Algorithmic Graph Minors: Turning Combinatorics to Algorithms Page 1/1111 Antichain: an Inﬁnite Sequence on Non- ≤-Comparable Elements

Graph Minors and algorithms The irrelevant vertex technique

Algorithmic Graph Minors:

turning Combinatorics to Algorithms

Dimitrios M. Thilikos Department of Mathematics – µQλ∀ UoA - National and Kapodistrian University of Athens Laboratoire d’Informatique, de Robotique et de Micro´electroniquede Montpellier (LIRMM) Montpellier, February 2, 2012

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 1/1111 Antichain: an inﬁnite sequence on non- ≤-comparable elements.

We say that X is Well-Quasi-Ordered by ≤ if it has no inﬁnite

antichain

Examples:

I 2N is not W.Q.O. by set inclusion.

I N is not W.Q.O. by divisibility. k I N is W.Q.O. by by component-wise ordering.

Graph Minors and algorithms The irrelevant vertex technique

Well Quasi Ordering Theory

Let X be a set and let “≤” be a partial ordering relation on X .

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 2/1112 We say that X is Well-Quasi-Ordered by ≤ if it has no inﬁnite

antichain

Examples:

I 2N is not W.Q.O. by set inclusion.

I N is not W.Q.O. by divisibility. k I N is W.Q.O. by by component-wise ordering.

Graph Minors and algorithms The irrelevant vertex technique

Well Quasi Ordering Theory

Let X be a set and let “≤” be a partial ordering relation on X .

Antichain: an inﬁnite sequence on non- ≤-comparable elements.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 3/1112 Examples:

I 2N is not W.Q.O. by set inclusion.

I N is not W.Q.O. by divisibility. k I N is W.Q.O. by by component-wise ordering.

Graph Minors and algorithms The irrelevant vertex technique

Well Quasi Ordering Theory

Let X be a set and let “≤” be a partial ordering relation on X .

Antichain: an inﬁnite sequence on non- ≤-comparable elements.

We say that X is Well-Quasi-Ordered by ≤ if it has no inﬁnite

antichain

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 4/1112 Graph Minors and algorithms The irrelevant vertex technique

Well Quasi Ordering Theory

Let X be a set and let “≤” be a partial ordering relation on X .

Antichain: an inﬁnite sequence on non- ≤-comparable elements.

We say that X is Well-Quasi-Ordered by ≤ if it has no inﬁnite

antichain

Examples:

I 2N is not W.Q.O. by set inclusion.

I N is not W.Q.O. by divisibility. k I N is W.Q.O. by by component-wise ordering.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 5/1112 The theory of Well-quasi-ordering was ﬁrst developed by

Graham Higman and Erd˝os& Rado

under the name “ﬁnite basis property”

Remind: This talk is about graphs and algorithms!

Graph Minors and algorithms The irrelevant vertex technique

Well Quasi Ordering Theory

General question: Given a set X and an ordering relation ≤ on it,

is X W.Q.O. by to ≤?

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 6/1113 Remind: This talk is about graphs and algorithms!

Graph Minors and algorithms The irrelevant vertex technique

Well Quasi Ordering Theory

General question: Given a set X and an ordering relation ≤ on it,

is X W.Q.O. by to ≤?

The theory of Well-quasi-ordering was ﬁrst developed by

Graham Higman and Erd˝os& Rado

under the name “ﬁnite basis property”

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 7/1113 Graph Minors and algorithms The irrelevant vertex technique

Well Quasi Ordering Theory

General question: Given a set X and an ordering relation ≤ on it,

is X W.Q.O. by to ≤?

The theory of Well-quasi-ordering was ﬁrst developed by

Graham Higman and Erd˝os& Rado

under the name “ﬁnite basis property”

Remind: This talk is about graphs and algorithms!

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 8/1113 v 1 \v: vertex removal: e 2 \e: edge removal: e 3 /e: edge contraction

Minor Relation: H ≤ G if H can be obtained from G after a sequence of the above operations

Graph Minors and algorithms The irrelevant vertex technique

The minor relation on Graphs

Graph Minors We deﬁne3 local operations on graphs:

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 9/1114 e 2 \e: edge removal: e 3 /e: edge contraction

Minor Relation: H ≤ G if H can be obtained from G after a sequence of the above operations

Graph Minors and algorithms The irrelevant vertex technique

The minor relation on Graphs

Graph Minors We deﬁne3 local operations on graphs:

v 1 \v: vertex removal:

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 10/1114 e 3 /e: edge contraction

Minor Relation: H ≤ G if H can be obtained from G after a sequence of the above operations

Graph Minors and algorithms The irrelevant vertex technique

The minor relation on Graphs

Graph Minors We deﬁne3 local operations on graphs:

v 1 \v: vertex removal: e 2 \e: edge removal:

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 11/1114 Minor Relation: H ≤ G if H can be obtained from G after a sequence of the above operations

Graph Minors and algorithms The irrelevant vertex technique

The minor relation on Graphs

Graph Minors We deﬁne3 local operations on graphs:

v 1 \v: vertex removal: e 2 \e: edge removal: e 3 /e: edge contraction

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 12/1114 Graph Minors and algorithms The irrelevant vertex technique

The minor relation on Graphs

Graph Minors We deﬁne3 local operations on graphs:

v 1 \v: vertex removal: e 2 \e: edge removal: e 3 /e: edge contraction

Minor Relation: H ≤ G if H can be obtained from G after a sequence of the above operations

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 13/1114 Graph Minors and algorithms The irrelevant vertex technique

Wagner’s Conjecture

Wagner’s Conjecture:

I The set of all graphs is W.Q.O. by the minor relation [formulated by Klaus Wagner in the 1930s (?)]

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 14/1115 Graph Minors and algorithms The irrelevant vertex technique

Wagner’s Conjecture

Wagner’s Conjecture:

I The set of all graphs is W.Q.O. by the minor relation [formulated by Klaus Wagner in the 1930s (?)]

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 15/1115 Now it is known as the Robertson & Seymour Theorem. Width of the proof: < 10 cm (23 papers)

10/11 Fulkerson Prize (2006) (2/3 for N.R. & 4/5 for P.S.)

