from fvs to f-deletion
DESCRIPTION
TRANSCRIPT
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From to F-deletionFVS
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From to F-deletiona simple constant-factor randomized
approximation algorithm
FVS
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From to F-deletiona simple constant-factor randomized
approximation algorithm
VC
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A Generic Algorithm
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A Generic Algorithm
Special Cases
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THE BLUEPRINT
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Every
intersects
at at least one endpoint.
Vertex Cover
every edge
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Every
intersects
at at least one endpoint.
every edge
Solution
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Every
intersects
at at least one endpoint.
Solution
some subset of edges?
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Every
intersects
at at least one endpoint.
Solution
a good fraction of edges
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Every
intersects
at at least one endpoint.
Solution
a good fraction of edges
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Pick an edge e, uniformly at random.
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Pick an endpoint of e, uniformly at random.
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Repeat until a solution is obtained.
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Pick an edge e, uniformly at random.
Repeat until a solution is obtained.
Pick an endpoint of e, uniformly at random.
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Pick an edge e, uniformly at random.
Repeat until a solution is obtained.
Pick an endpoint of e, uniformly at random.
Pick a good edge with probability (1/c)
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Pick an edge e, uniformly at random.
Repeat until a solution is obtained.
Pick an endpoint of e, uniformly at random.
Pick a good edge with probability (1/c)
Pick a good endpoint with probability (1/2)
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Pick an edge e, uniformly at random.
Repeat until a solution is obtained.
Pick an endpoint of e, uniformly at random.
Pick a good edge with probability (1/c)
Pick a good endpoint with probability (1/2)
The expected solution size: 2c(OPT)
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Pick an edge e, uniformly at random.
Repeat until a solution is obtained.
Pick an endpoint of e, uniformly at random.
Pick a good edge with probability (1/c)
Pick a good endpoint with probability (1/2)
The expected solution size: 2c(OPT)
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S
G \ S
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S
G \ S
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#cross edges + #edges within S � (1/c) ·m
S
G \ S
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#cross edges + #edges within S � (1/c) ·m
Pv2S d(v)
2
S
G \ S
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� (1/c) ·mP
v2S d(v)
2
S
G \ S
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� (1/c) ·mX
v2S
d(v) ·2
S
G \ S
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� (1/c) ·mX
v2S
d(v) ·2
S
G \ S
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X
v2S
d(v) � (1/c) ·X
v2G
d(v)
� (1/c) ·mX
v2S
d(v) ·2
S
G \ S
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X
v2S
d(v) � (1/c) ·X
v2G
d(v)
S
G \ S
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SPECIAL CASES
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G\S is an independent set.
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G\S is a matching
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S
G \ S
X
v2S
d(v) � (1/c) ·X
v2G
d(v)
Preprocess: Delete isolated edges.
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S
G \ S
Preprocess: Delete isolated edges.
X
v2S
d(v) � (1/4) ·X
v2G
d(v)
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G\S is an acyclic graph
(forest)
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G\S is an acyclic graph
(forest)
cÉÉÇÄ~Åâ=sÉêíÉñ=pÉí
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S
G \ S
X
v2S
d(v) � (1/c) ·X
v2G
d(v)
Preprocess: ???
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S
When can we say that every leaf “contributes” a cross-edge?
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S
Preprocess: Delete pendant vertices.
