P arameterized Algorithms II PAR T I St. Petersburg, 2011 · Graph Minors I Some consequences of...
Transcript of P arameterized Algorithms II PAR T I St. Petersburg, 2011 · Graph Minors I Some consequences of...
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PART ISt. Petersburg, 2011
FEDOR V. FOMIN
Parameterized Algorithms II
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Graph Minors
FPT algorithmic techniques
Significant advances in the past 20 years or so (especially in recent years).
Powerful toolbox for designing FPT algorithms:
Iterative compressionTreewidth
Bounded Search Tree
Graph Minors TheoremColor coding
Kernelization
Fixed Parameter Algorithms – p.8/98
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Graph MinorsFPT algorithmic techniques
Significant advances in the past 20 years or so (especially in recent years).
Powerful toolbox for designing FPT algorithms:
Iterative compressionTreewidth
Bounded Search Tree
Graph Minors TheoremColor coding
Kernelization
Fixed Parameter Algorithms – p.8/98
Neil Robertson Paul Seymour
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Graph Minors
I Some consequences of the Graph Minors Theorem give aquick way of showing that certain problems are FPT.
I However, the function f (k) in the resulting FPT algorithmscan be HUGE, completely impractical.
I History: motivation for FPT.
I Parts and ingredients of the theory are useful for algorithmdesign.
I New algorithmic results are still being developed.
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Graph Minors
Definition: Graph H is a minor G (H ≤ G ) if H can be obtainedfrom G by deleting edges, deleting vertices, and contracting edges.
Graph Minors
Definition: Graph H is a minor G (H ! G ) if H can be obtained from G by
deleting edges, deleting vertices, and contracting edges.
deleting uv
w
u
u
v
v
contracting uv
Example: A triangle is a minor of a graph G if and only if G has a cycle (i.e., it is
not a forest).
Fixed Parameter Algorithms – p.58/98
Example: A triangle is a minor of a graph G if and only if G has acycle (i.e., it is not a forest).
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Graph minors
Equivalent definition: Graph H is a minor of G if there is amapping φ that maps each vertex of H to a connected subset of Gsuch that
I φ(u) and φ(v) are disjoint if u 6= v , and
I if uv ∈ E (G ), then there is an edge between φ(u) and φ(v).
Graph minors
Equivalent definition: Graph H is a minor of G if there is a mapping ! that maps
each vertex of H to a connected subset of G such that
!(u) and !(v) are disjoint if u != v , and
if uv " E (G), then there is an edge between !(u) and !(v).
3 4 5
6 7
1 2
46
77
32
57
5
541
7
6 6
Fixed Parameter Algorithms – p.59/98
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Minor closed properties
Definition: A set G of graphs is minor closed if whenever G ∈ Gand H ≤ G , then H ∈ G as well.
Examples of minor closed properties:planar graphsacyclic graphs (forests)graphs having no cycle longer than kempty graphs
Examples of not minor closed properties:complete graphsregular graphsbipartite graphs
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Forbidden minors
Let G be a minor closed set and let F be the set of “minimal badgraphs”: H ∈ F if H 6∈ G, but every proper minor of H is in G.
Characterization by forbidden minors:
G ∈ G ⇐⇒ ∀H ∈ F ,H 6≤ G
The set F is the obstruction set of property G.
Theorem: [Wagner] A graph is planar if and only if it does nothave a K5 or K3,3 minor.
In other words: the obstruction set of planarity is F = {K5,K3,3}.Does every minor closed property have such a finitecharacterization?
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Forbidden minors
Let G be a minor closed set and let F be the set of “minimal badgraphs”: H ∈ F if H 6∈ G, but every proper minor of H is in G.
Characterization by forbidden minors:
G ∈ G ⇐⇒ ∀H ∈ F ,H 6≤ G
The set F is the obstruction set of property G.
Theorem: [Wagner] A graph is planar if and only if it does nothave a K5 or K3,3 minor.
In other words: the obstruction set of planarity is F = {K5,K3,3}.Does every minor closed property have such a finitecharacterization?
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Graph Minors Theorem
Theorem: [Robertson and Seymour] Every minor closed propertyG has a finite obstruction set.
Note: The proof is contained in the paper series “Graph MinorsI–XX”.Note: The size of the obstruction set can be astronomical even forsimple properties.
Theorem: [Robertson and Seymour] For every fixed graph H,there is an O(n3) time algorithm for testing whether H is a minorof the given graph G .
Corollary: For every minor closed property G, there is anO(n3) timealgorithm for testing whether a given graph G is in G.
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Graph Minors Theorem
Theorem: [Robertson and Seymour] Every minor closed propertyG has a finite obstruction set.
Note: The proof is contained in the paper series “Graph MinorsI–XX”.Note: The size of the obstruction set can be astronomical even forsimple properties.
Theorem: [Robertson and Seymour] For every fixed graph H,there is an O(n3) time algorithm for testing whether H is a minorof the given graph G .
Corollary: For every minor closed property G, there is anO(n3) timealgorithm for testing whether a given graph G is in G.
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ApplicationsPlanar Face Cover: Given a graph G and an integer k , findan embedding of planar graph G such that there are k faces thatcover all the vertices.
Applications
PLANAR FACE COVER: Given a graph G and an integer k , find an embedding of
planar graph G such that there are k faces that cover all the vertices.
One line argument:
For every fixed k , the class Gk of graphs of yes-instances is minor closed.
!For every fixed k , there is a O(n3) time algorithm for PLANAR FACE COVER.
Note: non-uniform FPT.
Fixed Parameter Algorithms – p.63/98
One line argument:
For every fixed k , the class Gk of graphs of yes-instances is minorclosed.
⇓
For every fixed k , there is a O(n3) time algorithm for PlanarFace Cover.
Note: non-uniform FPT.
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Applicationsk-Leaf Spanning Tree: Given a graph G and an integer k , finda spanning tree with at least k leaves.
Applications
k -LEAF SPANNING TREE: Given a graph G and an integer k , find a spanning tree
with at least k leaves.
Technical modification: Is there such a spanning tree for at least one component of
G?
One line argument:
For every fixed k , the class Gk of no-instances is minor closed.
!For every fixed k , k -LEAF SPANNING TREE can be solved in time O(n3).
Fixed Parameter Algorithms – p.64/98
Technical modification: Is there such a spanning tree for at leastone component of G?
One line argument:
For every fixed k , the class Gk of no-instances is minor closed.
⇓
For every fixed k , k-Leaf Spanning Tree can be solved in timeO(n3).
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G + k verticesLet G be a graph property, and let G + kv contain graph G if thereis a set S ⊆ V (G ) of k vertices such that G \ S ∈ G.
G + k vertices
Let G be a graph property, and let G + kv contain graph G if there is a set
S ! V (G) of k vertices such that G \ S " G.
S
Lemma: If G is minor closed, then G + kv is minor closed for every fixed k .
# It is (nonuniform) FPT to decide if G can be transformed into a member of G by
deleting k vertices.
Fixed Parameter Algorithms – p.65/98
Lemma: If G is minor closed, then G + kv is minor closed forevery fixed k.⇒ It is (nonuniform) FPT to decide if G can be transformed into amember of G by deleting k vertices.
I If G = forests ⇒ G + kv = graphs that can be made acyclicby the deletion of k vertices ⇒ Feedback Vertex Set isFPT.
I If G = planar graphs ⇒ G + kv = graphs that can be madeplanar by the deletion of k vertices (k-apex graphs) ⇒k-Apex Graph is FPT.
I If G = empty graphs ⇒ G + kv = graphs with vertex covernumber at most k ⇒ Vertex Cover is FPT.
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G + k verticesLet G be a graph property, and let G + kv contain graph G if thereis a set S ⊆ V (G ) of k vertices such that G \ S ∈ G.
G + k vertices
Let G be a graph property, and let G + kv contain graph G if there is a set
S ! V (G) of k vertices such that G \ S " G.
S
Lemma: If G is minor closed, then G + kv is minor closed for every fixed k .
# It is (nonuniform) FPT to decide if G can be transformed into a member of G by
deleting k vertices.
Fixed Parameter Algorithms – p.65/98
Lemma: If G is minor closed, then G + kv is minor closed forevery fixed k.⇒ It is (nonuniform) FPT to decide if G can be transformed into amember of G by deleting k vertices.
I If G = forests ⇒ G + kv = graphs that can be made acyclicby the deletion of k vertices ⇒ Feedback Vertex Set isFPT.
