Introduction
First result
Conclusion 1
Second result
Conclusion 2
FPT results through potential maximal cliques
Pedro Montealegre
in collaboration with
Fedor V. Fomin, Mathieu Liedloff, Ioan Todinca
Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, Orléans , France
JGA 2015, Orléans, France
November 4, 2015
1 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Minimal separators
Definition
S ⊆ V is a minimal
a, b-separator if S separates a
and b and it is inclusion-minimal
for this property.
Definition
S is a minimal separator of G
there exist vertices a, b s.t. S is a
minimal a, b-separator.
2 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Minimal separators
Definition
S ⊆ V is a minimal
a, b-separator if S separates a
and b and it is inclusion-minimal
for this property.
Definition
S is a minimal separator of G
there exist vertices a, b s.t. S is a
minimal a, b-separator.
2 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Potential maximal cliques
Definition
A set of vertices Ω is a potential maximal clique of G if is a
maximal clique in some minimal triangulation of G .
Proposition (Bouchitté, Todinca 2001)
The number of potential maximal cliques is polynomial in the
number of minimal separators.
3 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Minimal separators and Gpoly
Definition
For a polynomial poly , let Gpoly be the class of graphs such that
G ∈ Gpoly if G has at most poly (n) minimal separators.
Class
Weakly chordal
Chordal
Polygon-circle
Circle
Circular arc
d-trapezoid
minimal separators
O(n2 )
O(n)
O(n2 )
O(n2 )
O(n2 )
O(nd )
potential maximal cliques
O(n2 )
O(n)
O(n3 )
O(n3 )
O(n3 )
O(nd+2 )
4 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
When the input graph belongs to Gpoly we can find in polynomial
time a Maximum Induced:
• Independent Set, Forest, Path, Matching
• Subgraph With no cycles ≥ l, Outerplanar
• Subgraph With no cycles of length 0 mod l
5 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
When the input graph belongs to Gpoly we can find in polynomial
time a Maximum Induced:
• Independent Set, Forest, Path, Matching
• Subgraph With no cycles ≥ l, Outerplanar
• Subgraph With no cycles of length 0 mod l
For t ≥ 0 and P a CMSO property:
Optimal Induced Subgraph for P and t
Input: A graph G = (V , E )
Output A largest induced subgraph G [F ] s.t.
- tw (G [F ]) ≤ t,
- G [F ] satisfies P.
5 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
When the input graph belongs to Gpoly we can find in polynomial
time a Maximum Induced:
• Independent Set, Forest, Path, Matching
• Subgraph With no cycles ≥ l, Outerplanar
• Subgraph With no cycles of length 0 mod l
For t ≥ 0 and P a CMSO property:
Optimal Induced Subgraph for P and t
Input: A graph G = (V , E )
Output A largest induced subgraph G [F ] s.t.
- tw (G [F ]) ≤ t,
- G [F ] satisfies P.
Theorem (Fomin, Todinca, Villanger, 2015)
Optimal Induced Subgraph for P and t on Gpoly is
solvable in polynomial time.
5 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
When the input graph belongs to Gpoly we can find in polynomial
time a Maximum Induced:
• Independent Set, Forest, Path, Matching
• Subgraph With no cycles ≥ l, Outerplanar
• Subgraph With no cycles of length 0 mod l
For t ≥ 0 and P a CMSO property:
Optimal Induced Subgraph for P and t
Input: A graph G = (V , E )
Output A largest induced subgraph G [F ] s.t.
- tw (G [F ]) ≤ t,
- G [F ] satisfies P.
Theorem (Fomin, Todinca, Villanger, 2015)
Optimal Induced Subgraph for P and t is solvable time
O(#Potential maximal cliques · nt+cst · f (P, t)).
5 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
In this talk, two extensions:
1) Theorem (Fomin, Liedloff, M., Todinca. SWAT 2014)
Optimal Induced Subgraph for P and t can be solved in
time O∗ (4vc ) and O∗ (1.7347mw ).
2) Definition (Gpoly + kv )
G ∈ Gpoly + kv if there exists a set M ⊂ V called modulator,
|M| ≤ k s.t. G − M ∈ Gpoly .
Theorem (Liedloff, M. and Todinca. WG 2015)
Optimal Induced Subgraph for P and t on Gpoly + kv
with parameter k is fixed-parameter tractable, when the modulator
is also part of the input.
6 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
First result
Theorem (Fomin, Todinca, Villanger, 2015)
Optimal Induced Subgraph for P and t is solvable time
O(#Potential maximal cliques · nt+cst · f (P, t)).