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#of cross edges #of leaves�
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#of edges in the tree #of leaves + #internal nodes
#of cross edges #of leaves�
= - 1
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#of leaves #internal nodes�
(minimum degree at least three)
#of edges in the tree #of leaves + #internal nodes
#of cross edges #of leaves�
= - 1
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#of leaves #internal nodes�
(minimum degree at least three)
#of edges in the tree #of leaves +
#of cross edges #of leaves�
� - 1#of leaves
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#of leaves #internal nodes�
(minimum degree at least three)
#of edges in the tree #of leaves +
#of cross edges #of leaves�
� - 1#of leaves
#of edges in the tree 2(#of leaves) -1�
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#of leaves #internal nodes�
(minimum degree at least three)
#of edges in the tree #of leaves +
#of cross edges #of leaves�
� - 1#of leaves
#of edges in the tree 2(#of leaves) -1�
#of edges in the tree 2(#of cross edges) -1�
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#of edges in the tree 2(#of cross edges) -1�
�
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#of edges in the tree 2(#of cross edges) -1�
X
v2S
d(v) � (1/c) ·X
v2G
d(v)
�
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#of edges in the tree 2(#of cross edges) -1�
X
v2S
d(v) � (1/c) ·X
v2G
d(v)
X
v2G
d(v) = 2(#of edges in the tree) + 2(#of cross edges)
�
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#of edges in the tree 2(#of cross edges) -1�
X
v2S
d(v) � (1/c) ·X
v2G
d(v)
X
v2G
d(v) = 2(#of edges in the tree) + 2(#of cross edges)
2(2#cross edges - 1) + 2(#of cross edges)�
�
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#of edges in the tree 2(#of cross edges) -1�
X
v2S
d(v) � (1/c) ·X
v2G
d(v)
X
v2G
d(v) = 2(#of edges in the tree) + 2(#of cross edges)
2(2#cross edges - 1) + 2(#of cross edges)�
�
�6(#cross edges)
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#of edges in the tree 2(#of cross edges) -1�
X
v2S
d(v) � (1/c) ·X
v2G
d(v)
X
v2G
d(v) = 2(#of edges in the tree) + 2(#of cross edges)
2(2#cross edges - 1) + 2(#of cross edges)�
�
�6(#cross edges)
�6 X
v2S
d(v)
!
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#of leaves #internal nodes�
(minimum degree at least three)
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#of leaves #internal nodes�
(minimum degree at least three)
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#of leaves #internal nodes�
(minimum degree at least three)
More preprocessing!
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#of leaves #internal nodes�
(minimum degree at least three)
#of edges in the tree #of leaves +
#of cross edges 2(#of leaves)�
= - 1#of leaves
#of edges in the tree 2(#of leaves) -1�
#of edges in the tree 2(#of cross edges) -1�
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#of leaves #internal nodes�
(minimum degree at least three)
#of edges in the tree #of leaves +
#of cross edges 2(#of leaves)�
= - 1#of leaves
#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 �
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G\S is acyclic
G\S is independent
G\S is a matching
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G\S is acyclic
G\S is independent
G\S is a matching
Factor 2, for free.
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G\S is acyclic
G\S is independent
G\S is a matching
Factor 2, for free.
Factor 4, after removingisolated edges
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G\S is acyclic
G\S is independent
G\S is a matching
Factor 2, for free.
Factor 4, after removingisolated edges
Factor 4, after deleting degree 1and short-circuiting degree 2
vertices.
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WHAT’S NEXT?
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What is the most general problem forwhich the algorithm
“just works”?
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Beyond problem-specific reduction rules...
Is there a one-size-fits-all?
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Answer: mä~å~ê=cJÇÉäÉíáçå
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Remove at most k vertices such that theremaining graph has no minor models of graphs from F.
![Page 81: From FVS to F-Deletion](https://reader034.vdocuments.us/reader034/viewer/2022042713/547c70dab37959532b8b5026/html5/thumbnails/81.jpg)
qÜÉ=cJaÉäÉíáçå=mêçÄäÉãRemove at most k vertices such that the
remaining graph has no minor models of graphs from F.
![Page 82: From FVS to F-Deletion](https://reader034.vdocuments.us/reader034/viewer/2022042713/547c70dab37959532b8b5026/html5/thumbnails/82.jpg)
qÜÉ=cJaÉäÉíáçå=mêçÄäÉãRemove at most k vertices such that the
remaining graph has no minor models of graphs from F.
mä~å~ê
(Where F contains a planar graph.)
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Independent = no edges
Forbid an edge as a minor
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Acyclic = no cycles
Forbid a triangle as a minor
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Pathwidth-one graphs
Forbid T2, K3 as a minor
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Turns out that when you want to killminor models of planar graphs,
G\S must have bounded treewidth.
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This can be exploited to framesome very general reduction rules.
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This can be exploited to framesome very general reduction rules.
http://arxiv.org/abs/1204.4230
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Thank You!http://neeldhara.com/planar-f-deletion-1/
A brief summary of this discussion
![Page 90: From FVS to F-Deletion](https://reader034.vdocuments.us/reader034/viewer/2022042713/547c70dab37959532b8b5026/html5/thumbnails/90.jpg)
Thank You!
http://neeldhara.com/planar-f-deletion-1/
A brief summary of this discussion