I If G = planar graphs ⇒ G + kv = graphs that can be madeplanar by the deletion of k vertices (k-apex graphs) ⇒k-Apex Graph is FPT.
I If G = empty graphs ⇒ G + kv = graphs with vertex covernumber at most k ⇒ Vertex Cover is FPT.
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Treewidth
FPT algorithmic techniques
Significant advances in the past 20 years or so (especially in recent years).
Powerful toolbox for designing FPT algorithms:
Iterative compressionTreewidth
Bounded Search Tree
Graph Minors TheoremColor coding
Kernelization
Fixed Parameter Algorithms – p.8/98
![Page 17: P arameterized Algorithms II PAR T I St. Petersburg, 2011 · Graph Minors I Some consequences of the Graph Minors Theorem give a quick way of showing that certain problems are FPT.](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc9f29ad6a402d66675ddc/html5/thumbnails/17.jpg)
TREEWIDTH
Introduction and definition
Part I: Algorithms for bounded treewidth graphs.
Part II: Graph-theoretic properties of treewidth.
Part III: Applications for general graphs.
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The Party Problem
Party ProblemProblem: Invite some colleagues for a party.
Maximize: The total fun factor of the invited people.Constraint: Everyone should be having fun.
No fun with your direct boss!
I Input: A tree with weightson the vertices.
I Task: Find an independentset of maximum weight.
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The Party Problem
Party ProblemProblem: Invite some colleagues for a party.
Maximize: The total fun factor of the invited people.Constraint: Everyone should be having fun.
No fun with your direct boss!
I Input: A tree with weightson the vertices.
I Task: Find an independentset of maximum weight.
![Page 20: P arameterized Algorithms II PAR T I St. Petersburg, 2011 · Graph Minors I Some consequences of the Graph Minors Theorem give a quick way of showing that certain problems are FPT.](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc9f29ad6a402d66675ddc/html5/thumbnails/20.jpg)
The Party Problem
Party ProblemProblem: Invite some colleagues for a party.
Maximize: The total fun factor of the invited people.Constraint: Everyone should be having fun.
No fun with your direct boss!
The Party Problem
PARTY PROBLEM
Problem: Invite some colleagues for a party.
Maximize: The total fun factor of the invited people.
Constraint: Everyone should be having fun.
Do not invite a colleague and his direct boss at the same time!
6
644
5
2
Input: A tree with weights on the
vertices.
Task: Find an independent set of maxi-
mum weight.
Fixed Parameter Algorithms – p.5/48
I Input: A tree with weightson the vertices.
I Task: Find an independentset of maximum weight.
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The Party Problem
Party ProblemProblem: Invite some colleagues for a party.
Maximize: The total fun factor of the invited people.Constraint: Everyone should be having fun.
No fun with your direct boss!
The Party Problem
PARTY PROBLEM
Problem: Invite some colleagues for a party.
Maximize: The total fun factor of the invited people.
Constraint: Everyone should be having fun.
Do not invite a colleague and his direct boss at the same time!
2
5
4 4 6
6Input: A tree with weights on the
vertices.
Task: Find an independent set of maxi-
mum weight.
Fixed Parameter Algorithms – p.5/48
I Input: A tree with weightson the vertices.
I Task: Find an independentset of maximum weight.
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Solving the Party ProblemDynamic programming paradigm: We solve a large number ofsubproblems that depend on each other. The answer is a singlesubproblem.
Tv : the subtree rooted at v .A[v ]: max. weight of an independent set in Tv
B[v ]: max. weight of an independent set in Tv that doesnot contain v
Goal: determine A[r ] for the root r .
Method:Assume v1, . . . , vk are the children of v . Use the recurrencerelations
B[v ] =∑k
i=1 A[vi ]
A[v ] = max{B[v ] , w(v) +∑k
i=1 B[vi ]}The values A[v ] and B[v ] can be calculated in a bottom-up order(the leaves are trivial).
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Solving the Party ProblemDynamic programming paradigm: We solve a large number ofsubproblems that depend on each other. The answer is a singlesubproblem.
Tv : the subtree rooted at v .A[v ]: max. weight of an independent set in Tv
B[v ]: max. weight of an independent set in Tv that doesnot contain v
Goal: determine A[r ] for the root r .
Method:Assume v1, . . . , vk are the children of v . Use the recurrencerelations
B[v ] =∑k
i=1 A[vi ]
A[v ] = max{B[v ] , w(v) +∑k
i=1 B[vi ]}The values A[v ] and B[v ] can be calculated in a bottom-up order(the leaves are trivial).
![Page 24: P arameterized Algorithms II PAR T I St. Petersburg, 2011 · Graph Minors I Some consequences of the Graph Minors Theorem give a quick way of showing that certain problems are FPT.](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc9f29ad6a402d66675ddc/html5/thumbnails/24.jpg)
TreewidthTreewidth: A measure of how “tree-like”the graph is.(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are ar-ranged in a tree structure satisfying the fol-lowing properties:
1. If u and v are neighbors, then there isa bag containing both of them.
2. For every vertex v , the bagscontaining v form a connectedsubtree.
Width of the decomposition: largest bagsize −1.
treewidth: width of the best decomposi-tion.
Fact: treewidth = 1 iff graph is a forest
Treewidth
Treewidth: A measure of how “tree-like” the graph is.
(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are arranged in a tree
structure satisfying the following properties:
1. If u and v are neighbors, then there is a bag con-
taining both of them.
2. For every vertex v , the bags containing v form a
connected subtree.
h
dcb
a
e f g
a, b, c g , hb, e, f
d , f , gb, c, f
c, d , f
Fixed Parameter Algorithms – p.8/48
Treewidth
Treewidth: A measure of how “tree-like” the graph is.
(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are arranged in a tree
structure satisfying the following properties:
1. If u and v are neighbors, then there is a bag con-
taining both of them.
2. For every vertex v , the bags containing v form a
connected subtree.
h
dcb
a
e f g
a, b, c g , hb, e, f
d , f , gb, c, f
c, d , f
Fixed Parameter Algorithms – p.8/48
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TreewidthTreewidth: A measure of how “tree-like”the graph is.(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are ar-ranged in a tree structure satisfying the fol-lowing properties:
1. If u and v are neighbors, then there isa bag containing both of them.
2. For every vertex v , the bagscontaining v form a connectedsubtree.
Width of the decomposition: largest bagsize −1.
treewidth: width of the best decomposi-tion.
Fact: treewidth = 1 iff graph is a forest
Treewidth
Treewidth: A measure of how “tree-like” the graph is.
(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are arranged in a tree
structure satisfying the following properties:
1. If u and v are neighbors, then there is a bag con-
taining both of them.
2. For every vertex v , the bags containing v form a
connected subtree.
h
dcb
a
e f g
a, b, c g , hb, e, f
d , f , gb, c, f
c, d , f
Fixed Parameter Algorithms – p.8/48
Treewidth
Treewidth: A measure of how “tree-like” the graph is.
(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are arranged in a tree
structure satisfying the following properties:
1. If u and v are neighbors, then there is a bag con-
taining both of them.
2. For every vertex v , the bags containing v form a
connected subtree.
h
dcb
a
e f g
a, b, c b, e, f
b, c, f
c, d , f
d , f , g
g , h
Fixed Parameter Algorithms – p.8/48
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TreewidthTreewidth: A measure of how “tree-like”the graph is.(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are ar-ranged in a tree structure satisfying the fol-lowing properties:
1. If u and v are neighbors, then there isa bag containing both of them.
2. For every vertex v , the bagscontaining v form a connectedsubtree.
Width of the decomposition: largest bagsize −1.
treewidth: width of the best decomposi-tion.
Fact: treewidth = 1 iff graph is a forest
Treewidth
Treewidth: A measure of how “tree-like” the graph is.
(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are arranged in a tree
structure satisfying the following properties:
1. If u and v are neighbors, then there is a bag con-
taining both of them.
2. For every vertex v , the bags containing v form a
connected subtree.
h
dcb
a
e f g
a, b, c g , hb, e, f
d , f , gb, c, f
c, d , f
Fixed Parameter Algorithms – p.8/48
Treewidth
Treewidth: A measure of how “tree-like” the graph is.
(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are arranged in a tree
structure satisfying the following properties:
1. If u and v are neighbors, then there is a bag con-
taining both of them.
