A survey of graph modification problems in terms of forbidden minors, in a mostly algorithmic context.

1. Parameterized Graph Modification: A Modern Perspective
Neeldhara Misra
Department of Computer Science and Automation,
Indian Institute of Science

2. Parameterized Graph Modification: A Modern Perspective

3. The General Setup
A brief introduction to parameterized algorithmics
Parameterized Graph Modification: A Modern Perspective

4. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
The General Setup
A brief introduction to parameterized algorithmics
Parameterized Graph Modification: A Modern Perspective

5. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
The General Setup
A brief introduction to parameterized algorithmics
Parameterized Graph Modification: A Modern Perspective
The general notion of a “graph modification problem”.

6. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
The General Setup
A brief introduction to parameterized algorithmics
Parameterized Graph Modification: A Modern Perspective
Implications of the Graph Minor Theorem.
The general notion of a “graph modification problem”.

7. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
The General Setup
A brief introduction to parameterized algorithmics
Parameterized Graph Modification: A Modern Perspective
Implications of the Graph Minor Theorem.
Vertex Cover, Revisited.
The general notion of a “graph modification problem”.

8. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
The General Setup
A brief introduction to parameterized algorithmics
Parameterized Graph Modification: A Modern Perspective
Implications of the Graph Minor Theorem.
Vertex Cover, Revisited.
Beyond Vertex Cover: Feedback Vertex Set
(on graphs of constant degree).
The general notion of a “graph modification problem”.

9. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
The General Setup
A brief introduction to parameterized algorithmics
Parameterized Graph Modification: A Modern Perspective
Implications of the Graph Minor Theorem.
Vertex Cover, Revisited.
Beyond Vertex Cover: Feedback Vertex Set
(on graphs of constant degree).
Planar F-Deletion.
The general notion of a “graph modification problem”.

10. The General Setup
A brief introduction to parameterized algorithmics
Parameterized Graph Modification: A Modern Perspective

11. The General Setup
A brief introduction to parameterized algorithmics

12. The General Setup
A brief introduction to parameterized algorithmics
The Classical Perspective

13. The General Setup
A brief introduction to parameterized algorithmics
The Classical Perspective
Evaluate running times of algorithms as a function of
the size of input alone.

14. The General Setup
A brief introduction to parameterized algorithmics
The Classical Perspective
Evaluate running times of algorithms as a function of
the size of input alone.
The Parameterized Perspective

15. The General Setup
A brief introduction to parameterized algorithmics
The Classical Perspective
Evaluate running times of algorithms as a function of
the size of input alone.
The Parameterized Perspective
Evaluate running times of algorithms as a function of
the size of input and additional structural parameters.

16. Does there exist a subset S of at most k vertices
whose removal makes the graph independent?
Vertex Cover

17. Does there exist a subset S of at most k vertices
whose removal makes the graph independent?
Vertex Cover

18. Does there exist a subset S of at most k vertices
whose removal makes the graph independent?
Vertex Cover
Brute Force

19. Does there exist a subset S of at most k vertices
whose removal makes the graph independent?
nk
Vertex Cover
Brute Force

20. Pruned Search Tree
Does there exist a subset S of at most k vertices
whose removal makes the graph independent?
nk
Vertex Cover
Brute Force

21. Pruned Search Tree
2k
nO(1)
Does there exist a subset S of at most k vertices
whose removal makes the graph independent?
nk
Vertex Cover
Brute Force

22. Pruned Search Tree
2k
nO(1)
Does there exist a subset S of at most k vertices
whose removal makes the graph independent?
nk
(Implemented by CBRG, ETH Zurich)
Vertex Cover
Brute Force

23. For real-world computations, we always know more about the
problem.
Exploiting Structure

24. For real-world computations, we always know more about the
problem.
Small Diameter
Exploiting Structure

25. For real-world computations, we always know more about the
problem.
Planar graphs
Small Diameter
Exploiting Structure

26. For real-world computations, we always know more about the
problem.
Planar graphs
Small Nesting depth
Small Diameter
Exploiting Structure

27. For real-world computations, we always know more about the
problem.
Planar graphs
Graphs without cycles
Small Nesting depth
Small Diameter
Exploiting Structure