Graph Minors and algorithms The irrelevant vertex technique

The Graph Minors Series

This conjecture was proven by Neil Robertson and Paul Seymour in their Graph Minor series of papers.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 16/1116 Now it is known as the Robertson & Seymour Theorem. Width of the proof: < 10 cm (23 papers)

10/11 Fulkerson Prize (2006) (2/3 for N.R. & 4/5 for P.S.)

Graph Minors and algorithms The irrelevant vertex technique

The Graph Minors Series

This conjecture was proven by Neil Robertson and Paul Seymour in their Graph Minor series of papers.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 17/1116 Now it is known as the Robertson & Seymour Theorem. Width of the proof: < 10 cm (23 papers)

10/11 Fulkerson Prize (2006) (2/3 for N.R. & 4/5 for P.S.)

Graph Minors and algorithms The irrelevant vertex technique

The Graph Minors Series

This conjecture was proven by Neil Robertson and Paul Seymour in their Graph Minor series of papers.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 18/1116 Width of the proof: < 10 cm (23 papers)

10/11 Fulkerson Prize (2006) (2/3 for N.R. & 4/5 for P.S.)

Graph Minors and algorithms The irrelevant vertex technique

The Graph Minors Series

This conjecture was proven by Neil Robertson and Paul Seymour in their Graph Minor series of papers.

Now it is known as the Robertson & Seymour Theorem.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 19/1116 (23 papers)

10/11 Fulkerson Prize (2006) (2/3 for N.R. & 4/5 for P.S.)

Graph Minors and algorithms The irrelevant vertex technique

The Graph Minors Series

This conjecture was proven by Neil Robertson and Paul Seymour in their Graph Minor series of papers.

Now it is known as the Robertson & Seymour Theorem. Width of the proof: < 10 cm

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 20/1116 10/11 Fulkerson Prize (2006) (2/3 for N.R. & 4/5 for P.S.)

Graph Minors and algorithms The irrelevant vertex technique

The Graph Minors Series

This conjecture was proven by Neil Robertson and Paul Seymour in their Graph Minor series of papers.

Now it is known as the Robertson & Seymour Theorem. Width of the proof: < 10 cm (23 papers)

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 21/1116 Graph Minors and algorithms The irrelevant vertex technique

The Graph Minors Series

This conjecture was proven by Neil Robertson and Paul Seymour in their Graph Minor series of papers.

Now it is known as the Robertson & Seymour Theorem. Width of the proof: < 10 cm (23 papers)

10/11 Fulkerson Prize (2006) (2/3 for N.R. & 4/5 for P.S.)

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 22/1116 some of

Graph Minors and algorithms The irrelevant vertex technique

The Graph Minors Series

In this talk we will present ( ) the algorithmic applications of the Graph Minors Series.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 23/1117 Graph Minors and algorithms The irrelevant vertex technique

The Graph Minors Series

In this talk we will present (some of) the algorithmic applications of the Graph Minors Series.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 24/1117 I Not all parameterized problems admit FPT-algorithms. There are parameterized complexity classes like W[1], W[2], or W[P] and

adequate reductions such that when a problem is hard for them is not

expected to have an FPT-algorithm.

Graph Minors and algorithms The irrelevant vertex technique

Parameterized complexity

We say that a parameterized (by k) problem belongs in the

parameterized complexity class FPT if it can be solved by an

FPT-algorithm, that is an algorithm that runs in

O(f(k) · nO(1)) steps

(n is the size of the input, f depends only one the parameter k.)

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 25/1118 There are parameterized complexity classes like W[1], W[2], or W[P] and

adequate reductions such that when a problem is hard for them is not

expected to have an FPT-algorithm.

Graph Minors and algorithms The irrelevant vertex technique

Parameterized complexity

We say that a parameterized (by k) problem belongs in the

parameterized complexity class FPT if it can be solved by an

FPT-algorithm, that is an algorithm that runs in

O(f(k) · nO(1)) steps

(n is the size of the input, f depends only one the parameter k.)

I Not all parameterized problems admit FPT-algorithms.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 26/1118 Graph Minors and algorithms The irrelevant vertex technique

Parameterized complexity

We say that a parameterized (by k) problem belongs in the

parameterized complexity class FPT if it can be solved by an

FPT-algorithm, that is an algorithm that runs in

O(f(k) · nO(1)) steps

(n is the size of the input, f depends only one the parameter k.)

I Not all parameterized problems admit FPT-algorithms. There are parameterized complexity classes like W[1], W[2], or W[P] and

adequate reductions such that when a problem is hard for them is not

expected to have an FPT-algorithm. Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 27/1118 A meta-problem:

k-Parameter p-Checking Instance: a graph G and an integer k ≥ 0. Parameter: k Question: p(G)≤ k?

p can be the minimum Vertex Cover, Dominating Set, Edge Dominating

Set, Chromatic Number, Feedback Vertex Set, e.t.c.

I Holy grail (meta)-question For which functions p it holds that k-Parameter p-Checking∈FPT? (i.e., there is an f(k)·nO(1)-step algorithm checking whether p(G) ≤ k?)

Graph Minors and algorithms The irrelevant vertex technique

Parameters on graphs

graph parameter: a function p that maps graphs to integers.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 28/1119 p can be the minimum Vertex Cover, Dominating Set, Edge Dominating

Set, Chromatic Number, Feedback Vertex Set, e.t.c.

I Holy grail (meta)-question For which functions p it holds that k-Parameter p-Checking∈FPT? (i.e., there is an f(k)·nO(1)-step algorithm checking whether p(G) ≤ k?)

Graph Minors and algorithms The irrelevant vertex technique

Parameters on graphs

graph parameter: a function p that maps graphs to integers. A meta-problem:

k-Parameter p-Checking Instance: a graph G and an integer k ≥ 0. Parameter: k Question: p(G)≤ k?

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 29/1119 I Holy grail (meta)-question For which functions p it holds that k-Parameter p-Checking∈FPT? (i.e., there is an f(k)·nO(1)-step algorithm checking whether p(G) ≤ k?)