2. For every vertex v , the bags containing v form a
connected subtree.
h
dcb
a
e f g
b, c, f
g , ha, b, c
c, d , f
d , f , g
b, e, f
Fixed Parameter Algorithms – p.8/48
![Page 27: P arameterized Algorithms II PAR T I St. Petersburg, 2011 · Graph Minors I Some consequences of the Graph Minors Theorem give a quick way of showing that certain problems are FPT.](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc9f29ad6a402d66675ddc/html5/thumbnails/27.jpg)
TreewidthTreewidth: A measure of how “tree-like”the graph is.(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are ar-ranged in a tree structure satisfying the fol-lowing properties:
1. If u and v are neighbors, then there isa bag containing both of them.
2. For every vertex v , the bagscontaining v form a connectedsubtree.
Width of the decomposition: largest bagsize −1.
treewidth: width of the best decomposi-tion.
Fact: treewidth = 1 iff graph is a forest
Treewidth
Treewidth: A measure of how “tree-like” the graph is.
(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are arranged in a tree
structure satisfying the following properties:
1. If u and v are neighbors, then there is a bag con-
taining both of them.
2. For every vertex v , the bags containing v form a
connected subtree.
h
dcb
a
e f g
a, b, c g , hb, e, f
d , f , gb, c, f
c, d , f
Fixed Parameter Algorithms – p.8/48
Treewidth
Treewidth: A measure of how “tree-like” the graph is.
(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are arranged in a tree
structure satisfying the following properties:
1. If u and v are neighbors, then there is a bag con-
taining both of them.
2. For every vertex v , the bags containing v form a
connected subtree.
h
dcb
a
e f g
b, c, f
g , ha, b, c
c, d , f
d , f , g
b, e, f
Fixed Parameter Algorithms – p.8/48
![Page 28: P arameterized Algorithms II PAR T I St. Petersburg, 2011 · Graph Minors I Some consequences of the Graph Minors Theorem give a quick way of showing that certain problems are FPT.](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc9f29ad6a402d66675ddc/html5/thumbnails/28.jpg)
TreewidthTreewidth: A measure of how “tree-like”the graph is.(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are ar-ranged in a tree structure satisfying the fol-lowing properties:
1. If u and v are neighbors, then there isa bag containing both of them.
2. For every vertex v , the bagscontaining v form a connectedsubtree.
Width of the decomposition: largest bagsize −1.
treewidth: width of the best decomposi-tion.
Fact: treewidth = 1 iff graph is a forest
Treewidth
Treewidth: A measure of how “tree-like” the graph is.
(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are arranged in a tree
structure satisfying the following properties:
1. If u and v are neighbors, then there is a bag con-
taining both of them.
2. For every vertex v , the bags containing v form a
connected subtree.
Width of the decomposition: largest bag size !1.
treewidth: width of the best decomposition.
Fact: treewidth = 1 "# graph is a forest
c
a
f g he
db
d , f , g
g , ha, b, c b, e, f
b, c, f
c, d , f
Fixed Parameter Algorithms – p.8/48
Treewidth
Treewidth: A measure of how “tree-like” the graph is.
(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are arranged in a tree
structure satisfying the following properties:
1. If u and v are neighbors, then there is a bag con-
taining both of them.
2. For every vertex v , the bags containing v form a
connected subtree.
Width of the decomposition: largest bag size !1.
treewidth: width of the best decomposition.
Fact: treewidth = 1 "# graph is a forest
c
a
f g he
db
d , f , g
g , ha, b, c b, e, f
b, c, f
c, d , f
Fixed Parameter Algorithms – p.8/48
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TreewidthTreewidth: A measure of how “tree-like”the graph is.(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are ar-ranged in a tree structure satisfying the fol-lowing properties:
1. If u and v are neighbors, then there isa bag containing both of them.
2. For every vertex v , the bagscontaining v form a connectedsubtree.
Width of the decomposition: largest bagsize −1.
treewidth: width of the best decomposi-tion.
Fact: treewidth = 1 iff graph is a forest
Treewidth
Treewidth: A measure of how “tree-like” the graph is.
(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are arranged in a tree
structure satisfying the following properties:
1. If u and v are neighbors, then there is a bag con-
taining both of them.
2. For every vertex v , the bags containing v form a
connected subtree.
Width of the decomposition: largest bag size !1.
treewidth: width of the best decomposition.
Fact: treewidth = 1 "# graph is a forest
d
hgfe
a
b c
c, d , f
g , hb, e, fa, b, c
d , f , gb, c, f
Fixed Parameter Algorithms – p.8/48
Treewidth
Treewidth: A measure of how “tree-like” the graph is.
(Introduced by Robertson and Seymour.)
Tree decomposition: Vertices are arranged in a tree
structure satisfying the following properties:
1. If u and v are neighbors, then there is a bag con-
taining both of them.
2. For every vertex v , the bags containing v form a
connected subtree.
Width of the decomposition: largest bag size !1.
treewidth: width of the best decomposition.
Fact: treewidth = 1 "# graph is a forest
d
hgfe
a
b c
c, d , f
g , hb, e, fa, b, c
d , f , gb, c, f
Fixed Parameter Algorithms – p.8/48
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Finding tree decompositions
Fact: It is NP-hard to determine the treewidth of a graph (given agraph G and an integer w , decide if the treewidth of G is at mostw), but there is a polynomial-time algorithm for every fixed w .
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Finding tree decompositions
Fact: [Bodlaender’s Theorem] For every fixed w , there is alinear-time algorithm that finds a tree decomposition of width w(if exists).
⇒ Deciding if treewidth is at most w is fixed-parameter tractable.⇒ If we want an FPT algorithm parameterized by treewidth w ofthe input graph, then we can assume that a tree decomposition ofwidth w is available.
Running time is 2O(w3) · n. Sometimes it is better to use thefollowing results instead:
Fact: There is a O(33w · w · n2) time algorithm that finds a treedecomposition of width 4w + 1, if the treewidth of the graph is atmost w .
Fact: There is a polynomial-time algorithm that finds a treedecomposition of width O(w
√log w), if the treewidth of the graph
is at most w .
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Finding tree decompositions
Fact: [Bodlaender’s Theorem] For every fixed w , there is alinear-time algorithm that finds a tree decomposition of width w(if exists).
⇒ Deciding if treewidth is at most w is fixed-parameter tractable.⇒ If we want an FPT algorithm parameterized by treewidth w ofthe input graph, then we can assume that a tree decomposition ofwidth w is available.
Running time is 2O(w3) · n. Sometimes it is better to use thefollowing results instead:
Fact: There is a O(33w · w · n2) time algorithm that finds a treedecomposition of width 4w + 1, if the treewidth of the graph is atmost w .
Fact: There is a polynomial-time algorithm that finds a treedecomposition of width O(w
√log w), if the treewidth of the graph
is at most w .
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Part I:
Algoritmhs forbounded-treewidth graphs
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Weighted Max Independent Setand tree decompositions
Fact: Given a tree decomposition of width w , Weighted MaxIndependent Set can be solved in time O(2w · n).
Bx : vertices appearing in node x .Vx : vertices appearing in the subtreerooted at x .
Generalizing our solution for trees:
Instead of computing 2 values A[v ],B[v ] for each vertex of the graph, wecompute 2|Bx | ≤ 2w+1 values for eachbag Bx .
M[x ,S ]: the maximum weight of anindependent set I ⊆ Vx with I ∩ Bx =S .
WEIGHTED MAX INDEPENDENT SET
and tree decompositions
Fact: Given a tree decomposition of width w , WEIGHTED MAX INDEPENDENT SET can
be solved in time O(2w · n).
Bx : vertices appearing in node x .
Vx : vertices appearing in the subtree rooted at x .
Generalizing our solution for trees:
Instead of computing 2 values A[v ], B[v ] for each
vertex of the graph, we compute 2|Bx | ! 2w+1
values for each bag Bx .
M[x , S]: the maximum weight of an independent
set I " Vx with I # Bx = S .
b, e, f g , h
c, d , f
b, c, f d , f , g
a, b, c
$ =? bc =?b =? cf =?c =? bf =?f =? bcf =?
How to determine M[x , S ] if all the values are known forthe children of x?
Fixed Parameter Algorithms – p.11/48
How to determine M[x , S ] if all the values are known forthe children of x?