28. Acknowledge the presence of additional structure.
Exploiting Structure

29. Acknowledge the presence of additional structure.
This will be a secondary measurement, a parameter.
Exploiting Structure

30. Acknowledge the presence of additional structure.
This will be a secondary measurement, a parameter.
Design algorithms that restrict the
combinatorial explosion to a function of the parameter.
Exploiting Structure

31. f(k)p(n)
Fixed-parameter Tractable Algorithms

32. f(k)p(n)
a structural parameter
Fixed-parameter Tractable Algorithms

33. f(k)p(n)
a structural parameter
input size
Fixed-parameter Tractable Algorithms

34. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
Parameterized Graph Modification: A Modern Perspective

35. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
The general notion of a “graph modification problem”.

36. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
Implications of the Graph Minor Theorem.
The general notion of a “graph modification problem”.

37. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
Implications of the Graph Minor Theorem.
Vertex Cover, Revisited.
The general notion of a “graph modification problem”.

38. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
Implications of the Graph Minor Theorem.
Vertex Cover, Revisited.
Beyond Vertex Cover: Feedback Vertex Set
(on graphs of constant degree).
The general notion of a “graph modification problem”.

39. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
Implications of the Graph Minor Theorem.
Vertex Cover, Revisited.
Beyond Vertex Cover: Feedback Vertex Set
(on graphs of constant degree).
Planar F-Deletion.
The general notion of a “graph modification problem”.

40. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
The general notion of a “graph modification problem”.
Implications of the Graph Minor Theorem.
Vertex Cover, Revisited.
Beyond Vertex Cover: Feedback Vertex Set
(on graphs of constant degree).
Planar F-Deletion.

42. Input Graph, G

43. Input Graph, G Graph Class or “property”, ᴨ

44. Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”.

45. Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”.

46. Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”.

47. Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”.
vertex deletions

48. Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”.
edge deletionsvertex deletions

49. Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”.
edge deletions edge additionsvertex deletions

50. Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”.
edge deletions edge additions edge contractionsvertex deletions

51. Input Graph, G Graph Class or “property”, ᴨ
Goal: Transform G so that it belongs to ᴨ with “minimum damages”.
edge deletions edge additions edge contractions edge editingvertex deletions

52. Input Graph, G
Goal: Transform G so that it belongs to ᴨ with “minimum damages”.
edge deletions edge additions edge contractions edge editingvertex deletions
Vertex Cover
Edgeless Graphs

53. Input Graph, G
Goal: Transform G so that it belongs to ᴨ with “minimum damages”.
edge deletions edge additions edge contractions edge editingvertex deletions
Feedback Vertex Set
Acyclic Graphs

54. Input Graph, G
Goal: Transform G so that it belongs to ᴨ with “minimum damages”.
edge deletions edge additions edge contractions edge editingvertex deletions
Minimum Fill-In
Chordal Graphs

55. Input Graph, G
Goal: Transform G so that it belongs to ᴨ with “minimum damages”.
edge deletions edge additions edge contractions edge editingvertex deletions
Cluster Editing
Cluster Graphs

56. Graph Class or “property”, ᴨ

57. Graph Class or “property”, ᴨ
Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs.

58. Graph Class or “property”, ᴨ
Monotone If G belongs to ᴨ, then all subgraphs of G also belong to ᴨ.
Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs.

59. Graph Class or “property”, ᴨ
Monotone
Hereditary
If G belongs to ᴨ, then all subgraphs of G also belong to ᴨ.
If G belongs to ᴨ, then all induced subgraphs of G also belong to ᴨ.
Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs.

60. Graph Class or “property”, ᴨ
Monotone
Hereditary
If G belongs to ᴨ, then all subgraphs of G also belong to ᴨ.
If G belongs to ᴨ, then all induced subgraphs of G also belong to ᴨ.
Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs.
(Bipartite, Planar)

61. Graph Class or “property”, ᴨ
Monotone
Hereditary
If G belongs to ᴨ, then all subgraphs of G also belong to ᴨ.
If G belongs to ᴨ, then all induced subgraphs of G also belong to ᴨ.
Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs.
(Bipartite, Planar)
(Complete graphs, Forests)

62. Graph Class or “property”, ᴨ
Monotone
Hereditary
If G belongs to ᴨ, then all subgraphs of G also belong to ᴨ.
If G belongs to ᴨ, then all induced subgraphs of G also belong to ᴨ.
Non-Trivial ᴨ admits infinitely many graphs, and also excludes infinitely many graphs.
(Bipartite, Planar)
(Complete graphs, Forests)
Connectivity, Biconnectivity, Trees, Stars, Eulerian, etc.