Graph Minors and algorithms The irrelevant vertex technique

Parameters on graphs

graph parameter: a function p that maps graphs to integers. A meta-problem:

k-Parameter p-Checking Instance: a graph G and an integer k ≥ 0. Parameter: k Question: p(G)≤ k?

p can be the minimum Vertex Cover, Dominating Set, Edge Dominating

Set, Chromatic Number, Feedback Vertex Set, e.t.c.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 30/1119 For which functions p it holds that k-Parameter p-Checking∈FPT? (i.e., there is an f(k)·nO(1)-step algorithm checking whether p(G) ≤ k?)

Graph Minors and algorithms The irrelevant vertex technique

Parameters on graphs

graph parameter: a function p that maps graphs to integers. A meta-problem:

k-Parameter p-Checking Instance: a graph G and an integer k ≥ 0. Parameter: k Question: p(G)≤ k?

p can be the minimum Vertex Cover, Dominating Set, Edge Dominating

Set, Chromatic Number, Feedback Vertex Set, e.t.c.

I Holy grail (meta)-question

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 31/1119 Graph Minors and algorithms The irrelevant vertex technique

Parameters on graphs

graph parameter: a function p that maps graphs to integers. A meta-problem:

k-Parameter p-Checking Instance: a graph G and an integer k ≥ 0. Parameter: k Question: p(G)≤ k?

p can be the minimum Vertex Cover, Dominating Set, Edge Dominating

Set, Chromatic Number, Feedback Vertex Set, e.t.c.

I Holy grail (meta)-question For which functions p it holds that k-Parameter p-Checking∈FPT? (i.e., there is an f(k)·nO(1)-step algorithm checking whether p(G) ≤ k?)

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 32/1119 Minor-closed parameters:

I vertex cover, vc(G) I feedback vertex set, fvs(G) I branchwidth/treewidth/pathwidth/tree-depth, bw(G)/tw(G)/pw(G)/td(G)

I minimum maximal matching, mmm(G) I p(G) = |V (G)| − α(G) (α(G) is the max independent set size) I the genus of a graph, γ(G) I the apex number of a graph, apx(G)

I p(G) = min{k | Pk 6≤ G} (Longest Path)

Graph Minors and algorithms The irrelevant vertex technique

Parameters on graphs

A parameter p is minor closed if H ≤ G ⇒ p(H) ≤p(G).

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 33/111 10 Graph Minors and algorithms The irrelevant vertex technique

Parameters on graphs

A parameter p is minor closed if H ≤ G ⇒ p(H) ≤p(G). Minor-closed parameters:

I vertex cover, vc(G) I feedback vertex set, fvs(G) I branchwidth/treewidth/pathwidth/tree-depth, bw(G)/tw(G)/pw(G)/td(G)

I minimum maximal matching, mmm(G) I p(G) = |V (G)| − α(G) (α(G) is the max independent set size) I the genus of a graph, γ(G) I the apex number of a graph, apx(G)

I p(G) = min{k | Pk 6≤ G} (Longest Path)

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 34/111 10 I If p is minor closed then k-Parameter p-Checking ∈FPT.

In other words,

3 I p(G) ≤ k can be checked in f(k) · n steps. 3 I Every minor-closed graph class G can be recognized in O(n ). [Take p(G) = 0 if G ∈ G and p(G) = 1, otherwise]

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithmic consequence of GMT

Main meta-algorithmic consequence of GMT:

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 35/111 11 In other words,

3 I p(G) ≤ k can be checked in f(k) · n steps. 3 I Every minor-closed graph class G can be recognized in O(n ). [Take p(G) = 0 if G ∈ G and p(G) = 1, otherwise]

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithmic consequence of GMT

Main meta-algorithmic consequence of GMT:

I If p is minor closed then k-Parameter p-Checking ∈FPT.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 36/111 11 Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithmic consequence of GMT

Main meta-algorithmic consequence of GMT:

I If p is minor closed then k-Parameter p-Checking ∈FPT.

In other words,

3 I p(G) ≤ k can be checked in f(k) · n steps. 3 I Every minor-closed graph class G can be recognized in O(n ). [Take p(G) = 0 if G ∈ G and p(G) = 1, otherwise]

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 37/111 11 3 I p(G) ≤ k can be checked in f(k) · n steps.

What is f?

3 I Every minor-closed graph class G can be recognized in O(n ).

What is hidden in the O-notation?

A-question: Is there any practical algorithm here?

... go back to the proofs!

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithmic consequence of GMT

Questions on the last two versions of the meta-algorithmic result:

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 38/111 12 What is f?

3 I Every minor-closed graph class G can be recognized in O(n ).

What is hidden in the O-notation?

A-question: Is there any practical algorithm here?

... go back to the proofs!

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithmic consequence of GMT

Questions on the last two versions of the meta-algorithmic result:

3 I p(G) ≤ k can be checked in f(k) · n steps.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 39/111 12 3 I Every minor-closed graph class G can be recognized in O(n ).

What is hidden in the O-notation?

A-question: Is there any practical algorithm here?

... go back to the proofs!

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithmic consequence of GMT

Questions on the last two versions of the meta-algorithmic result:

3 I p(G) ≤ k can be checked in f(k) · n steps.

What is f?

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 40/111 12 What is hidden in the O-notation?

A-question: Is there any practical algorithm here?

... go back to the proofs!

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithmic consequence of GMT

Questions on the last two versions of the meta-algorithmic result:

3 I p(G) ≤ k can be checked in f(k) · n steps.

What is f?

3 I Every minor-closed graph class G can be recognized in O(n ).

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 41/111 12 A-question: Is there any practical algorithm here?

... go back to the proofs!

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithmic consequence of GMT

Questions on the last two versions of the meta-algorithmic result:

3 I p(G) ≤ k can be checked in f(k) · n steps.

What is f?

3 I Every minor-closed graph class G can be recognized in O(n ).

What is hidden in the O-notation?

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 42/111 12 ... go back to the proofs!

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithmic consequence of GMT

Questions on the last two versions of the meta-algorithmic result:

3 I p(G) ≤ k can be checked in f(k) · n steps.

What is f?