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Nice tree decompositions
Definition: A rooted tree decomposition is nice if every node x isone of the following 4 types:
I Leaf: no children, |Bx | = 1
I Introduce: 1 child y , Bx = By ∪ {v} for some vertex v
I Forget: 1 child y , Bx = By \ {v} for some vertex v
I Join: 2 children y1, y2 with Bx = By1 = By2
Nice tree decompositions
Definition: A rooted tree decomposition is nice if every node x is one of the following
4 types:
Leaf: no children, |Bx | = 1
Introduce: 1 child y , Bx = By ! {v} for some vertex v
Forget: 1 child y , Bx = By \ {v} for some vertex v
Join: 2 children y1, y2 with Bx = By1 = By2
Leaf Forget JoinIntroduce
u, v , w
u, v , w u, w
u, w u, v , w
u, v , w
u, v , w
v
Fact: A tree decomposition of width w and n nodes can be turned into a nice tree
decomposition of width w and O(wn) nodes in time O(w 2n).Fixed Parameter Algorithms – p.12/48Fact: A tree decomposition of width w and n nodes can be turned
into a nice tree decomposition of width w and O(wn) nodes intime O(w2n).
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Weighted Max Independent Setand nice tree decompositions
I Leaf: no children, |Bx | = 1Trivial!
I Introduce: 1 child y , Bx = By ∪ {v} for some vertex v
m[x ,S ] =
m[y ,S ] if v 6∈ S ,
m[y ,S \ {v}] + w(v) if v ∈ S but v has no neighbor in S ,
−∞ if S contains v and its neighbor.
I Forget: 1 child y , Bx = By \ {v} for some vertex v
m[x ,S ] = max{m[y ,S ],m[y , S ∪ {v}]}
I Join: 2 children y1, y2 with Bx = By1 = By2
m[x ,S ] = m[y1,S ] + m[y2,S ]− w(S)
Nice tree decompositions
Definition: A rooted tree decomposition is nice if every node x is one of the following
4 types:
Leaf: no children, |Bx | = 1
Introduce: 1 child y , Bx = By ! {v} for some vertex v
Forget: 1 child y , Bx = By \ {v} for some vertex v
Join: 2 children y1, y2 with Bx = By1 = By2
Leaf Forget JoinIntroduce
u, v , w
u, v , w u, w
u, w u, v , w
u, v , w
u, v , w
v
Fact: A tree decomposition of width w and n nodes can be turned into a nice tree
decomposition of width w and O(wn) nodes in time O(w 2n).Fixed Parameter Algorithms – p.12/48
There are at most 2w+1 · n subproblems m[x ,S ] and eachsubproblem can be solved in constant time (assuming the childrenare already solved) ⇒ Running time is O(2w · n).
⇒ Weighted Max Independent Set is FPT parameterizedby treewidth ⇒ Weighted Min Vertex Cover is FPTparameterized by treewidth.
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Weighted Max Independent Setand nice tree decompositions
I Forget: 1 child y , Bx = By \ {v} for some vertex v
m[x ,S ] = max{m[y , S ],m[y , S ∪ {v}]}
I Join: 2 children y1, y2 with Bx = By1 = By2
m[x ,S ] = m[y1,S ] + m[y2,S ]− w(S)
Nice tree decompositions
Definition: A rooted tree decomposition is nice if every node x is one of the following
4 types:
Leaf: no children, |Bx | = 1
Introduce: 1 child y , Bx = By ! {v} for some vertex v
Forget: 1 child y , Bx = By \ {v} for some vertex v
Join: 2 children y1, y2 with Bx = By1 = By2
Leaf Forget JoinIntroduce
u, v , w
u, v , w u, w
u, w u, v , w
u, v , w
u, v , w
v
Fact: A tree decomposition of width w and n nodes can be turned into a nice tree
decomposition of width w and O(wn) nodes in time O(w 2n).Fixed Parameter Algorithms – p.12/48
There are at most 2w+1 · n subproblems m[x ,S ] and eachsubproblem can be solved in constant time (assuming the childrenare already solved) ⇒ Running time is O(2w · n).
⇒ Weighted Max Independent Set is FPT parameterizedby treewidth ⇒ Weighted Min Vertex Cover is FPTparameterized by treewidth.
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Weighted Max Independent Setand nice tree decompositions
I Forget: 1 child y , Bx = By \ {v} for some vertex v
m[x ,S ] = max{m[y , S ],m[y , S ∪ {v}]}
I Join: 2 children y1, y2 with Bx = By1 = By2
m[x ,S ] = m[y1,S ] + m[y2,S ]− w(S)
There are at most 2w+1 · n subproblems m[x ,S ] and eachsubproblem can be solved in constant time (assuming the childrenare already solved) ⇒ Running time is O(2w · n).
⇒ Weighted Max Independent Set is FPT parameterizedby treewidth ⇒ Weighted Min Vertex Cover is FPTparameterized by treewidth.
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3-Coloring and tree decompositions
Fact: Given a tree decomposition of width w , 3-Coloring canbe solved in O(3w · n).
Bx : vertices appearing in node x .Vx : vertices appearing in the subtreerooted at x .
For every node x and coloring c : Bx →{1, 2, 3}, we compute the Booleanvalue E [x , c], which is true if and onlyif c can be extended to a proper 3-coloring of Vx .
3-COLORING andtree decompositions
Fact: Given a tree decomposition of width w , 3-COLORING can be solved in O(3w ·n).
Bx : vertices appearing in node x .
Vx : vertices appearing in the subtree rooted at x .
For every node x and coloring c : Bx ! {1, 2, 3},we compute the Boolean value E [x , c], which is
true if and only if c can be extended to a proper
3-coloring of Vx .
b, e, f g , h
c, d , f
b, c, f d , f , g
a, b, c
bcf=T bcf=Fbcf=T bcf=F
... ...
How to determine E [x , c] if all the values are known forthe children of x?
Fixed Parameter Algorithms – p.14/48
How to determine E [x , c] if all the values are known forthe children of x?
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3-Coloring and nice tree decompositions
I Leaf: no children, |Bx | = 1Trivial!
I Introduce: 1 child y , Bx = By ∪ {v} for some vertex vIf c(v) 6= c(u) for every neighbor u of v , thenE [x , c] = E [y , c ′], where c ′ is c restricted to By .
I Forget: 1 child y , Bx = By \ {v} for some vertex vE [x , c] is true if E [y , c ′] is true for one of the 3 extensions ofc to By .
I Join: 2 children y1, y2 with Bx = By1 = By2
E [x , c] = E [y1, c] ∧ E [y2, c]
Nice tree decompositions
Definition: A rooted tree decomposition is nice if every node x is one of the following
4 types:
Leaf: no children, |Bx | = 1
Introduce: 1 child y , Bx = By ! {v} for some vertex v
Forget: 1 child y , Bx = By \ {v} for some vertex v
Join: 2 children y1, y2 with Bx = By1 = By2
Leaf Forget JoinIntroduce
u, v , w
u, v , w u, w
u, w u, v , w
u, v , w
u, v , w
v
Fact: A tree decomposition of width w and n nodes can be turned into a nice tree
decomposition of width w and O(wn) nodes in time O(w 2n).Fixed Parameter Algorithms – p.12/48
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3-Coloring and nice tree decompositions
I Leaf: no children, |Bx | = 1Trivial!
I Introduce: 1 child y , Bx = By ∪ {v} for some vertex vIf c(v) 6= c(u) for every neighbor u of v , thenE [x , c] = E [y , c ′], where c ′ is c restricted to By .
I Forget: 1 child y , Bx = By \ {v} for some vertex vE [x , c] is true if E [y , c ′] is true for one of the 3 extensions ofc to By .
I Join: 2 children y1, y2 with Bx = By1 = By2
E [x , c] = E [y1, c] ∧ E [y2, c]
There are at most 3w+1 · n subproblems E [x , c] and eachsubproblem can be solved in constant time (assuming the childrenare already solved).
⇒ Running time is O(3w · n).
⇒ 3-Coloring is FPT parameterized by treewidth.
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Vertex coloring
More generally:
Fact: Given a tree decomposition of width w , c-Coloring canbe solved in O∗(cw ).
Exercise: Every graph of treewidth at most w can be colored withw + 1 colors.
Fact: Given a tree decomposition of width w , VertexColoring can be solved in time O∗(ww ).
⇒ Vertex Coloring is FPT parameterized by treewidth.
