshtanov, Saurabh. 2012] [Grigoriev, Koutsonas, Thilikos. 2013]

Further applications of Bidimensionality

1 Bidimensionality+DP ⇒ Subexponential FPT algorithms [Demaine, Fomin, Hajiaghayi, Thilikos. 2004-2005] [Fomin, Golovach, Thilikos. 2009]

2 Bidimensionality+ separation properties ⇒ (E)PTAS

[Demaine and Hajiaghayi. 2005] [Fomin, Lokshtanov, Raman, Saurabh. 2011]

56/57 4 Bidimensionality+ new Grid Theorems ⇒ Geometric graphs [Fomin, Lokshtanov, Saurabh. 2012] [Grigoriev, Koutsonas, Thilikos. 2013]

Further applications of Bidimensionality

1 Bidimensionality+DP ⇒ Subexponential FPT algorithms [Demaine, Fomin, Hajiaghayi, Thilikos. 2004-2005] [Fomin, Golovach, Thilikos. 2009]

2 Bidimensionality+ separation properties ⇒ (E)PTAS

[Demaine and Hajiaghayi. 2005] [Fomin, Lokshtanov, Raman, Saurabh. 2011]

3 Bidimensionality+ separation properties ⇒ Kernelization [Fomin, Lokshtanov, Saurabh, Thilikos. 2009-2010]

56/57 Further applications of Bidimensionality

1 Bidimensionality+DP ⇒ Subexponential FPT algorithms [Demaine, Fomin, Hajiaghayi, Thilikos. 2004-2005] [Fomin, Golovach, Thilikos. 2009]

2 Bidimensionality+ separation properties ⇒ (E)PTAS

[Demaine and Hajiaghayi. 2005] [Fomin, Lokshtanov, Raman, Saurabh. 2011]

3 Bidimensionality+ separation properties ⇒ Kernelization [Fomin, Lokshtanov, Saurabh, Thilikos. 2009-2010]

4 Bidimensionality+ new Grid Theorems ⇒ Geometric graphs [Fomin, Lokshtanov, Saurabh. 2012] [Grigoriev, Koutsonas, Thilikos. 2013]

56/57 Gr`acies!

CATALONIA, THE NEXT STATE IN EUROPE 57/57