3 I Every minor-closed graph class G can be recognized in O(n ).

What is hidden in the O-notation?

A-question: Is there any practical algorithm here?

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 43/111 12 Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithmic consequence of GMT

Questions on the last two versions of the meta-algorithmic result:

3 I p(G) ≤ k can be checked in f(k) · n steps.

What is f?

3 I Every minor-closed graph class G can be recognized in O(n ).

What is hidden in the O-notation?

A-question: Is there any practical algorithm here?

... go back to the proofs!

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 44/111 12 Graph Minors and algorithms The irrelevant vertex technique

Meta-algorithms from Graph Minors

I For any minor closed parameter p and any k, we deﬁne obk(p) as the set of minor-minimal elements in

{G | p(G) >k }

I we call obk(p) obstruction family of p.

I Observe: p(G) ≤ k ⇔ ∀H∈obk(p) H 6≤ G

I Observe: obk(p) is an antichain.

I GMT Consequence: obk(p) is ﬁnite!

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 45/111 13 The meta-problem is reduced to the k-Minor Containment problem. k-Minor Containment problem can be solved in O(g(k) · n3) steps

3 The whole algorithm takes O(|obk(p)| · g(k) · n ) steps.

Good news: g(k) is constructible!

A-news: g(k) is huge!

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithm

An algorithm for the k-Parameter p-Checking problem

1. for all H ∈ obk(p)

3 2. check (in O(g(k) · n ) steps) whether H ≤mn G 3. and if this holds, then output NO 4. output YES.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 46/111 14 k-Minor Containment problem can be solved in O(g(k) · n3) steps

3 The whole algorithm takes O(|obk(p)| · g(k) · n ) steps.

Good news: g(k) is constructible!

A-news: g(k) is huge!

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithm

An algorithm for the k-Parameter p-Checking problem

1. for all H ∈ obk(p)

3 2. check (in O(g(k) · n ) steps) whether H ≤mn G 3. and if this holds, then output NO 4. output YES.

The meta-problem is reduced to the k-Minor Containment problem.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 47/111 14 3 The whole algorithm takes O(|obk(p)| · g(k) · n ) steps.

Good news: g(k) is constructible!

A-news: g(k) is huge!

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithm

An algorithm for the k-Parameter p-Checking problem

1. for all H ∈ obk(p)

3 2. check (in O(g(k) · n ) steps) whether H ≤mn G 3. and if this holds, then output NO 4. output YES.

The meta-problem is reduced to the k-Minor Containment problem. k-Minor Containment problem can be solved in O(g(k) · n3) steps

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 48/111 14 Good news: g(k) is constructible!

A-news: g(k) is huge!

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithm

An algorithm for the k-Parameter p-Checking problem

1. for all H ∈ obk(p)

3 2. check (in O(g(k) · n ) steps) whether H ≤mn G 3. and if this holds, then output NO 4. output YES.

The meta-problem is reduced to the k-Minor Containment problem. k-Minor Containment problem can be solved in O(g(k) · n3) steps

3 The whole algorithm takes O(|obk(p)| · g(k) · n ) steps.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 49/111 14 A-news: g(k) is huge!

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithm

An algorithm for the k-Parameter p-Checking problem

1. for all H ∈ obk(p)

3 2. check (in O(g(k) · n ) steps) whether H ≤mn G 3. and if this holds, then output NO 4. output YES.

The meta-problem is reduced to the k-Minor Containment problem. k-Minor Containment problem can be solved in O(g(k) · n3) steps

3 The whole algorithm takes O(|obk(p)| · g(k) · n ) steps.

Good news: g(k) is constructible!

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 50/111 14 Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithm

An algorithm for the k-Parameter p-Checking problem

1. for all H ∈ obk(p)

3 2. check (in O(g(k) · n ) steps) whether H ≤mn G 3. and if this holds, then output NO 4. output YES.

The meta-problem is reduced to the k-Minor Containment problem. k-Minor Containment problem can be solved in O(g(k) · n3) steps

3 The whole algorithm takes O(|obk(p)| · g(k) · n ) steps.

Good news: g(k) is constructible!

A-news: g(k) is huge! Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 51/111 14 1. the above algorithm “exists” but cannot be constructed as we

do not know obk(p)

I There is no TM that, given a machine description of p, can

produce obk(p). [Fellows & Langston, JCSS 1994]

2. we know obk(p) for few parameters and for small values of k

3. when we have upper bounds for |obk(p)|, they are immense.

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithm

AA-facts on the main meta-algorithmic result of GMT.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 52/111 15 I There is no TM that, given a machine description of p, can

produce obk(p). [Fellows & Langston, JCSS 1994]

2. we know obk(p) for few parameters and for small values of k

3. when we have upper bounds for |obk(p)|, they are immense.

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithm

AA-facts on the main meta-algorithmic result of GMT.

1. the above algorithm “exists” but cannot be constructed as we

do not know obk(p)

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 53/111 15 2. we know obk(p) for few parameters and for small values of k

3. when we have upper bounds for |obk(p)|, they are immense.

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithm

AA-facts on the main meta-algorithmic result of GMT.

1. the above algorithm “exists” but cannot be constructed as we

do not know obk(p)

I There is no TM that, given a machine description of p, can

produce obk(p). [Fellows & Langston, JCSS 1994]

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 54/111 15 3. when we have upper bounds for |obk(p)|, they are immense.

Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithm

AA-facts on the main meta-algorithmic result of GMT.

1. the above algorithm “exists” but cannot be constructed as we

do not know obk(p)

I There is no TM that, given a machine description of p, can

produce obk(p). [Fellows & Langston, JCSS 1994]

2. we know obk(p) for few parameters and for small values of k

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 55/111 15 Graph Minors and algorithms The irrelevant vertex technique

The meta-algorithm

AA-facts on the main meta-algorithmic result of GMT.