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Hamiltonian cycle andtree decompositions
Fact: Given a tree decomposition of width w , Hamiltoniancycle can be solved in time wO(w) · n.
Bx : vertices appearing in node x .Vx : vertices appearing in the subtreerooted at x .
If H is a Hamiltonian cycle, then thesubgraph H[Vx ] is a set of paths withendpoints in Bx .
Hamiltonian cycle andtree decompositions
Fact: Given a tree decomposition of width w , HAMILTONIAN CYCLE can be solved in
time wO(w) · n.
Bx : vertices appearing in node x .
Vx : vertices appearing in the subtree rooted at x .
If H is a Hamiltonian cycle, then the subgraph
H[Vx ] is a set of paths with endpoints in Bx .
What are the important properties of H[Vx ] “seen
from the outside world”?
The subsets B0x , B
1x , B
2x of Bx having degree
0, 1, and 2.
The matching M of B1x .
B0x B1
x B2x
x
Vx
Number of subproblems (B0x , B1
x ,B2x , M) for each node x : at most 3w · ww .
Fixed Parameter Algorithms – p.17/48
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Hamiltonian cycle andtree decompositions
Bx : vertices appearing in node x .Vx : vertices appearing in the subtreerooted at x .
If H is a Hamiltonian cycle, then thesubgraph H[Vx ] is a set of paths withendpoints in Bx .
What are the important properties ofH[Vx ] “seen from the outside world”?
I The subsets B0x , B1
x , B2x of Bx
having degree 0, 1, and 2.
I The matching M of B1x .
Hamiltonian cycle andtree decompositions
Fact: Given a tree decomposition of width w , HAMILTONIAN CYCLE can be solved in
time wO(w) · n.
Bx : vertices appearing in node x .
Vx : vertices appearing in the subtree rooted at x .
If H is a Hamiltonian cycle, then the subgraph
H[Vx ] is a set of paths with endpoints in Bx .
What are the important properties of H[Vx ] “seen
from the outside world”?
The subsets B0x , B
1x , B
2x of Bx having degree
0, 1, and 2.
The matching M of B1x .
B0x B1
x B2x
x
Vx
Number of subproblems (B0x , B1
x ,B2x , M) for each node x : at most 3w · ww .
Fixed Parameter Algorithms – p.17/48
Number of subproblems (B0x ,B
1x ,B
2x ,M) for each node x : at most
3w · ww .
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Hamiltonian cycle andnice tree decompositions
For each subproblem (B0x ,B
1x ,B
2x ,M), we have to determine if
there is a set of paths with this pattern.
How to do this for the different types of nodes?(Assuming that all the subproblems are solved for the children.)
Leaf: no children, |Bx | = 1
Trivial!
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Hamiltonian cycle andnice tree decompositions
Solving subproblem (B0x ,B
1x ,B
2x ,M) of node x .
Forget: 1 child y , Bx = By \ {v} for some vertex v
In a solution H of (B0x ,B
1x ,B
2x ,M), vertex v has degree 2. Thus
subproblem (B0x ,B
1x ,B
2x ,M) of x is equivalent to subproblem
(B0x ,B
1x ,B
2x ∪ {v},M) of y .
Hamiltonian cycle andnice tree decompositions
Solving subproblem (B0x , B1
x , B2x , M) of node x .
Forget: 1 child y , Bx = By \ {v} for some vertex v
In a solution H of (B0x , B1
x , B2x , M), vertex v has degree 2. Thus subproblem
(B0x , B1
x ,B2x , M) of x is equivalent to subproblem (B0
x , B1x ,B2
x ! {v},M) of y .
B2xB0
x B1x
Fixed Parameter Algorithms – p.19/48
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Hamiltonian cycle andnice tree decompositions
Solving subproblem (B0x ,B
1x ,B
2x ,M) of node x .
Forget: 1 child y , Bx = By \ {v} for some vertex v
In a solution H of (B0x ,B
1x ,B
2x ,M), vertex v has degree 2. Thus
subproblem (B0x ,B
1x ,B
2x ,M) of x is equivalent to subproblem
(B0x ,B
1x ,B
2x ∪ {v},M) of y .
Hamiltonian cycle andnice tree decompositions
Solving subproblem (B0x , B1
x , B2x , M) of node x .
Forget: 1 child y , Bx = By \ {v} for some vertex v
In a solution H of (B0x , B1
x , B2x , M), vertex v has degree 2. Thus subproblem
(B0x , B1
x ,B2x , M) of x is equivalent to subproblem (B0
x , B1x ,B2
x ! {v},M) of y .
B0x B2
xB1x
v
Fixed Parameter Algorithms – p.19/48
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Hamiltonian cycle andnice tree decompositions
Solving subproblem (B0x ,B
1x ,B
2x ,M) of node x .
Forget: 1 child y , Bx = By \ {v} for some vertex v
In a solution H of (B0x ,B
1x ,B
2x ,M), vertex v has degree 2. Thus
subproblem (B0x ,B
1x ,B
2x ,M) of x is equivalent to subproblem
(B0x ,B
1x ,B
2x ∪ {v},M) of y .
Hamiltonian cycle andnice tree decompositions
Solving subproblem (B0x , B1
x , B2x , M) of node x .
Forget: 1 child y , Bx = By \ {v} for some vertex v
In a solution H of (B0x , B1
x , B2x , M), vertex v has degree 2. Thus subproblem
(B0x , B1
x ,B2x , M) of x is equivalent to subproblem (B0
x , B1x ,B2
x ! {v},M) of y .
B2xB1
xB0x
v
Fixed Parameter Algorithms – p.19/48
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Hamiltonian cycle andnice tree decompositions
Solving subproblem (B0x ,B
1x ,B
2x ,M) of node x .
Introduce: 1 child y , Bx = By ∪ {v} for some vertex v
Case 1: v ∈ B0x .
Subproblem is equivalent with (B0x \ {v},B1
x ,B2x ,M) for node y .
Hamiltonian cycle andnice tree decompositions
Solving subproblem (B0x , B1
x , B2x , M) of node x .
Introduce: 1 child y , Bx = By ! {v} for some vertex v
Case 1: v " B0x . Subproblem is equivalent with (B0
x \ {v},B1x , B2
x , M) for node y .
v x
B0x B2
xB1x
! y
B0y B2
yB1y
Fixed Parameter Algorithms – p.20/48
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Hamiltonian cycle andnice tree decompositions
Solving subproblem (B0x ,B
1x ,B
2x ,M) of node x .
Introduce: 1 child y , Bx = By ∪ {v} for some vertex v
Case 2:v ∈ B1
x . Every neighbor of v in Vx is in Bx . Thus v has to beadjacent with one other vertex of Bx .
Is there a subproblem (B0y ,B
1y ,B
2y ,M
′) of node y such that making a
vertex of By adjacent to v makes it a solution for subproblem
(B0x ,B
1x ,B
2x ,M) of node x?
Hamiltonian cycle andnice tree decompositions
Solving subproblem (B0x , B1
x , B2x , M) of node x .
Introduce: 1 child y , Bx = By ! {v} for some vertex v
Case 2: v " B1x . Every neighbor of v in Vx is in Bx . Thus v has to be adjacent with one
other vertex of Bx .
Is there a subproblem (B0y , B1
y ,B2y , M !) of node y such that making a vertex of By
adjacent to v makes it a solution for subproblem (B0x , B1
x , B2x , M) of node x?
x
B0x B1
x B2x
vy
B0y B2
yB1y
Fixed Parameter Algorithms – p.20/48
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Hamiltonian cycle andnice tree decompositions
Solving subproblem (B0x ,B
1x ,B
2x ,M) of node x .
Introduce: 1 child y , Bx = By ∪ {v} for some vertex v
Case 3: v ∈ B1x . Similar to Case 2, but 2 vertices of By are
adjacent with v .
Hamiltonian cycle andnice tree decompositions
Solving subproblem (B0x , B1
x , B2x , M) of node x .
Introduce: 1 child y , Bx = By ! {v} for some vertex v
Case 3: v " B1x . Similar to Case 2, but 2 vertices of By are adjacent with v .
vx
B2xB0
x B1x
! y
B1yB0
y
Fixed Parameter Algorithms – p.20/48
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Hamiltonian cycle andnice tree decompositions
Solving subproblem (B0x ,B
1x ,B
2x ,M) of node x .
