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.
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)
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
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!
66.
67.
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
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.
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.)