Our result:
Theorem
The number of potential maximal cliques is bounded by
• O ∗ (4vc )
• O ∗ (1.7347mw )
7 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Vertex cover
Definition
The vertex cover of a graph G , denoted by vc(G ), is the minimum
number of vertices that cover all edges of the graph.
• The number of minimal separators is bounded by 3vc
• The number of potential maximal cliques is bounded by
O∗ (4vc )
8 / 28
Introduction
First result
Conclusion 1
Second result
Vertex cover and minimal separators
G
Conclusion 2
Introduction
First result
Conclusion 1
Second result
Vertex cover and minimal separators
W
G
Conclusion 2
Introduction
First result
Conclusion 1
Second result
Vertex cover and minimal separators
S
W
G
Conclusion 2
Introduction
First result
Conclusion 1
Second result
Vertex cover and minimal separators
S
D1
D2
W
G
Conclusion 2
Introduction
First result
Conclusion 1
Second result
Vertex cover and minimal separators
S
D1
D2
W
G
Conclusion 2
Introduction
First result
Conclusion 1
Second result
Vertex cover and minimal separators
S
D1
D2
W
G
Conclusion 2
Introduction
First result
Conclusion 1
Second result
Vertex cover and minimal separators
S
D1
D2
W
G
Conclusion 2
Introduction
First result
Conclusion 1
Second result
Vertex cover and minimal separators
S
D1
D2
W
G
Conclusion 2
Introduction
First result
Conclusion 1
Second result
Vertex cover and minimal separators
S
D1
D2
W
G
Conclusion 2
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Vertex cover and minimal separators
S
D1
D2
W
G
9 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Vertex cover and minimal separators
For any vertex cover W
S → (S W , D1 , D2 ) = W
S = S W ∪ {x ∈ V \ W | N(x) intersects both D1 and D2 }
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Introduction
First result
Conclusion 1
Second result
Conclusion 2
Vertex cover and minimal separators
For any vertex cover W
S → (S W , D1 , D2 ) = W
S = S W ∪ {x ∈ V \ W | N(x) intersects both D1 and D2 }
Theorem
• Number of minimal separators is O(3vc )
• They can be listed in time O ∗ (3vc )
10 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Potential maximal cliques and minimal separators
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Potential maximal cliques and minimal separators
Ω
G
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Potential maximal cliques and minimal separators
C
S
G \(C ∪ Ω)
y
Ω
x
G
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Potential maximal cliques and minimal separators
C
S
G \(C ∪ Ω)
y
Ω
x
C ∪ {x, y }
G
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Potential maximal cliques and minimal separators
S
C
G \(C ∪ Ω)
y
T
Ω
x
C ∪ {x, y }
G
11 / 28
Introduction
First result
Conclusion 1
Second result
Vertex cover and pmc
G
Conclusion 2
Introduction
First result
Conclusion 1
Second result
Vertex cover and pmc
W
G
Conclusion 2
Introduction
First result
Conclusion 1
Second result
Vertex cover and pmc
Ω
W
G
Conclusion 2
Introduction
First result
Conclusion 1
Second result
Vertex cover and pmc
DU
Ω
DL
DR
W
G
Conclusion 2
Introduction
First result
Conclusion 1
Second result
Vertex cover and pmc
DU
Ω
DL
DR
W
G
Conclusion 2
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Vertex cover and pmc
DU
Ω
DL
DR
W
G
12 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Vertex cover and pmc
For any vertex cover W
Ω → (ΩW , DR , DU , DL ) = W
Ω = ΩW
∪{x ∈ V \ W | N(x) intersects both DR and (DU ∪ DL )}
∪{x ∈ V \ W | N(x) intersects both DU and DL }
13 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Vertex cover and pmc
For any vertex cover W
Ω → (ΩW , DR , DU , DL ) = W
Ω = ΩW
∪{x ∈ V \ W | N(x) intersects both DR and (DU ∪ DL )}
∪{x ∈ V \ W | N(x) intersects both DU and DL }
Theorem
• Number of potential maximal cliques is O ∗ (4vc )
• They can be listed in time O ∗ (4vc )
13 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Modular width
Definition
The modular width mw(G ) can be defined as the maximum degree
of a prime node in the modular decomposition tree of G
31
11
21
41
1
2
3
5
6
4
32
51
61
52
53
• The number of minimal separators is bounded by
O∗ (1.6181mw ),
• The number of potential maximal cliques is bounded by
O∗ (1.7347mw ).