Join: 2 children y1, y2 with Bx = By1 = By2
A solution H is the union of a subgraph H1 ⊆ G [Vy1 ] and asubgraph H2 ⊆ G [Vy2 ].
If H1 is a solution for (B0y1,B1
y1,B2
y1,M1) of node y1 and H2 is a
solution for (B0y2,B1
y2,B2
y2,M2) of node y2, then we can check if
H1 ∪ H2 is a solution for (B0x ,B
1x ,B
2x ,M) of node x .
For any two subproblems of y1 and y2, we check if they havesolutions and if their union is a solution for (B0
x ,B1x ,B
2x ,M) of
node x .
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Monadic Second Order Logic
Extended Monadic Second Order Logic (EMSO)
A logical language on graphs consisting of the following:
I Logical connectives ∧, ∨, →, ¬, =, 6=I quantifiers ∀, ∃ over vertex/edge variables
I predicate adj(u, v): vertices u and v are adjacent
I predicate inc(e, v): edge e is incident to vertex v
I quantifiers ∀, ∃ over vertex/edge set variables
I ∈, ⊆ for vertex/edge sets
Example: The formula∃C ⊆ V ∀v ∈ C ∃u1, u2 ∈ C (u1 6= u2 ∧ adj(u1, v) ∧ adj(u2, v)) istrue . . .
if graph G (V ,E ) has a cycle.
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Monadic Second Order Logic
Extended Monadic Second Order Logic (EMSO)
A logical language on graphs consisting of the following:
I Logical connectives ∧, ∨, →, ¬, =, 6=I quantifiers ∀, ∃ over vertex/edge variables
I predicate adj(u, v): vertices u and v are adjacent
I predicate inc(e, v): edge e is incident to vertex v
I quantifiers ∀, ∃ over vertex/edge set variables
I ∈, ⊆ for vertex/edge sets
Example: The formula∃C ⊆ V ∀v ∈ C ∃u1, u2 ∈ C (u1 6= u2 ∧ adj(u1, v) ∧ adj(u2, v)) istrue if graph G (V ,E ) has a cycle.
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Courcelle’s Theorem
Courcelle’s Theorem: If a graph property can be expressed inEMSO, then for every fixed w ≥ 1, there is a linear-time algorithmfor testing this property on graphs having treewidth at most w .
Note: The constant depending on w can be very large (double,triple exponential etc.), therefore a direct dynamic programmingalgorithm can be more efficient.
If we can express a property in EMSO, then we immediately getthat testing this property is FPT parameterized by the treewidth wof the input graph.
Can we express 3-Coloring and Hamiltonian Cycle inEMSO?
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Courcelle’s Theorem
Courcelle’s Theorem: If a graph property can be expressed inEMSO, then for every fixed w ≥ 1, there is a linear-time algorithmfor testing this property on graphs having treewidth at most w .
Note: The constant depending on w can be very large (double,triple exponential etc.), therefore a direct dynamic programmingalgorithm can be more efficient.
If we can express a property in EMSO, then we immediately getthat testing this property is FPT parameterized by the treewidth wof the input graph.
Can we express 3-Coloring and Hamiltonian Cycle inEMSO?
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Using Courcelle’s Theorem
3-Coloring
∃C1,C2,C3 ⊆ V(∀v ∈ V (v ∈ C1 ∨ v ∈ C2 ∨ v ∈ C3)
)∧(∀u, v ∈
V adj(u, v)→ (¬(u ∈ C1 ∧ v ∈ C1) ∧ ¬(u ∈ C2 ∧ v ∈ C2) ∧ ¬(u ∈C3 ∧ v ∈ C3))
)
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Using Courcelle’s Theorem
3-Coloring Hamiltonian Cycle
∃H ⊆ E(spanning(H) ∧ (∀v ∈ V degree2(H, v))
)degree0(H, v) := ¬∃e ∈ H inc(e, v)
degree1(H, v) := ¬degree0(H, v) ∧(¬∃e1, e2 ∈ H (e1 6=
e2 ∧ inc(e1, v) ∧ inc(e2, v)))
degree2(H, v) := ¬degree0(H, v) ∧ ¬degree1(H, v) ∧(¬∃e1, e2, e3 ∈
H (e1 6= e2 ∧ e2 6= e3 ∧ e1 6= e3 ∧ inc(e1, v) ∧ inc(e2, v) ∧ inc(e3, v))))
spanning(H) := ∀u, v ∈ V ∃P ⊆ H ∀x ∈ V(((x = u ∨ x =
v)∧ degree1(P, x))∨ (x 6= u ∧ x 6= v ∧ (degree0(P, x)∨ degree2(P, x))))
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Using Courcelle’s TheoremTwo ways of using Courcelle’s Theorem:
1. The problem can be described by a single formula (e.g,3-Coloring, Hamiltonian Cycle).
⇒ Problem can be solved in time f (w) · n for graphs of treewidthat most w .
⇒ Problem is FPT parameterized by the treewidth w of the inputgraph.
2. The problem can be described by a formula for each value ofthe parameter k .
Example: For each k , having a cycle of length exactly k can beexpressed as
∃v1, . . . , vk ∈ V (adj(v1, v2)∧adj(v2, v3)∧· · ·∧adj(vk−1, vk)∧adj(vk , v1)).
⇒ Problem can be solved in time f (k,w) · n for graphs oftreewidth w .
⇒ Problem is FPT parameterized with combined parameter k andtreewidth w .
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Using Courcelle’s TheoremTwo ways of using Courcelle’s Theorem:
1. The problem can be described by a single formula (e.g,3-Coloring, Hamiltonian Cycle).
⇒ Problem can be solved in time f (w) · n for graphs of treewidthat most w .
⇒ Problem is FPT parameterized by the treewidth w of the inputgraph.
2. The problem can be described by a formula for each value ofthe parameter k .
Example: For each k , having a cycle of length exactly k can beexpressed as
∃v1, . . . , vk ∈ V (adj(v1, v2)∧adj(v2, v3)∧· · ·∧adj(vk−1, vk)∧adj(vk , v1)).
⇒ Problem can be solved in time f (k,w) · n for graphs oftreewidth w .
⇒ Problem is FPT parameterized with combined parameter k andtreewidth w .
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Subgraph Isomorphism
Subgraph Isomorphism: given graphs H and G , find a copy ofH in G as subgraph. Parameter k := |V (H)| (size of the smallgraph).
For each H, we can construct a formula φH that expresses “G hasa subgraph isomorphic to H” (similarly to the k-cycle on theprevious slide).
⇒ By Courcelle’s Theorem, Subgraph Isomorphism can besolved in time f (H,w) · n if G has treewidth at most w .
⇒ Since there is only a finite number of simple graphs on kvertices, Subgraph Isomorphism can be solved in timef (k ,w) · n if H has k vertices and G has treewidth at most w .
⇒ Subgraph Isomorphism is FPT parameterized by combinedparameter k := |V (H)| and the treewidth w of G .
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Subgraph Isomorphism
Subgraph Isomorphism: given graphs H and G , find a copy ofH in G as subgraph. Parameter k := |V (H)| (size of the smallgraph).
For each H, we can construct a formula φH that expresses “G hasa subgraph isomorphic to H” (similarly to the k-cycle on theprevious slide).
⇒ By Courcelle’s Theorem, Subgraph Isomorphism can besolved in time f (H,w) · n if G has treewidth at most w .
⇒ Since there is only a finite number of simple graphs on kvertices, Subgraph Isomorphism can be solved in timef (k ,w) · n if H has k vertices and G has treewidth at most w .
⇒ Subgraph Isomorphism is FPT parameterized by combinedparameter k := |V (H)| and the treewidth w of G .
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Subgraph Isomorphism
Subgraph Isomorphism: given graphs H and G , find a copy ofH in G as subgraph. Parameter k := |V (H)| (size of the smallgraph).
For each H, we can construct a formula φH that expresses “G hasa subgraph isomorphic to H” (similarly to the k-cycle on theprevious slide).
⇒ By Courcelle’s Theorem, Subgraph Isomorphism can besolved in time f (H,w) · n if G has treewidth at most w .
⇒ Since there is only a finite number of simple graphs on kvertices, Subgraph Isomorphism can be solved in timef (k ,w) · n if H has k vertices and G has treewidth at most w .
⇒ Subgraph Isomorphism is FPT parameterized by combinedparameter k := |V (H)| and the treewidth w of G .
