Graph Minors, Bidimensionality and Algorithmsphdopen.mimuw.edu.pl/zima10/fomin3.pdf · Graph...
Transcript of Graph Minors, Bidimensionality and Algorithmsphdopen.mimuw.edu.pl/zima10/fomin3.pdf · Graph...
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PART IIIWarsaw, 2011
FEDOR V. FOMIN
Graph Minors, Bidimensionality and Algorithms
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Reminder
Back to the YEAR 2010...
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Branch Decompositions
Definition
A branch decomposition of a graph G = (V,E) is a tuple (T, µ)
where
I T is a tree with degree 3 for all internal nodes.
I µ is a bijection between the leaves of T and E(G).
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Example of Branch Decomposition
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Edge e ∈ T partitions the edge set of G in Ae and Be
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Edge e ∈ T partitions the edge set of G in Ae and Be
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4 6
1 3a b
f g
c d m
h i j k
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j
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Middle set mid(e) = V (Ae) ∩V (Be)
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Branchwidth
I The width of a branch decomposition is maxe∈T |mid(e)|.
I The branchwidth of a graph G is the minimum width over all
branch decompositions of G.
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Grid Theorem
Theorem (Robertson, Seymour & Thomas, 1994)
Let ` ≥ 1 be an integer. Every planar graph of branchwidth ≥ 4`
contains ` as a minor.
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Subexponential algorithms on planar graphs: What is the
main idea?
Dynamic programming andGrid Theorem
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Meta conditions
(A) For every graph G ∈ G, bw(G) ≤ α ·√P (G) +O(1)
(B) For every graph G ∈ G and given a branch decomposition
(T, µ) of G, the value of P (G) can be computed in
f(bw(T, µ)) · nO(1) steps.
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Algorithm
(A) For every graph G ∈ G, bw(G) ≤ α ·√P (G) +O(1)
(B) For every graph G ∈ G and given a branch decomposition
(T, µ) of G, the value of P (G) can be computed in
f(bw(T, µ)) · nO(1) steps.
If bw(T, µ) > α ·√k, then by (A) the answer is clear
Else, by (B), P (G) can be computed in f(α ·√k) · nO(1) steps.
When f(k) = 2O(k), the running time is 2O(√k) · nO(1)
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Bidimensionality: The main idea
If the graph parameter is closed under taking minors or
contractions, the only thing needed for the proof
branchwidth/parameter bound is to understand how this parameter
behaves on a (partially triangulated) grid.
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Bidimensionality: Demaine, FF, Hajiaghayi, Thilikos, 2005
Definition
A parameter P is minor bidimensional with density δ if
1. P is closed under taking of minors, and
2. for the (r × r)-grid R, P (R) = (δr)2 + o((δr)2).
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Bidimensionality: Demaine, FF, Hajiaghayi, Thilikos, 2005
Definition
A parameter P is called contraction bidimensional with density δ if
1. P is closed under contractions,
2. for any partially triangulated (r × r)-grid R,
P (R) = (δRr)2 + o((δRr)2), and
3. δ is the smallest δR among all paritally triangulated
(r × r)-grids.
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Bidimensionality
(A) For every graph G ∈ G, bw(G) ≤ α ·√P (G) +O(1)
Lemma
If P is a bidimensional parameter with density δ then P satisfies
property (A) for α = 4/δ, on planar graphs.
Proof.
Let R be an (r × r)-grid.
P (R) ≥ (δRr)2.
If G contains R as a minor, then bw(G) ≤ 4r ≤ 4/δ√P (G).
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Examples of bidimensional problems
Vertex cover
Dominating Set
Independent Set
(k, r)-center
Feedback Vertex Set
Minimum Maximal Matching
Planar Graph TSP
Longest Path ...
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Bidimensional theory
If P is a parameter that
(A) is minor (contraction) bidimensional
(B) can be computed in f(bw(G)) · nO(1) steps.
then there is a f(O(√k)) · nO(1) step algorithm for checking
whether P(G) ≤ k for H (apex) -minor free graphs.
We now fix our attention to property (B) and function f .
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PART III
Dynamic programming and Catalan
structures
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Dynamic programming for branch decompositions
I We root the tree T of the branch decomposition (T, τ),
I We define a partial solution for each cut-set of an edge e of T
I We compute all partial solutions bottom-up (using the partial
solutions corresponding to the children edges).