14 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Modular width and minimal separators
H
G
31
11
21
41
1
2
3
5
6
32
51
61
52
53
4
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Modular width and minimal separators
H
G
31
11
21
41
1
2
3
5
6
4
32
51
61
52
53
15 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Modular width and minimal separators
H
G
31
21
11
41
1
2
3
5
6
4
32
61
51
52
53
15 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Modular width and minimal separators
H
G
31
21
11
41
1
2
3
5
6
4
32
61
51
52
53
Theorem (Fomin, Villanger (2010))
Every n-vertex graph has O(1.6181n ) minimal separators and
O(1.7347n ) potential maximal cliques.
15 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Conclusion
Theorem
Optimal Induced Subgraph for P and t can be solved in
time O∗ (4vc ) and O∗ (1.7347mw )
16 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Conclusion
Theorem
Optimal Induced Subgraph for P and t can be solved in
time O∗ (4vc ) and O∗ (1.7347mw )
Also..
• (Weighted) Treewidth,
• (Weighted) Minimum fill in,
• Treelength.
Are solvable in time O∗ (# pmc) ([Gysel], [Lokshtanov],..)
16 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Conclusion
Theorem
Optimal Induced Subgraph for P and t can be solved in
time O∗ (4vc ) and O∗ (1.7347mw )
Also..
• (Weighted) Treewidth,
• (Weighted) Minimum fill in,
• Treelength.
Are solvable in time O∗ (# pmc) ([Gysel], [Lokshtanov],..)
(then in time O∗ (4vc ) and O∗ (1.7347mw )).
16 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Conclusion
Theorem
Optimal Induced Subgraph for P and t can be solved in
time O∗ (4vc ) and O∗ (1.7347mw )
Also..
• (Weighted) Treewidth,
• (Weighted) Minimum fill in,
• Treelength.
Are solvable in time O∗ (# pmc) ([Gysel], [Lokshtanov],..)
(then in time O∗ (4vc ) and O∗ (1.7347mw )).
• Running times that are single exponential in the parameter.
• This result covers both sparse and dense families of graphs.
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Introduction
First result
Conclusion 1
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Conclusion 2
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Introduction
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Conclusion 1
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Introduction
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Introduction
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3O(n) minimal separators
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Introduction
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Conclusion 1
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Conclusion 2
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Introduction
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Introduction
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O(n) minimal separators
17 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Second result
Definition (Gpoly + kv )
G ∈ Gpoly + kv if there exists a set M ⊂ V called modulator,
|M| ≤ k s.t. G − M ∈ Gpoly .
Theorem (Liedloff, M. and Todinca. WG 2015)
Optimal Induced Subgraph for P and t on Gpoly + kv
with parameter k is fixed-parameter tractable, when the modulator
is also part of the input.
18 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
The tools
• Tools from [Fomin, Villanger, 2010] and [Fomin, Todinca, Villanger,
2015] for computing a Maximum induced subgraph of
treewidth t:
• Decompositions with minimal separators.
• Potential maximal cliques
• Dynamic programming over all minimal tree decompositions of
the input graphs, using potential maximal cliques.
• Results [Bodlaender, Kloks, 1996]
• Given a graph G and a tree decomposition of width at most
k + t, determine if G has treewidth t in time O(f (t + k)n),
3
with f (x) = 2O(x log(x)) .
19 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Decomposing with minimal separators
20 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Decomposing with minimal separators
20 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Decomposing with minimal separators
20 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Decomposing with minimal separators
20 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Decomposing with minimal separators
20 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Decomposing with minimal separators
20 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Decomposing with minimal separators
20 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Decomposing with minimal separators
20 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Decomposing with minimal separators
20 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Decomposing with minimal separators
20 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Decomposing with minimal separators
20 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Decomposing with minimal separators
Theorem (Parra, Schaeffler 97)
Decomposing through minimal
separators → minimal tree
decompositions.
20 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Potential maximal cliques
Definition
A set of vertices Ω is a potential maximal clique of G if is a
maximal clique in some minimal triangulation of G .
Definition
A set of vertices Ω is a potential maximal clique of G if there is a
minimal tree decomposition TG of G such that Ω is a bag in TG .
Proposition (Bouchitté, Todinca 2001)
The number of potential maximal cliques is polynomial in the
number of minimal separators.
21 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Dynamic programming over minimal separators. . .