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Part II:
Graph-theoretical properties
of treewidth
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The Robber and Cops game
Game: k cops try to capture a robber in the graph.
I In each step, the cops can move from vertex to vertexarbitrarily with helicopters.
I The robber moves infinitely fast on the edges, and sees wherethe cops will land.
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The Robber and Cops game (cont.)
Example: 2 cops have a winning strategy in a tree.
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The Robber and Cops game (cont.)
Example: 2 cops have a winning strategy in a tree.
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The Robber and Cops game (cont.)
Example: 2 cops have a winning strategy in a tree.
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The Robber and Cops game (cont.)
Example: 2 cops have a winning strategy in a tree.
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The Robber and Cops game (cont.)
Example: 2 cops have a winning strategy in a tree.
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The Robber and Cops game (cont.)
Example: 2 cops have a winning strategy in a tree.
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The Robber and Cops game (cont.)
Example: 2 cops have a winning strategy in a tree.
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The Robber and Cops game (cont.)
Example: 2 cops have a winning strategy in a tree.
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The Robber and Cops game
Fact:k + 1 cops can win the game iff the treewidth of the graph is atmost k .
The winner of the game can be determined in time nO(k) usingstandard techniques (there are at most nk positions for the cops)
⇓
For every fixed k, it can be checked in polynomial-time if treewidthis at most k.
Exercise 1: Show that the treewidth of the k × k grid is at leastk − 1.Exercise 2: Show that the treewidth of the k × k grid is at least k .
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The Robber and Cops game
Fact:k + 1 cops can win the game iff the treewidth of the graph is atmost k .
The winner of the game can be determined in time nO(k) usingstandard techniques (there are at most nk positions for the cops)
⇓
For every fixed k, it can be checked in polynomial-time if treewidthis at most k.
Exercise 1: Show that the treewidth of the k × k grid is at leastk − 1.Exercise 2: Show that the treewidth of the k × k grid is at least k .
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The Robber and Cops game
Fact:k + 1 cops can win the game iff the treewidth of the graph is atmost k .
The winner of the game can be determined in time nO(k) usingstandard techniques (there are at most nk positions for the cops)
⇓
For every fixed k, it can be checked in polynomial-time if treewidthis at most k.
Exercise 1: Show that the treewidth of the k × k grid is at leastk − 1.Exercise 2: Show that the treewidth of the k × k grid is at least k .
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Properties of treewidth
Fact: For every k ≥ 2, the treewidth of the k × k grid is exactly k .
Fact: Treewidth does not increase if we delete edges, deletevertices, or contract edges.
⇒ If F is a minor of G , then the treewidth of F is at most thetreewidth of G .
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Properties of treewidth
Fact: For every k ≥ 2, the treewidth of the k × k grid is exactly k .
Fact: Treewidth does not increase if we delete edges, deletevertices, or contract edges.
⇒ If F is a minor of G , then the treewidth of F is at most thetreewidth of G .
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Excluded Grid Theorem
Fact: [Excluded Grid Theorem] If the treewidth of G is at leastk4k2(k+2), then G has a k × k grid minor.
A large grid minor is a “witness” that treewidth is large.
Fact: Every planar graph with treewidth at least 4k has k × kgrid minor.
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Outerplanar graphs
Definition: A planar graph is outerplanar if it has a planarembedding where every vertex is on the infinite face.
Fact: Every outerplanar graph has treewidth at most 2.
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k-outerplanar graphs
Given a planar embedding, we can define layers by iterativelyremoving the vertices on the infinite face.
Definition: A planar graph is k-outerplanar if it has a planarembedding having at most k layers.
k-outerplanar graphs
Given a planar embedding, we can define layers by iteratively removing the vertices
on the infinite face.
Definition: A planar graph is k-outerplanar if it has a planar embedding having at
most k layers.
1
1 1
1
2
2
1
2
3
3
2
3
3
3
3
2
2
2
32
2
2
1
Fact: Every k-outerplanar graph has treewidth at most 3k + 1.Fixed Parameter Algorithms – p.33/48
Fact: Every k-outerplanar graph has treewidth at most 3k + 1.
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Part III:
Applications
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Baker’s shifting strategySubgraph Isomorphism for planar graphs: given planar graphs H and G , find acopy of H in G as subgraph. Parameter k := |V (H)|.
Layers of the planargraph: (as in the defini-tion of k-outerplanar):
I For a fixed 0 ≤ s < k + 1, delete every layer Li with i = s(mod k + 1)
I The resulting graph is k-outerplanar, hence it has treewidth at most3k + 1.
I Using the f (k,w) · n time algorithm for Subgraph Isomorphism,the problem can be solved in time f (k , 3k + 1) · n.
I We do this for every 0 ≤ s < k + 1: for at least one value of s, wedo not delete any of the k vertices of the solution ⇒ we find a copyof H in G if there is one.
I Subgraph Isomorphism for planar graphs is FPT parameterizedby k := |V (H)|.
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Baker’s shifting strategySubgraph Isomorphism for planar graphs: given planar graphs H and G , find acopy of H in G as subgraph. Parameter k := |V (H)|.
Layers of the planargraph: (as in the defini-tion of k-outerplanar):
I For a fixed 0 ≤ s < k + 1, delete every layer Li with i = s(mod k + 1)
I The resulting graph is k-outerplanar, hence it has treewidth at most3k + 1.
I Using the f (k,w) · n time algorithm for Subgraph Isomorphism,the problem can be solved in time f (k, 3k + 1) · n.
I We do this for every 0 ≤ s < k + 1: for at least one value of s, wedo not delete any of the k vertices of the solution ⇒ we find a copyof H in G if there is one.
I Subgraph Isomorphism for planar graphs is FPT parameterizedby k := |V (H)|.
![Page 85: P arameterized Algorithms II PAR T I St. Petersburg, 2011 · Graph Minors I Some consequences of the Graph Minors Theorem give a quick way of showing that certain problems are FPT.](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc9f29ad6a402d66675ddc/html5/thumbnails/85.jpg)
Baker’s shifting strategySubgraph Isomorphism for planar graphs: given planar graphs H and G , find acopy of H in G as subgraph. Parameter k := |V (H)|.
Layers of the planargraph: (as in the defini-tion of k-outerplanar):
I For a fixed 0 ≤ s < k + 1, delete every layer Li with i = s(mod k + 1)
I The resulting graph is k-outerplanar, hence it has treewidth at most3k + 1.
I Using the f (k ,w) · n time algorithm for Subgraph Isomorphism,the problem can be solved in time f (k , 3k + 1) · n.
I We do this for every 0 ≤ s < k + 1: for at least one value of s, wedo not delete any of the k vertices of the solution ⇒ we find a copyof H in G if there is one.
I Subgraph Isomorphism for planar graphs is FPT parameterizedby k := |V (H)|.
![Page 86: P arameterized Algorithms II PAR T I St. Petersburg, 2011 · Graph Minors I Some consequences of the Graph Minors Theorem give a quick way of showing that certain problems are FPT.](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc9f29ad6a402d66675ddc/html5/thumbnails/86.jpg)
Baker’s shifting strategySubgraph Isomorphism for planar graphs: given planar graphs H and G , find acopy of H in G as subgraph. Parameter k := |V (H)|.
Layers of the planargraph: (as in the defini-tion of k-outerplanar):
I For a fixed 0 ≤ s < k + 1, delete every layer Li with i = s(mod k + 1)
I The resulting graph is k-outerplanar, hence it has treewidth at most3k + 1.
I Using the f (k ,w) · n time algorithm for Subgraph Isomorphism,the problem can be solved in time f (k , 3k + 1) · n.
I We do this for every 0 ≤ s < k + 1: for at least one value of s, wedo not delete any of the k vertices of the solution ⇒ we find a copyof H in G if there is one.
I Subgraph Isomorphism for planar graphs is FPT parameterizedby k := |V (H)|.
![Page 87: P arameterized Algorithms II PAR T I St. Petersburg, 2011 · Graph Minors I Some consequences of the Graph Minors Theorem give a quick way of showing that certain problems are FPT.](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc9f29ad6a402d66675ddc/html5/thumbnails/87.jpg)
Baker’s shifting strategySubgraph Isomorphism for planar graphs: given planar graphs H and G , find acopy of H in G as subgraph. Parameter k := |V (H)|.
Layers of the planargraph: (as in the defini-tion of k-outerplanar):
I For a fixed 0 ≤ s < k + 1, delete every layer Li with i = s(mod k + 1)
I The resulting graph is k-outerplanar, hence it has treewidth at most3k + 1.