This can be done in O(f(`) · n) if we have a branch decomposition
of width at most `.
f(`) depends on the number of partial solutions we have to
compute for each edge of T .
I To find a good bound for f(`) is important!
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Dynamic programming for branch decompositions
I We root the tree T of the branch decomposition (T, τ),
I We define a partial solution for each cut-set of an edge e of T
I We compute all partial solutions bottom-up (using the partial
solutions corresponding to the children edges).
This can be done in O(f(`) · n) if we have a branch decomposition
of width at most `.
f(`) depends on the number of partial solutions we have to
compute for each edge of T .
I To find a good bound for f(`) is important!
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For many problems, 2O(bw(G)) · nO(1) step algorithms exist.
Dynamic programming on graphs with small branchwidth gives
such algorithms for problems like
Vertex Cover,
Dominating Set, or
Edge Dominating Set, (and others...)
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However: There are (many) problems where no general
2O(bw(G)) · nO(1) step algorithm is known.
Such problems are
Longest Path, Longest Cycle, Connected Dominating
Set, Feedback Vertex Set, Hamiltonian Cycle, Max
Leaf Tree and Graph Metric TSP
For the natural parameterizations of these problems, no 2O(√
k) · nO(1)
step FPT-algorithm follows by just using bidimensionality theory and
dynamic programming.
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However: There are (many) problems where no general
2O(bw(G)) · nO(1) step algorithm is known.
Such problems are
Longest Path, Longest Cycle, Connected Dominating
Set, Feedback Vertex Set, Hamiltonian Cycle, Max
Leaf Tree and Graph Metric TSP
For the natural parameterizations of these problems, no 2O(√
k) · nO(1)
step FPT-algorithm follows by just using bidimensionality theory and
dynamic programming.
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However: There are (many) problems where no general
2O(bw(G)) · nO(1) step algorithm is known.
Such problems are
Longest Path, Longest Cycle, Connected Dominating
Set, Feedback Vertex Set, Hamiltonian Cycle, Max
Leaf Tree and Graph Metric TSP
For the natural parameterizations of these problems, no 2O(√
k) · nO(1)
step FPT-algorithm follows by just using bidimensionality theory and
dynamic programming.
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Example: k- Path
The k-Path problem is to decide, given a graph G and a
positive integer k, whether G contains a path of length k.
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k- Path
On general graphs k-Path can be solved Color Coding technique
in time O((2e)kn) [Alon, Yuster, Zwick, 1995].
A chain of improvements up to recent O∗(1.66k) algorithm of
[Bjorklund, Husfeldt, Kaski, Koivisto, 2010]
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k-Path
On planar graphs k-Path has a
2O(√k·log k) · nO(1) step algorithm.
Because
(A) The parameter is minor bidimensional
(B) to find a longest path in a graph G takes
2O(bw(G)·log bw(G)) · n steps
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k-Path
On planar graphs k-Path has a
2O(√k·log k) · nO(1) step algorithm.
Because
(A) The parameter is minor bidimensional
(B) to find a longest path in a graph G takes
2O(bw(G)·log bw(G)) · n steps
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k-Path
On planar graphs k-Path has a
2O(√k·log k) · nO(1) step algorithm.
Because
(A) The parameter is minor bidimensional
(B) to find a longest path in a graph G takes
2O(bw(G)·log bw(G)) · n steps
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Dynamic programming: k-Path
Left Right
Left Right
Remaining
Remaining
e
f
gmid(e)
mid(g)
mid(f)
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Dynamic programming: k-Path
Left Right
Remainingmid(e)
mid(g)
mid(f)
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k-Path
Fact: Given a branch decomposition of width w, k-Path can be
solved in time wO(w) · n.
Ge: subgraph formed by edges of the subtree rooted at e.
If P is a path, then the subgraph
Pe = Ge ∩ P is a set of paths with
endpoints in mid(e).
mid(e)
B^0
B^1
B^2
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k-Path
What are the important properties of Pe“seen from the outside
world”?
I The subsets B0e , B1
e , B2e of
mid(e) having degree 0, 1, and
2.
I Disjoint pairs pairs(e) of B1e .
mid(e)
B^0
B^1
B^2
Number of subproblems (B0e , B
1e , B
2e , pairs(e)) for each edge e:
at most 3w · ww.
