The document describes a generic algorithm for the F-deletion problem, where the goal is to remove at most k vertices from a graph such that the remaining graph does not contain graphs from F as minors. It shows that when F contains only planar graphs, the algorithm provides a constant-factor approximation. It analyzes special cases where the algorithm works with different constants, such as when the graph minus the solution is independent, a matching, or acyclic. It then discusses how the algorithm extends to more general graphs by exploiting that the graph minus solution must have bounded treewidth when F contains planar graphs.

1. From FVS to F-deletion

2. From FVS to F-deletion
a simple constant-factor randomized
approximation algorithm

3. From VC to F-deletion
a simple constant-factor randomized
approximation algorithm

5. A Generic Algorithm

6. A Generic Algorithm
Special Cases

8. THE BLUEPRINT

17. Every
Vertex Cover
intersects
every edge
at at least one
endpoint.

18. Every
Solution
intersects
every edge
at at least one
endpoint.

19. Every
Solution
intersects
some subset of edges?
at at least one
endpoint.

20. Every
Solution
intersects
a good fraction of edges
at at least one
endpoint.

21. Every
Solution
intersects
a good fraction of edges
at at least one
endpoint.

22. Pick an edge e, uniformly at random.

23. Pick an endpoint of e, uniformly at random.

24. Repeat until a solution is obtained.

25. Pick an edge e, uniformly at random.
Pick an endpoint of e, uniformly at random.
Repeat until a solution is obtained.

26. Pick an edge e, uniformly at random.
Pick a good edge with probability (1/c)
Pick an endpoint of e, uniformly at random.
Repeat until a solution is obtained.

27. Pick an edge e, uniformly at random.
Pick a good edge with probability (1/c)
Pick an endpoint of e, uniformly at random.
Pick a good endpoint with probability (1/2)
Repeat until a solution is obtained.

28. Pick an edge e, uniformly at random.
Pick a good edge with probability (1/c)
Pick an endpoint of e, uniformly at random.
Pick a good endpoint with probability (1/2)
Repeat until a solution is obtained.
The expected solution size: 2c(OPT)

29. Pick an edge e, uniformly at random.
Pick a good edge with probability (1/c)
Pick an endpoint of e, uniformly at random.
Pick a good endpoint with probability (1/2)
Repeat until a solution is obtained.
The expected solution size: 2c(OPT)

30. S
GS

31. S
GS

32. S
GS
#cross edges + #edges within S (1/c) · m

33. S
GS
#cross edges + #edges within S (1/c) · m

P
v2S d(v)
2

34. S
GS
P
v2S d(v) (1/c) · m
2

35. S
GS
X
d(v) (1/c) · m ·2
v2S

36. S
GS
X
d(v) (1/c) · m ·2
v2S

37. S
GS
X
d(v) (1/c) · m ·2
v2S
X X
d(v) (1/c) · d(v)
v2S v2G

38. S
GS
X X
d(v) (1/c) · d(v)
v2S v2G

40. SPECIAL CASES

41. GS is
an independent set.

42. GS is
a matching

43. S
GS
Preprocess: Delete isolated edges.
X X
d(v) (1/c) · d(v)
v2S v2G

44. S
GS
Preprocess: Delete isolated edges.
X X
d(v) (1/4) · d(v)
v2S v2G

46. GS is
an acyclic graph
(forest)

47. GS is
an acyclic graph
(forest)
cÉÉÇÄ~Åâ=sÉêíÉñ=pÉí

48. S
GS
Preprocess: ???
X X
d(v) (1/c) · d(v)
v2S v2G

49. S
When can we say that every leaf
“contributes” a cross-edge?

50. S
Preprocess: Delete pendant vertices.

51. #of cross edges #of leaves

52. #of cross edges #of leaves
#of edges in the tree = #of leaves + #internal nodes - 1

53. #of leaves #internal nodes
(minimum degree at least three)
#of cross edges #of leaves
#of edges in the tree = #of leaves + #internal nodes - 1

54. #of leaves #internal nodes
(minimum degree at least three)
#of cross edges #of leaves
#of edges in the tree #of leaves + #of leaves -1

55. #of leaves #internal nodes
(minimum degree at least three)
#of cross edges #of leaves
#of edges in the tree #of leaves + #of leaves -1
#of edges in the tree 2(#of leaves) -1