64. Lewis &
Yannakakis
[1980]
The vertex-deletion problem is NP-complete for non-trivial,
hereditary properties.

65. Lewis &
Yannakakis
[1980]
The vertex-deletion problem is NP-complete for non-trivial,
hereditary properties.
(Edgeless, Acyclic, etc.)

67. Edge-Deletion or Completion problems are sometimes easier than their
vertex-deletion counterparts.

68. Edge-Deletion or Completion problems are sometimes easier than their
vertex-deletion counterparts.
Add edges to make the input graph a cluster graph.

69. Edge-Deletion or Completion problems are sometimes easier than their
vertex-deletion counterparts.
Add edges to make the input graph a cluster graph.
Remove edges to make the input graph edgeless.

70. Edge-Deletion or Completion problems are sometimes easier than their
vertex-deletion counterparts.
Add edges to make the input graph a cluster graph.
Remove edges to make the input graph edgeless.
Remove edges to make the input graph acyclic.

71. Edge-Deletion or Completion problems are sometimes easier than their
vertex-deletion counterparts.
Add edges to make the input graph a cluster graph.
Remove edges to make the input graph edgeless.
Remove edges to make the input graph acyclic.
Remove edges to make the input graph bipartite.

72. Edge-Deletion or Completion problems are sometimes easier than their
vertex-deletion counterparts.
Add edges to make the input graph a cluster graph.
Remove edges to make the input graph edgeless.
Remove edges to make the input graph acyclic.

73. Edge-Deletion or Completion problems are sometimes easier than their
vertex-deletion counterparts.
Add edges to make the input graph a cluster graph.
Remove edges to make the input graph edgeless.
Remove edges to make the input graph acyclic.
Alon, Shapira
& Sudakov
[2009]
Given a property ᴨ such that ᴨ holds for every
bipartite graph, the Minimum ᴨ-Deletion problem is NP-hard.

74. Edge-Deletion or Completion problems are sometimes easier than their
vertex-deletion counterparts.
Add edges to make the input graph a cluster graph.
Remove edges to make the input graph edgeless.
Remove edges to make the input graph acyclic.
Alon, Shapira
& Sudakov
[2009]
Given a property ᴨ such that ᴨ holds for every
bipartite graph, the Minimum ᴨ-Deletion problem is NP-hard.
(Triangle-­free)

75. Edge-Deletion or Completion problems are sometimes easier than their
vertex-deletion counterparts.
Add edges to make the input graph a cluster graph.
Remove edges to make the input graph edgeless.
Remove edges to make the input graph acyclic.
Alon, Shapira
& Sudakov
[2009]
Given a property ᴨ such that ᴨ holds for every
bipartite graph, the Minimum ᴨ-Deletion problem is NP-hard.
Colbourn &
El-Mallah
[1988]
Making a graph Pk-free by deleting the minimum number of
edges is NP-hard for every k > 2.
(Triangle-­free)

76. Finding large induced subgraphs with property ᴨ.
Khot and Raman, (2002)

77. Finding large induced subgraphs with property ᴨ.
If ᴨ is a hereditary property that contains all independent sets and all cliques,
or if ᴨ excludes some independent sets and some cliques,
then the problem of finding an induced subgraph of size at least k, satisfying ᴨ,
is FPT.
Khot and Raman, (2002)

78. Finding large induced subgraphs with property ᴨ.
If ᴨ is a hereditary property that contains all independent sets and all cliques,
or if ᴨ excludes some independent sets and some cliques,
then the problem of finding an induced subgraph of size at least k, satisfying ᴨ,
is FPT.
If ᴨ is a hereditary property contains all independent sets but not all cliques
(or vice versa),
then the problem of finding an induced subgraph of size at least k, satisfying ᴨ,
is W[1]-hard.
Khot and Raman, (2002)