1. the above algorithm “exists” but cannot be constructed as we

do not know obk(p)

I There is no TM that, given a machine description of p, can

produce obk(p). [Fellows & Langston, JCSS 1994]

2. we know obk(p) for few parameters and for small values of k

3. when we have upper bounds for |obk(p)|, they are immense.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 56/111 15 k-Minor Containment Instance: two graphs G and H. Parameter: k = |V (H)| Question: H ≤ G?

k-Disjoint Paths Instance: A graph G and a sequence of pairs of terminals

k T = (s1, t1),..., (sk, tk) ∈ (V (G) × V (G)) . Parameter: k.

Question: Are there k pairwise vertex disjoint paths P1,...,Pk in G such that

for every i ∈ {1,...,k }, Pi has endpoints si and ti?

Graph Minors and algorithms The irrelevant vertex technique

Two problems

Robertson& Seymour, proved the following:

Theorem (GM-VI, GM-VII, GM-XIII, GM-XXI, GM-XII) The following two problems can be solved in O(g(k) · n3) steps:

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 57/111 16 k-Disjoint Paths Instance: A graph G and a sequence of pairs of terminals

k T = (s1, t1),..., (sk, tk) ∈ (V (G) × V (G)) . Parameter: k.

Question: Are there k pairwise vertex disjoint paths P1,...,Pk in G such that

for every i ∈ {1,...,k }, Pi has endpoints si and ti?

Graph Minors and algorithms The irrelevant vertex technique

Two problems

Robertson& Seymour, proved the following:

Theorem (GM-VI, GM-VII, GM-XIII, GM-XXI, GM-XII) The following two problems can be solved in O(g(k) · n3) steps:

k-Minor Containment Instance: two graphs G and H. Parameter: k = |V (H)| Question: H ≤ G?

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 58/111 16 Graph Minors and algorithms The irrelevant vertex technique

Two problems

Robertson& Seymour, proved the following:

Theorem (GM-VI, GM-VII, GM-XIII, GM-XXI, GM-XII) The following two problems can be solved in O(g(k) · n3) steps:

k-Minor Containment Instance: two graphs G and H. Parameter: k = |V (H)| Question: H ≤ G?

k-Disjoint Paths Instance: A graph G and a sequence of pairs of terminals

k T = (s1, t1),..., (sk, tk) ∈ (V (G) × V (G)) . Parameter: k.

Question: Are there k pairwise vertex disjoint paths P1,...,Pk in G such that

for every i ∈ {1,...,k }, Pi has endpoints si and ti?

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 59/111 16 Graph Minors and algorithms The irrelevant vertex technique

Two problems

Both k-Minor Containment and k-Disjoint Paths

Problem where solved using the

The irrelevant vertex Technique

introduced in [GM XIII]

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 60/111 17 Idea: I Find irrelevant vertex and recurse!

We give a outline of the idea for the case of k-Disjoint Paths.

Graph Minors and algorithms The irrelevant vertex technique

Two problems

Given an instance (G,T,k ) of the k-Disjoint Paths problem, a vertex v ∈ V (G) is an irrelevant vertex of G if

(G,T,k ) and (G \ v, T,k ) are equivalent instances of the problem.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 61/111 18 We give a outline of the idea for the case of k-Disjoint Paths.

Graph Minors and algorithms The irrelevant vertex technique

Two problems

Given an instance (G,T,k ) of the k-Disjoint Paths problem, a vertex v ∈ V (G) is an irrelevant vertex of G if

(G,T,k ) and (G \ v, T,k ) are equivalent instances of the problem.

Idea: I Find irrelevant vertex and recurse!

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 62/111 18 Graph Minors and algorithms The irrelevant vertex technique

Two problems

Given an instance (G,T,k ) of the k-Disjoint Paths problem, a vertex v ∈ V (G) is an irrelevant vertex of G if

(G,T,k ) and (G \ v, T,k ) are equivalent instances of the problem.

Idea: I Find irrelevant vertex and recurse!

We give a outline of the idea for the case of k-Disjoint Paths.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 63/111 18 Here Gk represents some structural condition for the problem input.

Graph Minors and algorithms The irrelevant vertex technique

The general scheme

The general scheme of the algorithm in [GM XIII] is the following:

Input: An instance (G, T, k) of k-Disjoint Paths Output: An equivalent instance (G,T,k ) k-Disjoint Paths

1. while G 6∈G k, 2. ﬁnd an irrelevant vertex v in G 3. set G ← G \ v 4. output (G,T,k )

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 64/111 19 Graph Minors and algorithms The irrelevant vertex technique

The general scheme

The general scheme of the algorithm in [GM XIII] is the following:

Input: An instance (G, T, k) of k-Disjoint Paths Output: An equivalent instance (G,T,k ) k-Disjoint Paths

1. while G 6∈G k, 2. ﬁnd an irrelevant vertex v in G 3. set G ← G \ v 4. output (G,T,k )

Here Gk represents some structural condition for the problem input.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 65/111 19 Graph Minors and algorithms The irrelevant vertex technique

The general scheme

The algorithmic scheme depends on the parameterized class Gk and

creates an equivalent instance that belongs in Gk.

I It is applied ﬁrst for

Gk = {G | G is a Kh(k)-minor free graph}

and then for

Gk = {G | G is a Γj(k)-minor free graph},

for some suitable choice of recursive functions h, j : N → N.

I Γr is the (r × r)-grid.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 66/111 20 Graph Minors and algorithms The irrelevant vertex technique

The general scheme

Phase 1: What to do with a ”big” clique minor?

(technical) details omitted...

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 67/111 21 Two answers:

I Weak Structure Theorem [GM XIII] I Strong Structure Theorem [GM XVI]

Graph Minors and algorithms The irrelevant vertex technique

The general scheme

Assume now that the input graph excludes the clique Kh(k) as a minor. Combinatorial question: How such graphs look like?