56. #of leaves #internal nodes
(minimum degree at least three)
#of cross edges #of leaves
#of edges in the tree #of leaves + #of leaves -1
#of edges in the tree 2(#of leaves) -1
#of edges in the tree 2(#of cross edges) -1

57. #of edges in the tree 2(#of cross edges) -1

58. #of edges in the tree 2(#of cross edges) -1
X X
d(v) (1/c) · d(v)
v2S v2G

59. #of edges in the tree 2(#of cross edges) -1
X X
d(v) (1/c) · d(v)
v2S v2G
X
d(v) = 2(#of edges in the tree) + 2(#of cross edges)
v2G

60. #of edges in the tree 2(#of cross edges) -1
X X
d(v) (1/c) · d(v)
v2S v2G
X
d(v) = 2(#of edges in the tree) + 2(#of cross edges)
v2G
2(2#cross edges - 1) + 2(#of cross edges)

61. #of edges in the tree 2(#of cross edges) -1
X X
d(v) (1/c) · d(v)
v2S v2G
X
d(v) = 2(#of edges in the tree) + 2(#of cross edges)
v2G
2(2#cross edges - 1) + 2(#of cross edges)
6(#cross edges)

62. #of edges in the tree 2(#of cross edges) -1
X X
d(v) (1/c) · d(v)
v2S v2G
X
d(v) = 2(#of edges in the tree) + 2(#of cross edges)
v2G
2(2#cross edges - 1) + 2(#of cross edges)
6(#cross edges)
!
X
6 d(v)
v2S

63. #of leaves #internal nodes
(minimum degree at least three)

64. #of leaves #internal nodes
(minimum degree at least three)

65. #of leaves #internal nodes
(minimum degree at least three)
More preprocessing!

68. #of leaves #internal nodes
(minimum degree at least three)
#of cross edges 2(#of leaves)
#of edges in the tree = #of leaves + #of leaves -1
#of edges in the tree 2(#of leaves) -1
#of edges in the tree 2(#of cross edges) -1

69. #of leaves #internal nodes
(minimum degree at least three)
#of cross edges 2(#of leaves)
#of edges in the tree = #of leaves + #of leaves -1
#of edges in the tree 2(#of leaves) -1
#of edges in the tree 2(#of cross edges) -1
#of edges in the tree #of cross edges -1

71. GS is independent
GS is a matching
GS is acyclic

72. GS is independent Factor 2, for free.
GS is a matching
GS is acyclic

73. GS is independent Factor 2, for free.
Factor 4, after removing
GS is a matching isolated edges
GS is acyclic

74. GS is independent Factor 2, for free.
Factor 4, after removing
GS is a matching isolated edges
Factor 4, after deleting degree 1
GS is acyclic and short-circuiting degree 2
vertices.

76. WHAT’S NEXT?

77. What is the most general problem for
which the algorithm
“just works”?

78. Beyond problem-specific
reduction rules...
Is there a one-size-fits-all?

79. Answer: mä~å~ê=cJÇÉäÉíáçå

80. Remove at most k vertices such that the
remaining graph has no minor models of graphs from F.

81. qÜÉ=cJaÉäÉíáçå=mêçÄäÉã
Remove at most k vertices such that the
remaining graph has no minor models of graphs from F.

82. mä~å~ê
qÜÉ=cJaÉäÉíáçå=mêçÄäÉã
Remove at most k vertices such that the
remaining graph has no minor models of graphs from F.
(Where F contains a planar graph.)

83. Independent = no edges
Forbid an edge as a minor

84. Acyclic = no cycles
Forbid a triangle as a minor

85. Pathwidth-one graphs
Forbid T2, K3 as a minor

86. Turns out that when you want to kill
minor models of planar graphs,
GS must have bounded treewidth.

87. This can be exploited to frame
some very general reduction rules.

88. This can be exploited to frame
some very general reduction rules.
http://arxiv.org/abs/1204.4230

89. A brief summary of this discussion
http://neeldhara.com/planar-f-deletion-1/
Thank You!

90. A brief summary of this discussion
http://neeldhara.com/planar-f-deletion-1/
Thank You!