Maximum induced subgraph of treewidth t on Gpoly
S: minimal separator of G
C : component of G − S
T : a subset of S of size ≤ t + 1
OPT (S, C , T ) the size of the largest partial
solution G [F ] s.t.
• F ⊆S ∪C
S
T
C
• T =F ∩S
22 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Dynamic programming over minimal separators. . .
Maximum induced subgraph of treewidth t on Gpoly
S: minimal separator of G
C : component of G − S
T : a subset of S of size ≤ t + 1
OPT (S, C , T ) the size of the largest partial
solution G [F ] s.t.
• F ⊆S ∪C
S
T
C
• T =F ∩S
22 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
. . . and potential maximal cliques
S
T
OPT (S, C , T ):
C
23 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
. . . and potential maximal cliques
S
T
OPT (S, C , T ):
• guess the potential maximal
C
clique Ω splitting S ∪ C
Ω
23 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
. . . and potential maximal cliques
S
T
S2
S1
OPT (S, C , T ):
• guess the potential maximal
C
clique Ω splitting S ∪ C
Ω
C1
C2
23 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
. . . and potential maximal cliques
S
T
S1
T’
S2
OPT (S, C , T ):
• guess the potential maximal
C
clique Ω splitting S ∪ C
• and T ⊆ T 0 the bag of F
Ω
C1
C2
that intersects Ω,
|T 0 | ≤ t + 1.
23 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
. . . and potential maximal cliques
S
T
S1
T’
S2
OPT (S, C , T ):
• guess the potential maximal
C
clique Ω splitting S ∪ C
• and T ⊆ T 0 the bag of F
Ω
C1
C2
that intersects Ω,
|T 0 | ≤ t + 1.
OPT (S1 , C1 , T 0 ∩ S1 ) + OPT (S2 , C2 , T 0 ∩ S2 ) + |T 0 \{S1 ∪ S2 }|
23 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
. . . and potential maximal cliques
S
T
S1
T’
OPT (S, C , T ):
S2
• guess the potential maximal
C
clique Ω splitting S ∪ C
• and T ⊆ T 0 the bag of F
Ω
C1
OPT (S, C , T ) =
that intersects Ω,
|T 0 | ≤ t + 1.
C2
max
(OPT (S1 , C1 , T 0 ∩ S1 )
S⊂Ω⊂C ∪S,T ⊂T 0 ⊂Ω
+OPT (S2 , C2 , T 0
∩ S2 ) + |T 0 \{S1 ∪ S2 }|)
23 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
. . . and potential maximal cliques
S
T
S1
T’
OPT (S, C , T ):
S2
• guess the potential maximal
C
clique Ω splitting S ∪ C
• and T ⊆ T 0 the bag of F
Ω
C1
OPT (S, C , T ) =
that intersects Ω,
|T 0 | ≤ t + 1.
C2
max
(OPT (S1 , C1 , T 0 ∩ S1 )
S⊂Ω⊂C ∪S,T ⊂T 0 ⊂Ω
+OPT (S2 , C2 , T 0
∩ S2 ) + |T 0 \{S1 ∪ S2 }|)
Running time: O(nt+cst · #potential maximal cliques)
Key lemma [Fomin, Villanger 2010]: we don’t miss solutions.
23 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Dynamic programming over minimal separators. . .
Maximum induced subgraph of treewidth t on Gpoly + kv
F M : A subset of the modulator of size k 0 .
S: minimal separator of G 0
C : component of G 0 − S
T : a subset of S of size ≤ t + 1
OPT (S, C , T ) the size of the largest partial
solution G [F ] s.t.
M
S
C
FM
T
F
• FM = F ∩ M
• F \M ⊆ S ∪ C
• T =F ∩S
24 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Dynamic programming over minimal separators. . .
Maximum induced subgraph of treewidth t on Gpoly + kv
F M : A subset of the modulator of size k 0 .
S: minimal separator of G 0
C : component of G 0 − S
T : a subset of S of size ≤ t + 1
OPT (S, C , T ) the size of the largest partial
solution G [F ] s.t.