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 68/111 22 I Weak Structure Theorem [GM XIII] I Strong Structure Theorem [GM XVI]

Graph Minors and algorithms The irrelevant vertex technique

The general scheme

Assume now that the input graph excludes the clique Kh(k) as a minor. Combinatorial question: How such graphs look like? Two answers:

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 69/111 22 I Strong Structure Theorem [GM XVI]

Graph Minors and algorithms The irrelevant vertex technique

The general scheme

Assume now that the input graph excludes the clique Kh(k) as a minor. Combinatorial question: How such graphs look like? Two answers:

I Weak Structure Theorem [GM XIII]

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 70/111 22 Graph Minors and algorithms The irrelevant vertex technique

The general scheme

Assume now that the input graph excludes the clique Kh(k) as a minor. Combinatorial question: How such graphs look like? Two answers:

I Weak Structure Theorem [GM XIII] I Strong Structure Theorem [GM XVI]

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 71/111 22 (This is now excluded for r = h(k))

(This is what we want to achieve!)

and the compass of W has a rural division D such that each internal

ﬂap of D has treewidth at most g1(r, q).(irrelevant vertex wanted...)

Graph Minors and algorithms The irrelevant vertex technique

Weak structure theorem

Theorem (Weak Structure Theorem)

There exists recursive functions g1 : N × N → N and g2 : N → N, such that for every graph G and every r, q ∈ N, one of the following holds:

1 Kr is a minor of G,

2 Γg1(r,q) is not a minor of G,

3 ∃X ⊆ V (G) with |X| ≤ g2(r) such that G \ X contains as a subgraph a ﬂat subdivided wall W where W has height q

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 72/111 23 (This is now excluded for r = h(k))

(This is what we want to achieve!)

(irrelevant vertex wanted...)

Graph Minors and algorithms The irrelevant vertex technique

Weak structure theorem

Theorem (Weak Structure Theorem)

There exists recursive functions g1 : N × N → N and g2 : N → N, such that for every graph G and every r, q ∈ N, one of the following holds:

1 Kr is a minor of G,

2 Γg1(r,q) is not a minor of G,

3 ∃X ⊆ V (G) with |X| ≤ g2(r) such that G \ X contains as a subgraph a ﬂat subdivided wall W where W has height q and the compass of W has a rural division D such that each internal

ﬂap of D has treewidth at most g1(r, q).

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 73/111 23 (This is what we want to achieve!)

(irrelevant vertex wanted...)

Graph Minors and algorithms The irrelevant vertex technique

Weak structure theorem

Theorem (Weak Structure Theorem)

There exists recursive functions g1 : N × N → N and g2 : N → N, such that for every graph G and every r, q ∈ N, one of the following holds:

1 Kr is a minor of G, (This is now excluded for r = h(k))

2 Γg1(r,q) is not a minor of G,

3 ∃X ⊆ V (G) with |X| ≤ g2(r) such that G \ X contains as a subgraph a ﬂat subdivided wall W where W has height q and the compass of W has a rural division D such that each internal

ﬂap of D has treewidth at most g1(r, q).

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 74/111 23 (irrelevant vertex wanted...)

Graph Minors and algorithms The irrelevant vertex technique

Weak structure theorem

Theorem (Weak Structure Theorem)

There exists recursive functions g1 : N × N → N and g2 : N → N, such that for every graph G and every r, q ∈ N, one of the following holds:

1 Kr is a minor of G, (This is now excluded for r = h(k))

2 Γg1(r,q) is not a minor of G, (This is what we want to achieve!)

3 ∃X ⊆ V (G) with |X| ≤ g2(r) such that G \ X contains as a subgraph a ﬂat subdivided wall W where W has height q and the compass of W has a rural division D such that each internal

ﬂap of D has treewidth at most g1(r, q).

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 75/111 23 Graph Minors and algorithms The irrelevant vertex technique

Weak structure theorem

Theorem (Weak Structure Theorem)

There exists recursive functions g1 : N × N → N and g2 : N → N, such that for every graph G and every r, q ∈ N, one of the following holds:

1 Kr is a minor of G, (This is now excluded for r = h(k))

2 Γg1(r,q) is not a minor of G, (This is what we want to achieve!)

3 ∃X ⊆ V (G) with |X| ≤ g2(r) such that G \ X contains as a subgraph a ﬂat subdivided wall W where W has height q and the compass of W has a rural division D such that each internal

ﬂap of D has treewidth at most g1(r, q).(irrelevant vertex wanted...)

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 76/111 23 Graph Minors and algorithms The irrelevant vertex technique

Weak structure theorem

A subdivided Wall W of heigh 5:

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 77/111 24 Graph Minors and algorithms The irrelevant vertex technique

Weak structure theorem

X G \ X

The compass is the part of the G \ X that is “inside” the perimeter of the subdivided wall W . The perimeter is as a separator between the internal compass vertices and the part of G \ X that is outside the perimeter

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 78/111 25 Graph Minors and algorithms The irrelevant vertex technique

Weak structure theorem

X G \ X

The compass can be decomposed to graphs of bounded treewidth (ﬂaps) whose

“roots” have size ≤ 3 and form a planar hypergraph inside the disk bounded by

the perimeter Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 79/111 26 Graph Minors and algorithms The irrelevant vertex technique

Weak structure theorem

Weak structure theorem −→ sunny forest theorem!

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 80/111 27 Graph Minors and algorithms The irrelevant vertex technique

Weak structure theorem

We examine only the simpler case where X = ∅.

[the forest is dark!]

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 81/111 28 Graph Minors and algorithms The irrelevant vertex technique

Linkages

A solution to the k-Disjoint Paths Problem, for k = 12

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 82/111 29 Graph Minors and algorithms The irrelevant vertex technique

Linkages

The middle vertex of the subdivided wall W

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 83/111 30 Graph Minors and algorithms The irrelevant vertex technique

Linkages

A way to avoid the middle vertex

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 84/111 31 Graph Minors and algorithms The irrelevant vertex technique

Linkages

Is it always possible to avoid the middle vertex?

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 85/111 32 Graph Minors and algorithms The irrelevant vertex technique

Linkages

The answer is YES given that the height of W is at least λ(k)!

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 86/111 33 Graph Minors and algorithms The irrelevant vertex technique

Linkages

Therefore, if the height of W is ”big enough”, then we can safely

detect an irrelevant vertex (and remove it)!

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 87/111 34 I After the second phase, we have an equivalent instance satisfying cond.2 . I This means that G has treewidth bounded by some function of k: the problem can be solved using dynamic programming.