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The running time of dynamic programming is proportional to the
amount of all possible subproblems (B0e , B
1e , B
2e , pairs(e)), which
is 3w · ww = 2O(w logw).
I Issue: The same problem appears in many dynamic
programming algorithms
I Idea: as long as we care about sparse graph classes, we can take
their structure into consideration
I We want to show that on planar graphs, the amount of pairs
pairs(e) in mid(e) is 2O(w)
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The running time of dynamic programming is proportional to the
amount of all possible subproblems (B0e , B
1e , B
2e , pairs(e)), which
is 3w · ww = 2O(w logw).
I Issue: The same problem appears in many dynamic
programming algorithms
I Idea: as long as we care about sparse graph classes, we can take
their structure into consideration
I We want to show that on planar graphs, the amount of pairs
pairs(e) in mid(e) is 2O(w)
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The running time of dynamic programming is proportional to the
amount of all possible subproblems (B0e , B
1e , B
2e , pairs(e)), which
is 3w · ww = 2O(w logw).
I Issue: The same problem appears in many dynamic
programming algorithms
I Idea: as long as we care about sparse graph classes, we can take
their structure into consideration
I We want to show that on planar graphs, the amount of pairs
pairs(e) in mid(e) is 2O(w)
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Sphere-cut decomposition
Let G be a planar graph embedded on the sphere (or a plane) S0
A sphere-cut decomposition of G is a branch decomposition (T, τ)
where for every e ∈ E(T ), the vertices in mid(e) are the vertices in
a Jordan curve of S0 that meets no edges of G (a noose).
j
l
b m
k
g2, 5, 7
c
a d
fh
ie
c
8
4 6
1 3a b
f g
c d m
h i j k
l
5
7
2
Oe
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Seymour-Thomas 1994, Dorn-Penninkx-Bodlaender-FF
2005
Theorem
Every planar graph G of branchwidth ` has a sphere-cut
decomposition of width `. This decomposition can be constructed
in O(n3) steps.
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Proof ideas
I Carving-width
I Connectivity properties of carving-width decompositions
I Branch-width as the carving width of a radial graph
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Bounding (B0e , B
1e , B
2e , pairs(e))
We now have that
1: the vertices of mid(e) lay on the boundary of a disk and
2: the pairings pairs(e)) cannot be crossing because of planarity.
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It follows that pairs(e) = O(C(|mid(e)|)) = O(C(w))
Where C(w) is the w-th Catalan Number.
It is known that C(w) ∼ 4w
w3/2√π
= 2O(w)
We conclude that for planar graphs, there is an optimal branch
decomposition of width w such that for every edge e, the amount
of tuples (B0e , B
1e , B
2e , pairs(e)) is 2O(w)
Therefore: dynamic programming for finding a k-path of a planar
graph G on a sphere cut decompositions of G with width ≤ w
takes O(2O(w) · n) steps.
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It follows that pairs(e) = O(C(|mid(e)|)) = O(C(w))
Where C(w) is the w-th Catalan Number.
It is known that C(w) ∼ 4w
w3/2√π
= 2O(w)
We conclude that for planar graphs, there is an optimal branch
decomposition of width w such that for every edge e, the amount
of tuples (B0e , B
1e , B
2e , pairs(e)) is 2O(w)
Therefore: dynamic programming for finding a k-path of a planar
graph G on a sphere cut decompositions of G with width ≤ w
takes O(2O(w) · n) steps.
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It follows that pairs(e) = O(C(|mid(e)|)) = O(C(w))
Where C(w) is the w-th Catalan Number.
It is known that C(w) ∼ 4w
w3/2√π
= 2O(w)
We conclude that for planar graphs, there is an optimal branch
decomposition of width w such that for every edge e, the amount
of tuples (B0e , B
1e , B
2e , pairs(e)) is 2O(w)
Therefore: dynamic programming for finding a k-path of a planar
graph G on a sphere cut decompositions of G with width ≤ w
takes O(2O(w) · n) steps.
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Conclusion:
Planar k-Path can be solved in O(2O(√k) · nnO(1)) steps
Algorithm:
I Compute branch-width of G in polynomial time. If
bw(G) ≥ 4√k, then G contains
√k ×√k gris as a minor.
Thus it also contains a path of length k
I If bw(G) ≤ 4√k, use sphere-cut decomposition to find in
time 2O(√k)n if G has a path of length k.