79. Finding large induced subgraphs with property ᴨ.
If ᴨ is a hereditary property that contains all independent sets and all cliques,
or if ᴨ excludes some independent sets and some cliques,
then the problem of finding an induced subgraph of size at least k, satisfying ᴨ,
is FPT.
If ᴨ is a hereditary property contains all independent sets but not all cliques
(or vice versa),
then the problem of finding an induced subgraph of size at least k, satisfying ᴨ,
is W[1]-hard.
Khot and Raman, (2002)
Ramsey Numbers!
Reduction from Independent Set

80. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
Implications of the Graph Minor Theorem.
Vertex Cover, Revisited.
Beyond Vertex Cover: Feedback Vertex Set
(on graphs of constant degree).
Planar F-Deletion.
The general notion of a “graph modification problem”.

82. A graph G contains a graph H as a minor if H
can be obtained from G by a sequence of
vertex or edge deletions; and edge contractions.

83. A graph G contains a graph H as a minor if H
can be obtained from G by a sequence of
vertex or edge deletions; and edge contractions.

84. A graph G contains a graph H as a minor if H
can be obtained from G by a sequence of
vertex or edge deletions; and edge contractions.

85. A graph G contains a graph H as a minor if H
can be obtained from G by a sequence of
vertex or edge deletions; and edge contractions.

86. Graph Class or “property”, ᴨ

87. Graph Class or “property”, ᴨ
Forbidden Minor Characterization

88. Graph Class or “property”, ᴨ
Forbidden Minor Characterization
A graph G belongs to ᴨ if, and only if,
G does not contain any graph from Forb(ᴨ)
as a minor.

89. Graph Class or “property”, ᴨ
Forbidden Minor Characterization
A graph G belongs to ᴨ if, and only if,
G does not contain any graph from Forb(ᴨ)
as a minor.

90. Graph Class or “property”, ᴨ
Forbidden Minor Characterization
A graph G belongs to ᴨ if, and only if,
G does not contain any graph from Forb(ᴨ)
as a minor.
A graph admits a forbidden minor characterization
if, and only if,
it is closed under taking minors.

91. Graph Class or “property”, ᴨ
Forbidden Minor Characterization
A graph G belongs to ᴨ if, and only if,
G does not contain any graph from Forb(ᴨ)
as a minor.
A graph admits a forbidden minor characterization
if, and only if,
it is closed under taking minors.
In fact, there is always a finite forbidden set!
(Robertson-Seymour; Graph Minors Theorem)

92. Graph Class or “property”, ᴨ
Forbidden Minor Characterization
A graph G belongs to ᴨ if, and only if,
G does not contain any graph from Forb(ᴨ)
as a minor.
A graph admits a forbidden minor characterization
if, and only if,
it is closed under taking minors.
Suppose the class ᴨ + (modifications) is
closed under taking minors…
In fact, there is always a finite forbidden set!
(Robertson-Seymour; Graph Minors Theorem)

93. Graph Class or “property”, ᴨ
Forbidden Minor Characterization
A graph G belongs to ᴨ if, and only if,
G does not contain any graph from Forb(ᴨ)
as a minor.
A graph admits a forbidden minor characterization
if, and only if,
it is closed under taking minors.
Suppose the class ᴨ + (modifications) is
closed under taking minors…
In fact, there is always a finite forbidden set!
(Robertson-Seymour; Graph Minors Theorem)
+ (modifications)

94. Graph Class or “property”, ᴨ
Forbidden Minor Characterization
A graph G belongs to ᴨ if, and only if,
G does not contain any graph from Forb(ᴨ)
as a minor.
A graph admits a forbidden minor characterization
if, and only if,
it is closed under taking minors.
Suppose the class ᴨ + (modifications) is
closed under taking minors…
Now: the graph modification problem has boiled down
to a membership testing problem, which can be
done in cubic time, given the finite forbidden set.
In fact, there is always a finite forbidden set!
(Robertson-Seymour; Graph Minors Theorem)
+ (modifications)

95. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
Implications of the Graph Minor Theorem.
Vertex Cover, Revisited.
Beyond Vertex Cover: Feedback Vertex Set
(on graphs of constant degree).
Planar F-Deletion.
The general notion of a “graph modification problem”.

96. Can G be made edgeless by the removal of at most k vertices?
A simple randomized algorithm with an error probability of 2-k.