Graph Minors and algorithms The irrelevant vertex technique

Linkages

Theorem (Weak Structure Theorem)

There exists recursive functions g1 : N × N → N and g2 : N → N, such that for every graph G and every r, q ∈ N, one of the following holds:

1 Kr is a minor of G, (This is now excluded for r = h(k))

2 Γg1(r,q) is not a minor of G, (This is what we want to achieve!)

3 ∃X ⊆ V (G) with |X| ≤ g2(r) such that G \ X contains as a subgraph a ﬂat subdivided wall W where W has height q z and the compass of W has a rural division D such that each internal ﬂap of D

has treewidth at most g1(r, q). (irrelevant vertex wanted...)

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 88/111 35 I This means that G has treewidth bounded by some function of k: the problem can be solved using dynamic programming.

Graph Minors and algorithms The irrelevant vertex technique

Linkages

Theorem (Weak Structure Theorem)

There exists recursive functions g1 : N × N → N and g2 : N → N, such that for every graph G and every r, q ∈ N, one of the following holds:

1 Kr is a minor of G, (This is now excluded for r = h(k))

2 Γg1(r,q) is not a minor of G, (This is what we want to achieve!)

has treewidth at most g1(r, q). (irrelevant vertex wanted...)

I After the second phase, we have an equivalent instance satisfying cond.2 .

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 89/111 35 Graph Minors and algorithms The irrelevant vertex technique

Linkages

Theorem (Weak Structure Theorem)

There exists recursive functions g1 : N × N → N and g2 : N → N, such that for every graph G and every r, q ∈ N, one of the following holds:

1 Kr is a minor of G, (This is now excluded for r = h(k))

2 Γg1(r,q) is not a minor of G, (This is what we want to achieve!)

has treewidth at most g1(r, q). (irrelevant vertex wanted...)

I After the second phase, we have an equivalent instance satisfying cond.2 . I This means that G has treewidth bounded by some function of k: the problem can be solved using dynamic programming. Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 90/111 35 I Proved in [GM XXI]. I The proof uses the ”Vital Linkage Lemma” (proved in [GM XXI]) and the ”Strong Structural Theorem of GMT” (proved in [GM XVI]) A What is the estimation of λ(k)? Huge!

Graph Minors and algorithms The irrelevant vertex technique

Linkages

We can reroute the k disjoint path, given that the height of W is at least λ(k)!

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 91/111 36 I The proof uses the ”Vital Linkage Lemma” (proved in [GM XXI]) and the ”Strong Structural Theorem of GMT” (proved in [GM XVI]) A What is the estimation of λ(k)? Huge!

Graph Minors and algorithms The irrelevant vertex technique

Linkages

We can reroute the k disjoint path, given that the height of W is at least λ(k)!

I Proved in [GM XXI].

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 92/111 36 Huge!

Graph Minors and algorithms The irrelevant vertex technique

Linkages

We can reroute the k disjoint path, given that the height of W is at least λ(k)!

I Proved in [GM XXI]. I The proof uses the ”Vital Linkage Lemma” (proved in [GM XXI]) and the ”Strong Structural Theorem of GMT” (proved in [GM XVI]) A What is the estimation of λ(k)? Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 93/111 36 Graph Minors and algorithms The irrelevant vertex technique

Linkages

We can reroute the k disjoint path, given that the height of W is at least λ(k)!

I Proved in [GM XXI]. I The proof uses the ”Vital Linkage Lemma” (proved in [GM XXI]) and the ”Strong Structural Theorem of GMT” (proved in [GM XVI]) A What is the estimation of λ(k)? Huge! Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 94/111 36 David Johnson also estimated that just one constant in the parametric dependence of the strong structural Theorem to be roughly

↑2↑Θ(r) 22 2↑22 . . . 2 where 2↑r denotes a tower 22 involving r 2’s.

Graph Minors and algorithms The irrelevant vertex technique

Parameteric dependance

David Johnson mentioned in his ongoing guide on NP-completeness:

“for any instance G = (V,E) that one could ﬁt into the known universe, one would easily prefer |V |70 to even constant time, if that constant had to be one of Robertson and Seymour’s”.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 95/111 37 Graph Minors and algorithms The irrelevant vertex technique

Parameteric dependance

David Johnson mentioned in his ongoing guide on NP-completeness:

“for any instance G = (V,E) that one could ﬁt into the known universe, one would easily prefer |V |70 to even constant time, if that constant had to be one of Robertson and Seymour’s”.

David Johnson also estimated that just one constant in the parametric dependence of the strong structural Theorem to be roughly

↑2↑Θ(r) 22 2↑22 . . . 2 where 2↑r denotes a tower 22 involving r 2’s.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 96/111 37 Target 1: Prove that we can reroute the k disjoint paths without using the ”Strong Structural Theorem of GMT”. Achieved! By [Ken-ichi Kawarabayashi, Paul Wollan]: A shorter proof of the graph minor algorithm: the unique linkage theorem. STOC 2010

Graph Minors and algorithms The irrelevant vertex technique

Parameteric dependance

Improvements!

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 97/111 38 Achieved! By [Ken-ichi Kawarabayashi, Paul Wollan]: A shorter proof of the graph minor algorithm: the unique linkage theorem. STOC 2010

Graph Minors and algorithms The irrelevant vertex technique

Parameteric dependance

Improvements!

Target 1: Prove that we can reroute the k disjoint paths without using the ”Strong Structural Theorem of GMT”.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 98/111 38 Achieved! By [Ken-ichi Kawarabayashi, Paul Wollan]: A shorter proof of the graph minor algorithm: the unique linkage theorem. STOC 2010

Achieved! By [Ken-ichi Kawarabayashi, Paul Wollan: A simpler algorithm and shorter proof for the graph minor decomposition. STOC 2011] On going work by: [ Reed, Li, Ken-ichi Kawarabayashi]: On going work (proof: ∼ 100 pages)

Graph Minors and algorithms The irrelevant vertex technique

Parameteric dependance

Improvements!

Target 1: Prove that we can reroute the k disjoint paths without using the ”Strong Structural Theorem of GMT”.