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I Similar results hold for several other problems where an analogue of
pairs(e) can be defined for controlling the size of the tables in dynamic
programming.
I Like that one can design 2O(√
k) · nO(1) step algorithms for
parameterized planar versions of Cycle Cover, Path Cover,
Longest Cycle, Connected Dominating Set, Feedback
Vertex Set, Graph Metric TSP, Max Leaf Tree, etc.
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I Similar results hold for several other problems where an analogue of
pairs(e) can be defined for controlling the size of the tables in dynamic
programming.
I Like that one can design 2O(√
k) · nO(1) step algorithms for
parameterized planar versions of Cycle Cover, Path Cover,
Longest Cycle, Connected Dominating Set, Feedback
Vertex Set, Graph Metric TSP, Max Leaf Tree, etc.
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Planarity not a limit
I For H-minor free graphs, one can construct an algorithm that
solves the k-Path problem in 2O(√k) · nO(1) steps.
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More grids
Grids for other problems or walls
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t-spanners
Definition (t-spanner)
Let t be a positive integer. A subgraph S of G, such that
V (S) = V (G), is called a t-spanner, if distS(u, v) ≤ t · distG(u, v)
for every pair of vertices u and v. The parameter t is called the
stretch factor of S.
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Examples of spanners
3 and 2-spanners
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Examples of spanners
3 and 2-spanners
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Examples of spanners
3 and 2-spanners
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Spanners of bounded branchwidth
Problem (k-Branchwidth t-spanner)
Instance: A connected graph G and positive integers k and t.
Question: Is there a t-spanner of G of branchwidth at most k?
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Topological minors
Contracting edge can increase t
bounded genus graphs).
Finally, let us briefly mention the relevant work on parameterized al-gorithms on planar graphs and more general classes of graphs (we refer tobooks [21, 32, 43] for more information on parameterized complexity and al-gorithms). Alber et al. [2] initiated the study of subexponential parameter-ized algorithms for the dominating set problem and its di!erent variations.Demaine et al. [14, 15] gave a general framework called bidimensionality todesign parameterized algorithms for many problems on planar graphs andshowed how by making use of this framework to extend results from planargraphs to much more general graph classes including H-minor-free graphs.We refer to surveys [16, 20] for an overview of results and techniques relatedto bidimensionality (see also [33]). However, this framework cannot be useddirectly to solve the k-Treewidth t-spanner problem because the theoryof Demaine et al. is strongly based on the assumption that the parameter-ized problem should be minor or edge contraction closed, which is not thecase for spanners. In particular, it is easy to construct an example when bycontracting of an edge in a graph G with a t-spanner of treewidth k, onecan obtain a graph which does not have such a spanner (see Figure 1).
x y
Figure 1: Examples of graphs with tree 3-spanner and without such span-ners. The second graph is obtained from the first by contraction of (x, y).
Our results and organization of the paper. In this paper we resolvethe problem left open in [31] and extend solutions in several directions. Ourgeneral technique is combinatorial in nature and is based on the followingobservation. Let G be a class of graphs such that for every fixed t and everyG ! G, the treewidth of every t-spanner of G is "(treewidth(G)). Thenas an almost direct corollary of Bodlaender’s Algorithm and Courcelle’sTheorem (see Section 5 for details), we have that the k-Treewidth t-spanner problem is fixed parameter tractable on G. Our main combinatorialresult is the proof that the class of apex-minor-free graphs, which containsplanar and bounded genus graphs, is in G.
After preliminary Section 2, we start (Section 3) by proving the combi-natorial properties of t-spanners in planar graphs. Our main result here isthe proof that every t-spanner of a planar graph of treewidth k has treewidth"(k/t). The proof idea is based on a theorem due to Robertson, Seymour,
4
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Planar graphs
Theorem (Bounds for planar graphs)
Let G be a planar graph of branchwidth k and let S be a t-spanner
of G. Then the branchwidth of S is Ω(k/t).
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Walls
Bidimensionality for minors and contractions The irrelevant vertex technique
Walls and their subdivisions
Therefore, if bw(G) = !(k5/2), then!(k5/2)
!tp G.
This can be seen like a wall of height !(k5/2).
Dimitrios M. Thilikos !"#$-NKUA
Algorithmic Graph Minor Theory Part 2 89
A wall of height k.