97. Can G be made edgeless by the removal of at most k vertices?
A simple randomized algorithm with an error probability of 2-k.

98. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
Define a class of graph modification problems.
Implications of the Graph Minor Theorem.
Vertex Cover, Revisited.
Beyond Vertex Cover: Feedback Vertex Set
(on graphs of constant degree).
Planar F-Deletion.

99. Can G be made acyclic by the removal of at most k vertices?

100. Can G be made acyclic by the removal of at most k vertices?

101. Can G be made acyclic by the removal of at most k vertices?

102. Can G be made acyclic by the removal of at most k vertices?

103. Can G be made acyclic by the removal of at most k vertices?
Let us consider the structure of a YES-instance.
Let us also restrict ourselves to graphs of constant maximum degree, say five.

104. The k vertices corresponding to the FVS
(n-k) vertices and at most (n-k-1) edges “outside”.

105. The k vertices corresponding to the FVS
(n-k) vertices and at most (n-k-1) edges “outside”.

108. t “points of contact” between S and GS…

109. t “points of contact” between S and GS… at least t edges going across the two sets.

110. t “points of contact” between S and GS… at least t edges going across the two sets.
Delete the points of contact in GS to get at most 5t “pieces”.

111. t “points of contact” between S and GS… at least t edges going across the two sets.
Delete the points of contact in GS to get at most 5t “pieces”.
Suppose each piece has constant size, say c.

112. t “points of contact” between S and GS… at least t edges going across the two sets.
Delete the points of contact in GS to get at most 5t “pieces”.
Suppose each piece has constant size, say c.
The total number of edges in GS is at most 5ct.

113. t “points of contact” between S and GS… at least t edges going across the two sets.
Delete the points of contact in GS to get at most 5t “pieces”.
Suppose each piece has constant size, say c.
The total number of edges in GS is at most 5ct.
The number of good edges is at least t, and the number of bad edges is at most 5ct.

114. t “points of contact” between S and GS… at least t edges going across the two sets.
Delete the points of contact in GS to get at most 5t “pieces”.
Suppose each piece has constant size, say c.
The total number of edges in GS is at most 5ct.
The number of good edges is at least t, and the number of bad edges is at most 5ct.
In this scenario, a randomly chosen edge will be good with constant probability!

115. Bad scenario: one of the pieces is large.

116. These pieces have simple structure and bounded interaction with the outer world.

117. These pieces have simple structure and bounded interaction with the outer world.
Such pieces are called “protrusions”.

119. Finding protrusions? What do we replace them with?

120. One Size Fits All?
A meta-algorithm for a class of graph-theoretic problems
Implications of the Graph Minor Theorem.
Vertex Cover, Revisited.
Beyond Vertex Cover: Feedback Vertex Set
(on graphs of constant degree).
Planar F-Deletion.
The general notion of a “graph modification problem”.

121. The notion of treewidth — capturing “treelike” graphs
Pictures by Michal Pilipczuk

122. The notion of treewidth — capturing “treelike” graphs
Pictures by Michal Pilipczuk

123. The notion of treewidth — capturing “treelike” graphs
Pictures by Michal Pilipczuk

124. The notion of treewidth — capturing “treelike” graphs
Pictures by Michal Pilipczuk

126. Can G be made H-minor free, for all H in 𝓕,
by the removal of at most k vertices?

127. Can G be made H-minor free, for all H in 𝓕,
by the removal of at most k vertices?
Suppose G is a YES-instance, and further, suppose 𝓕 contains at least one planar graph.

128. Can G be made H-minor free, for all H in 𝓕,
by the removal of at most k vertices?
Suppose G is a YES-instance, and further, suppose 𝓕 contains at least one planar graph.
Then GS cannot have large grids.

129. Can G be made H-minor free, for all H in 𝓕,
by the removal of at most k vertices?
Suppose G is a YES-instance, and further, suppose 𝓕 contains at least one planar graph.
Then GS cannot have large grids.
Thus the treewidth of GS is bounded by a constant that depends only on 𝓕.