Target 2: Find an alternative proof of the ”Strong Structural Theorem of GMT” that has ”better constants”.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 99/111 38 Achieved! By [Ken-ichi Kawarabayashi, Paul Wollan: A simpler algorithm and shorter proof for the graph minor decomposition. STOC 2011] On going work by: [ Reed, Li, Ken-ichi Kawarabayashi]: On going work (proof: ∼ 100 pages)

Graph Minors and algorithms The irrelevant vertex technique

Parameteric dependance

Improvements!

Target 1: Prove that we can reroute the k disjoint paths without using the ”Strong Structural Theorem of GMT”. Achieved! By [Ken-ichi Kawarabayashi, Paul Wollan]: A shorter proof of the graph minor algorithm: the unique linkage theorem. STOC 2010

Target 2: Find an alternative proof of the ”Strong Structural Theorem of GMT” that has ”better constants”.

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 100/111 38 On going work by: [ Reed, Li, Ken-ichi Kawarabayashi]: On going work (proof: ∼ 100 pages)

Graph Minors and algorithms The irrelevant vertex technique

Parameteric dependance

Improvements!

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 101/111 38 Graph Minors and algorithms The irrelevant vertex technique

Parameteric dependance

Improvements!

Target 2: Find an alternative proof of the ”Strong Structural Theorem of GMT” that has ”better constants”. Achieved! By [Ken-ichi Kawarabayashi, Paul Wollan: A simpler algorithm and shorter proof for the graph minor decomposition. STOC 2011] On going work by: [ Reed, Li, Ken-ichi Kawarabayashi]: On going work Dimitrios(proof: M. Thilikos∼ 100 pages) Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 102/111 38 Ω(k) 22 This, still, gives an algorithm of 22 steps for the k-Disjoint Paths Problem

Graph Minors and algorithms The irrelevant vertex technique

Parameteric dependance

Fact: [Ken-ichi Kawarabayashi, Paul Wollan, STOC 2010]: Gives an

2k estimation of λ(k) = 22 in general gaphs!

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 103/111 39 Graph Minors and algorithms The irrelevant vertex technique

Parameteric dependance

Fact: [Ken-ichi Kawarabayashi, Paul Wollan, STOC 2010]: Gives an

2k estimation of λ(k) = 22 in general gaphs!

Ω(k) 22 This, still, gives an algorithm of 22 steps for the k-Disjoint Paths Problem

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 104/111 39 Planar graphs: By [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos: Tight Bounds for Linkages in Planar Graphs. ICALP 2011] it follows that λ(k) = 2O(k).

Ω(k) This gives an 22 · n3 algorithm for the k-Disjoint Paths Problem

A-Limits of the irrelevant vertex technique:

O(k) 22 · n3 steps is optimal as λ(k) = 2Ω(k)

I A non-A algorithm for the k-Disjoint Paths Problem (and related problems) would require radically diﬀerent techniques!

Graph Minors and algorithms The irrelevant vertex technique

Parameteric dependance

Can we make things better?

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 105/111 40 Ω(k) This gives an 22 · n3 algorithm for the k-Disjoint Paths Problem

A-Limits of the irrelevant vertex technique:

O(k) 22 · n3 steps is optimal as λ(k) = 2Ω(k)

I A non-A algorithm for the k-Disjoint Paths Problem (and related problems) would require radically diﬀerent techniques!

Graph Minors and algorithms The irrelevant vertex technique

Parameteric dependance

Can we make things better?

Planar graphs: By [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos: Tight Bounds for Linkages in Planar Graphs. ICALP 2011] it follows that λ(k) = 2O(k).

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 106/111 40 A-Limits of the irrelevant vertex technique:

O(k) 22 · n3 steps is optimal as λ(k) = 2Ω(k)

I A non-A algorithm for the k-Disjoint Paths Problem (and related problems) would require radically diﬀerent techniques!

Graph Minors and algorithms The irrelevant vertex technique

Parameteric dependance

Can we make things better?

Planar graphs: By [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos: Tight Bounds for Linkages in Planar Graphs. ICALP 2011] it follows that λ(k) = 2O(k).

Ω(k) This gives an 22 · n3 algorithm for the k-Disjoint Paths Problem

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 107/111 40 I A non-A algorithm for the k-Disjoint Paths Problem (and related problems) would require radically diﬀerent techniques!

Graph Minors and algorithms The irrelevant vertex technique

Parameteric dependance

Can we make things better?

Planar graphs: By [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos: Tight Bounds for Linkages in Planar Graphs. ICALP 2011] it follows that λ(k) = 2O(k).

Ω(k) This gives an 22 · n3 algorithm for the k-Disjoint Paths Problem

A-Limits of the irrelevant vertex technique:

O(k) 22 · n3 steps is optimal as λ(k) = 2Ω(k)

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 108/111 40 Graph Minors and algorithms The irrelevant vertex technique

Parameteric dependance

Can we make things better?

Planar graphs: By [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos: Tight Bounds for Linkages in Planar Graphs. ICALP 2011] it follows that λ(k) = 2O(k).

Ω(k) This gives an 22 · n3 algorithm for the k-Disjoint Paths Problem

A-Limits of the irrelevant vertex technique:

O(k) 22 · n3 steps is optimal as λ(k) = 2Ω(k)

I A non-A algorithm for the k-Disjoint Paths Problem (and related problems) would require radically diﬀerent techniques!

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 109/111 40 Graph Minors and algorithms The irrelevant vertex technique

Some problems

Last words on Algorithmic Graph Minors Theory...

Some recent FPT-Algorithms using the irrelevant vertex technique or variants of it:

Bipartite Contraction, Partial Vertex Cover, Partial Dominating Set,

Topological Minor Containment, Immersion Containment,

Bounded Genus Contraction Containment, Odd Cycle Induced Packing,

Odd Cycle Packing, Induced cycle, Optimal Embedding in a Surface

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 110/111 41 Graph Minors and algorithms The irrelevant vertex technique

Mondrian

Piet Mondrian, Composition with Yellow, Blue, and Red, 1921

Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 111/111 42