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Walls
Observation: If a graph contains a k × k grid as a minor, it
contains a wall of height k as a topological subgraph.
Bidimensionality for minors and contractions The irrelevant vertex technique
Walls and their subdivisions
Therefore, if bw(G) = !(k5/2), then!(k5/2)
!tp G.
Which can be seen as a subgraph that is a subdivided wall.
Dimitrios M. Thilikos !"#$-NKUA
Algorithmic Graph Minor Theory Part 2 90
Which can be seen as a subgraph that is a subdivided wall.
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Sketch of the proof
Walls and grids
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Sketch of the proof
Walls and grids
t
t
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Sketch of the proof
Walls and grids
t
t
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Sketch of the proof
Walls and grids
t
t
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Sketch of the proof
Walls and grids
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Sketch of the proof
Walls and grids
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Sketch of the proof
Walls and grids
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Sketch of the proof
Walls and grids
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Algorithmic consequences
Theorem (Dragan, FF, Golovach, 2008)
Deciding if a planar graph G has a t-spanner of treewidth at most
k is solvable in time O(f(k, t) · nO(1)).
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Disjoint Path Problem
p-Disjoint Paths Problem
Instance: a graph G and terminals (si, ti) ∈ V (G)2, i = 1, . . . , k.
Parameter: k
Question: Does G have k disjoint paths between si and ti, i = 1, . . . , k?
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Disjoint Path Problem
I To solve problem on planar graphs: If the branch-width is
small, do DP
I If the branch-widht is large, there is a wall of large height
I If there is a wall of large height, there is a wall of large height
containing no terminal vertices
I A vertex in the center of this wall is irrelevant. i.e. its removal
does not change the problem
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Disjoint Path Problem
I To solve problem on planar graphs: If the branch-width is
small, do DP
I If the branch-widht is large, there is a wall of large height
I If there is a wall of large height, there is a wall of large height
containing no terminal vertices
I A vertex in the center of this wall is irrelevant. i.e. its removal
does not change the problem
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Disjoint Path Problem
I To solve problem on planar graphs: If the branch-width is
small, do DP
I If the branch-widht is large, there is a wall of large height
I If there is a wall of large height, there is a wall of large height
containing no terminal vertices
I A vertex in the center of this wall is irrelevant. i.e. its removal
does not change the problem
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Disjoint Path Problem
I To solve problem on planar graphs: If the branch-width is
small, do DP
I If the branch-widht is large, there is a wall of large height
I If there is a wall of large height, there is a wall of large height
containing no terminal vertices
I A vertex in the center of this wall is irrelevant. i.e. its removal
does not change the problem
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Conclusion
I WIN/WIN approach via Excluding Grid Theorem
I Dynamic programming and Catalan structures
I Working on planar problems don’t afraid to hit the wall!
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Further reading. Subexponential algorithms and
bidimensionality
J. Alber, H. L. Bodlaender, H. Fernau, T. Kloks,
and R. Niedermeier, Fixed parameter algorithms for
dominating set and related problems on planar graphs,
Algorithmica, 33 (2002), pp. 461–493.
E. D. Demaine, F. V. Fomin, M. Hajiaghayi, and
D. M. Thilikos, Subexponential parameterized algorithms
on graphs of bounded genus and H-minor-free graphs, Journal
of the ACM, 52 (2005), pp. 866–893.
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Further reading. Catalan structures and dynamic
programming
F. Dorn, E. Penninkx, H. Bodlaender, and F. V.
Fomin, Efficient exact algorithms on planar graphs: Exploiting
sphere cut branch decompositions, Proceedings of the 13th
Annual European Symposium on Algorithms (ESA 2005),
vol. 3669 of LNCS, Springer, 2005, pp. 95–106.
F. Dorn, F. V. Fomin, and D. M. Thilikos, Catalan
structures and dynamic programming on H-minor-free graphs,
in Proceedings of the 19th Annual ACM-SIAM Symposium on
Discrete Algorithms (SODA 2008), ACM-SIAM, pp. 631–640.
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Further reading. Surveys
E. Demaine and M. Hajiaghayi, The bidimensionality
theory and its algorithmic applications, The Computer Journal,
(2007), pp. 332–337.
F. Dorn, F. V. Fomin, and D. M. Thilikos,
Subexponential parameterized algorithms, Computer Science
Review, 2 (2008), pp. 29–39.