130. Can G be made H-minor free, for all H in 𝓕,
by the removal of at most k vertices?
Suppose G is a YES-instance, and further, suppose 𝓕 contains at least one planar graph.
Then GS cannot have large grids.
Thus the treewidth of GS is bounded by a constant that depends only on 𝓕.
The Planar 𝓕-Deletion Problem

131. GS: no copies of graphs from 𝓕 here.

132. GS: no copies of graphs from 𝓕 here.

133. Planar 𝓕-Deletion: The Algorithm

134. Ensure that protrusions are removed.
Planar 𝓕-Deletion: The Algorithm

135. Ensure that protrusions are removed.
Pick an edge uniformly at random.
Planar 𝓕-Deletion: The Algorithm

136. Ensure that protrusions are removed.
Pick an edge uniformly at random.
Include both endpoints.
Planar 𝓕-Deletion: The Algorithm

137. Ensure that protrusions are removed.
Pick an edge uniformly at random.
Include both endpoints. Pick an endpoint u.a.r.
Planar 𝓕-Deletion: The Algorithm

138. Ensure that protrusions are removed.
Pick an edge uniformly at random.
Include both endpoints. Pick an endpoint u.a.r.
Repeat till G is 𝓕-free.
Planar 𝓕-Deletion: The Algorithm

139. Ensure that protrusions are removed.
Pick an edge uniformly at random.
Include both endpoints. Pick an endpoint u.a.r.
(Approximation algorithm.)
Repeat till G is 𝓕-free.
Planar 𝓕-Deletion: The Algorithm

140. Ensure that protrusions are removed.
Pick an edge uniformly at random.
Include both endpoints. Pick an endpoint u.a.r.
(Approximation algorithm.) (Randomized algorithm.)
Repeat till G is 𝓕-free.
Planar 𝓕-Deletion: The Algorithm

141. The Planar 𝓕-Deletion Problem
accounts for several specific problems…

142. The Planar 𝓕-Deletion Problem
accounts for several specific problems…
𝓕
K2
K3
((k+1)-by-(k+1)) Grids
K2,3, K4
K4
K3,T2
Corresponding Deletion Problem
Vertex Cover
Feedback Vertex Set
Treewidth k
Outerplanar
Series-Parallel
Pathwidth-One

144. Thanks to Graph Minors, we have a f(k)n2 algorithm for the 𝓕-Deletion problem.

145. Thanks to Graph Minors, we have a f(k)n2 algorithm for the 𝓕-Deletion problem.
For Planar 𝓕-Deletion, the starting point was a double-exponential algorithm.
[Bodlaender, 1997]

146. Thanks to Graph Minors, we have a f(k)n2 algorithm for the 𝓕-Deletion problem.
For Planar 𝓕-Deletion, the starting point was a double-exponential algorithm.
[Bodlaender, 1997]
Many single-exponential algorithms are known for special cases of 𝓕.

147. Thanks to Graph Minors, we have a f(k)n2 algorithm for the 𝓕-Deletion problem.
For Planar 𝓕-Deletion, the starting point was a double-exponential algorithm.
[Bodlaender, 1997]
Many single-exponential algorithms are known for special cases of 𝓕.
Feedback Vertex Set has a O*(3.83k) algorithm.Cao, Chen & Liu
[2010]

148. Thanks to Graph Minors, we have a f(k)n2 algorithm for the 𝓕-Deletion problem.
For Planar 𝓕-Deletion, the starting point was a double-exponential algorithm.
[Bodlaender, 1997]
Many single-exponential algorithms are known for special cases of 𝓕.
{K4}-Deletion has a O*(2O(k)) algorithm.
Kim, Paul & Philip
[2012]
Feedback Vertex Set has a O*(3.83k) algorithm.Cao, Chen & Liu
[2010]

149. Thanks to Graph Minors, we have a f(k)n2 algorithm for the 𝓕-Deletion problem.
For Planar 𝓕-Deletion, the starting point was a double-exponential algorithm.
[Bodlaender, 1997]
Many single-exponential algorithms are known for special cases of 𝓕.
Pathwidth-1-Deletion has a O*(4.65k) algorithm.
Cygan, Pilipczuk,
Pilipczuk & Wojtaszczyk
[2010]
{K4}-Deletion has a O*(2O(k)) algorithm.
Kim, Paul & Philip
[2012]
Feedback Vertex Set has a O*(3.83k) algorithm.Cao, Chen & Liu
[2010]

151. Fomin, Lokshtanov,
M. & Saurabh
[FOCS; 2012]
Planar 𝓕-Deletion admits a randomized (2O(k)n) algorithm
when every graph in 𝓕 is connected.