M
S
C
FM
T
F
• FM = F ∩ M
• F \M ⊆ S ∪ C
• T =F ∩S
This way we build a tree decomposition of G [F ] of width ≤ t + k 0
24 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
[Bodlaender, Kloks, 1996]:
• Algorithm that takes as input a graph with a tree
decomposition of width at most t + k and decides if this graph
3
has treewidth t in time O(f (t + k)n), with f (x) = 2O(x log(x))
25 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
[Bodlaender, Kloks, 1996]:
• Algorithm that takes as input a graph with a tree
decomposition of width at most t + k and decides if this graph
3
has treewidth t in time O(f (t + k)n), with f (x) = 2O(x log(x))
• Full set of characteristics: Set of constant size that encodes
the decision in a partial solution
25 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
[Bodlaender, Kloks, 1996]:
• Algorithm that takes as input a graph with a tree
decomposition of width at most t + k and decides if this graph
3
has treewidth t in time O(f (t + k)n), with f (x) = 2O(x log(x))
• Full set of characteristics: Set of constant size that encodes
the decision in a partial solution
• if two partial solutions are glued, then the characteristic of the
resulting graph can be computed from the characteristics of
each part.
25 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Dynamic programming over minimal separators. . .
F M : A subset of the modulator of size k 0 .
S: minimal separator of G 0
C : component of G 0 − S
T : a subset of S of size ≤ t + 1
c: a characteristic of (tw(G ) ≤ t)
OPT (S, C , T , c) the size of the largest partial
solution G [F ] s.t.
•
FM
M
S
C
FM
T
F
=F ∩M
• F \M ⊆ S ∪ C
• T = F ∩ S, |T | ≤ t
• G [F ] has characteristic c
26 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Dynamic programming over minimal separators. . .
F M : A subset of the modulator of size k 0 .
S: minimal separator of G 0
C : component of G 0 − S
T : a subset of S of size ≤ t + 1
c: a characteristic of (tw(G ) ≤ t)
OPT (S, C , T , c) the size of the largest partial
solution G [F ] s.t.
M
S
C
FM
T
F
• FM = F ∩ M
• F \M ⊆ S ∪ C
• T = F ∩ S, |T | ≤ t
• G [F ] has characteristic c
Running time: O(nt+cst · f (t + k)· #potential maximal cliques)
26 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Dynamic programming over minimal separators. . .
F M : A subset of the modulator of size k 0 .
S: minimal separator of G 0
C : component of G 0 − S
T : a subset of S of size ≤ t + 1
c: a characteristic of (tw(G ) ≤ t) ∧ P
OPT (S, C , T , c) the size of the largest partial
solution G [F ] s.t.
M
S
C
FM
T
F
• FM = F ∩ M
• F \M ⊆ S ∪ C
• T = F ∩ S, |T | ≤ t
• G [F ] has characteristic c
Running time: O(nt+cst · f (t + k, P)· #potential maximal cliques)
26 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
To sum up
Theorem
Optimal Induced Subgraph for P and t on Gpoly + kv is
solvable in time O(nt · poly 0 (n) · f (t + k, P)) when the modulator
is also part of the input.
t
any
any
0
1
P
any
none
none
none
f
tower of exponentials
3
2O((t+k) log(t+k))
2O(k)
2O(k log(k))
27 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
To sum up
Theorem
Optimal Induced Subgraph for P and t on Gpoly + kv is
solvable in time O(nt · poly 0 (n) · f (t + k, P)) when the modulator
is also part of the input.
t
any
any
0
1
P
any
be connected
none
be connected
f
tower of exponentials
3
2O((t+k) log(t+k))
2O(k)
2O(k log(k))
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Introduction
First result
Conclusion 1
Second result
Conclusion 2
Discussion
What if the modulator is not a part of the input?
28 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Discussion
What if the modulator is not a part of the input?
Deletion to Gpoly
Input: A graph G = (V , E ) and a constant k
Parameter: k
Output: A set M ⊆ V of size at most k s.t. G − M belongs to
Gpoly .
28 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Discussion
What if the modulator is not a part of the input?
Deletion to Gpoly
Input: A graph G = (V , E ) and a constant k
Parameter: k
Output: A set M ⊆ V of size at most k s.t. G − M belongs to
Gpoly .
[Heggernes, van’t Hof, Jansen, Kratsch, Villanger, 2013]
• Weakly chordal +kv is W [2]-Hard
[Cao, Marx, 2014]
• Chordal +kv is FPT
28 / 28
Introduction
First result
Conclusion 1
Second result
Conclusion 2
Discussion
What if the modulator is not a part of the input?
Deletion to Gpoly
Input: A graph G = (V , E ) and a constant k
Parameter: k
Output: A set M ⊆ V of size at most k s.t. G − M belongs to
Gpoly .
[Heggernes, van’t Hof, Jansen, Kratsch, Villanger, 2013]
• Weakly chordal +kv is W [2]-Hard
[Cao, Marx, 2014]
• Chordal +kv is FPT
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