I Using the f (k ,w) · n time algorithm for Subgraph Isomorphism,the problem can be solved in time f (k , 3k + 1) · n.
I We do this for every 0 ≤ s < k + 1: for at least one value of s, wedo not delete any of the k vertices of the solution ⇒ we find a copyof H in G if there is one.
I Subgraph Isomorphism for planar graphs is FPT parameterizedby k := |V (H)|.
![Page 88: P arameterized Algorithms II PAR T I St. Petersburg, 2011 · Graph Minors I Some consequences of the Graph Minors Theorem give a quick way of showing that certain problems are FPT.](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc9f29ad6a402d66675ddc/html5/thumbnails/88.jpg)
Baker’s shifting strategySubgraph Isomorphism for planar graphs: given planar graphs H and G , find acopy of H in G as subgraph. Parameter k := |V (H)|.
Layers of the planargraph: (as in the defini-tion of k-outerplanar):
I For a fixed 0 ≤ s < k + 1, delete every layer Li with i = s(mod k + 1)
I The resulting graph is k-outerplanar, hence it has treewidth at most3k + 1.
I Using the f (k ,w) · n time algorithm for Subgraph Isomorphism,the problem can be solved in time f (k , 3k + 1) · n.
I We do this for every 0 ≤ s < k + 1: for at least one value of s, wedo not delete any of the k vertices of the solution ⇒ we find a copyof H in G if there is one.
I Subgraph Isomorphism for planar graphs is FPT parameterizedby k := |V (H)|.
![Page 89: P arameterized Algorithms II PAR T I St. Petersburg, 2011 · Graph Minors I Some consequences of the Graph Minors Theorem give a quick way of showing that certain problems are FPT.](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc9f29ad6a402d66675ddc/html5/thumbnails/89.jpg)
Baker’s shifting strategySubgraph Isomorphism for planar graphs: given planar graphs H and G , find acopy of H in G as subgraph. Parameter k := |V (H)|.
Layers of the planargraph: (as in the defini-tion of k-outerplanar):
I For a fixed 0 ≤ s < k + 1, delete every layer Li with i = s(mod k + 1)
I The resulting graph is k-outerplanar, hence it has treewidth at most3k + 1.
I Using the f (k ,w) · n time algorithm for Subgraph Isomorphism,the problem can be solved in time f (k , 3k + 1) · n.
I We do this for every 0 ≤ s < k + 1: for at least one value of s, wedo not delete any of the k vertices of the solution ⇒ we find a copyof H in G if there is one.
I Subgraph Isomorphism for planar graphs is FPT parameterizedby k := |V (H)|.
![Page 90: P arameterized Algorithms II PAR T I St. Petersburg, 2011 · Graph Minors I Some consequences of the Graph Minors Theorem give a quick way of showing that certain problems are FPT.](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc9f29ad6a402d66675ddc/html5/thumbnails/90.jpg)
Detour to approximation. . .
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Detour toapproximation algorithms
Definition: A c-approximation algorithm for a maximizationproblem is a polynomial-time algorithm that finds a solution ofcost at least OPT/c.
Definition: A c-approximation algorithm for a minimizationproblem is a polynomial-time algorithm that finds a solution ofcost at most OPT · c .
There are well-known approximation algorithms for NP-hardproblems: 3
2 -approximation for Metric TSP, 2-approximation forVertex Cover, 8
7 -approximation for Max 3SAT, etc.
I For some problems, we have lower bounds: there is no(2− ε)-approximation for Vertex Cover or(87 − ε)-approximation for Max 3SAT (under suitable
complexity assumptions).I For some other problems, arbitrarily good approximation is
possible in polynomial time: for any c > 1 (say,c = 1.000001), there is a polynomial-time c-approximationalgorithm!
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Approximation schemes
Definition: A polynomial-time approximation scheme (PTAS)for a problem P is an algorithm that takes an instance of P and arational number ε > 0,
I always finds a (1 + ε)-approximate solution,
I the running time is polynomial in n for every fixed ε > 0.
Typical running times: 21/ε · n, n1/ε, (n/ε)2, n1/ε2 .
Some classical problems that have a PTAS:
I Independent Set for planar graphs
I TSP in the Euclidean plane
I Steiner Tree in planar graphs
I Knapsack
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Baker’s shifting strategy for EPTAS
Fact: There is a 2O(1/ε) · n time PTAS for Independent Set forplanar graphs.
I Let D := 1/ε. For a fixed 0 ≤ s < D, delete every layer Li
with i = s (mod D)
I The resulting graph is D-outerplanar, hence it has treewidthat most 3D + 1 = O(1/ε).
I Using the O(2w · n) time algorithm for Independent Set,the problem can be solved in time 2O(1/ε) · n.
I We do this for every 0 ≤ s < D: for at least one value of s,we delete at most 1/D = ε fraction of the solution ⇒ we geta (1 + ε)-approximate solution.
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Back to FPT. . .
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Bidimensionality
A powerful framework to obtain efficient algorithms on planargraphs.
Let x(G ) be some graph invariant (i.e., an integer associated witheach graph).Some typical examples:
I Maximum independent set size.
I Minimum vertex cover size.
I Length of the longest path.
I Minimum dominating set size
I Minimum feedback vertex set size.
Given G and k, we want to decide if x(G ) ≤ k (or x(G ) ≥ k).
For many natural invariants, we can do this in time 2O(√
k) · nO(1).
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Bidimensionality for Vertex Cover
Hr ,r for r = 10
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Bidimensionality for Vertex Cover
vc(Hr ,r ) ≥ r2
2
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Bidimensionality forVertex Cover
Observation: If the treewidth of a planar graph G is at least 4√
2k⇒ It contains a
√2k ×
√2k grid minor (Excluded Grid Theorem
for planar graphs)⇒ The vertex cover size of the grid is at least k in the grid⇒ Vertex cover size is at least k in G .
We use this observation to solve Vertex Cover on planargraphs as follows:
I Set w := 4√
2k .I Use the 4-approximation tree decomposition algorithm
(2O(w) · nO(1) = 2O(√
k) · nO(1) time).I If treewidth is at least w : we answer ’vertex cover is ≥ k’.I If we get a tree decomposition of width 4w , then we can solve
the problem in time 2w · nO(1) = 2O(√
k) · nO(1).
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Bidimensionality (cont.)
Definition: A graph invariant x(G ) is minor-bidimensional if
I x(G ′) ≤ x(G ) for every minor G ′ of G , and
I If Gk is the k × k grid, then x(Gk) ≥ ck2 (for some constantc > 0).
Examples: minimum vertex cover, length of the longest path,feedback vertex set are minor-bidimensional.
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Bidimensionality (cont.)
Definition: A graph invariant x(G ) is minor-bidimensional if
I x(G ′) ≤ x(G ) for every minor G ′ of G , and
I If Gk is the k × k grid, then x(Gk) ≥ ck2 (for some constantc > 0).
Examples: minimum vertex cover, length of the longest path,feedback vertex set are minor-bidimensional.
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Bidimensionality (cont.)
Definition: A graph invariant x(G ) is minor-bidimensional if
I x(G ′) ≤ x(G ) for every minor G ′ of G , and
I If Gk is the k × k grid, then x(Gk) ≥ ck2 (for some constantc > 0).
Examples: minimum vertex cover, length of the longest path,feedback vertex set are minor-bidimensional.
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Bidimensionality (cont.)
We can answer “x(G ) ≥ k?” for a minor-bidimensional parameterthe following way:
I Set w := c√
k for an appropriate constant c .I Use the 4-approximation tree decomposition algorithm.
I If treewidth is at least w : x(G ) is at least k.I If we get a tree decomposition of width 4w , then we can solve
the problem using dynamic programming on the treedecomposition.
Running time:
I If we can solve the problem using a width w treedecomposition in time 2O(w) · nO(1), then the running time is
2O(√
k) · nO(1).
I If we can solve the problem using a width w treedecomposition in time wO(w) · nO(1), then the running time is
2O(√
k log k) · nO(1).
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Summary
I Notion of treewidth allows us to generalize dynamicprogramming on trees to more general graphs.
I Standard techniques for designing algorithms on boundedtreewidth graphs: dynamic programming and Courcelle’sTheorem.
I Surprising uses of treewidth in other contexts (planar graphs).