152. Fomin, Lokshtanov,
M. & Saurabh
[FOCS; 2012]
Planar 𝓕-Deletion admits a randomized (2O(k)n) algorithm
when every graph in 𝓕 is connected.
Best possible under ETH.

153. Fomin, Lokshtanov,
M. & Saurabh
[FOCS; 2012]
Planar 𝓕-Deletion admits a randomized (2O(k)n) algorithm
when every graph in 𝓕 is connected.
Planar 𝓕-Deletion admits a deterministic (2O(k)n log2n) algorithm
when every graph in 𝓕 is connected.
Best possible under ETH.

154. Fomin, Lokshtanov,
M. & Saurabh
[FOCS; 2012]
Planar 𝓕-Deletion admits a randomized (2O(k)n) algorithm
when every graph in 𝓕 is connected.
Planar 𝓕-Deletion admits a deterministic (2O(k)n log2n) algorithm
when every graph in 𝓕 is connected.
Planar 𝓕-Deletion admits an O(nm) randomized algorithm
that leads us to a constant-factor approximation.
Can we get a deterministic
constant-­factor approximation algorithm?
Best possible under ETH.

155. Fomin, Lokshtanov,
M. & Saurabh
[FOCS; 2012]
Planar 𝓕-Deletion admits a randomized (2O(k)n) algorithm
when every graph in 𝓕 is connected.
Planar 𝓕-Deletion admits a deterministic (2O(k)n log2n) algorithm
when every graph in 𝓕 is connected.
Planar 𝓕-Deletion admits an O(nm) randomized algorithm
that leads us to a constant-factor approximation.
Planar 𝓕-Deletion admits an deterministic algorithm
that leads us to a O(log3/2(OPT)) approximation.
Can we get a deterministic
constant-­factor approximation algorithm?
Best possible under ETH.
Fomin, Lokshtanov,
M.,Philip & Saurabh
[STACS; 2011]

156. Fomin, Lokshtanov,
M. & Saurabh
[FOCS; 2012]
Planar 𝓕-Deletion admits a randomized (2O(k)n) algorithm
when every graph in 𝓕 is connected.
Planar 𝓕-Deletion admits a deterministic (2O(k)n log2n) algorithm
when every graph in 𝓕 is connected.
Planar 𝓕-Deletion admits an O(nm) randomized algorithm
that leads us to a constant-factor approximation.
Planar 𝓕-Deletion admits an deterministic algorithm
that leads us to a O(log3/2(OPT)) approximation.
Can we get a deterministic
constant-­factor approximation algorithm?
Best possible under ETH.
Fomin, Lokshtanov,
M.,Philip & Saurabh
[STACS; 2011]
Kim, Langer, Paul, Reidl,
Rossmanith, Sau, & Sikdar
[ICALP; 2013]
Planar 𝓕-Deletion admits a deterministic (2O(k)n2) algorithm.

157. Fomin, Lokshtanov,
M. & Saurabh
[FOCS; 2012]
Planar 𝓕-Deletion admits a randomized (2O(k)n) algorithm
when every graph in 𝓕 is connected.
Planar 𝓕-Deletion admits an O(nm) randomized algorithm
that leads us to a constant-factor approximation.
Planar 𝓕-Deletion admits an deterministic algorithm
that leads us to a O(log3/2(OPT)) approximation.
Can we get a deterministic
constant-­factor approximation algorithm?
Best possible under ETH.
Fomin, Lokshtanov,
M.,Philip & Saurabh
[STACS; 2011]
Kim, Langer, Paul, Reidl,
Rossmanith, Sau, & Sikdar
[ICALP; 2013]
Planar 𝓕-Deletion admits a deterministic (2O(k)n2) algorithm.
Planar 𝓕-Deletion admits a deterministic (2O(k)n) algorithm
when every graph in 𝓕 is connected.
Fomin, Lokshtanov,
M., Ramanujan & Saurabh
[SODA; 2015]

160. Parameterized Graph Modification: A Modern Perspective

161. Parameterized Graph Modification: A Modern Perspective