Parameterized complexity of generalized domination problems on ...
Parameterized complexity of generalized domination problems on bounded tree-width graphs
Mathieu Chapelle LIFO, University of Orléans, France
JGA 2010
CIRM, Luminy, France 9th November 2010
In brief
•
Very few problems are known to be W-hard (
i.e.
not FPT)
when parameterized by tree-width;
1/24
In brief
•
Very few problems are known to be W-hard (
•
Usual studied cases of
i.e.
not FPT)
when parameterized by tree-width;
∃[σ, %]-Dominating
Set are FPT when
parameterized by tree-width;
1/24
In brief
•
Very few problems are known to be W-hard (
•
Usual studied cases of
i.e.
not FPT)
when parameterized by tree-width;
∃[σ, %]-Dominating
Set are FPT when
parameterized by tree-width;
→
Is it always FPT?
1/24
In brief
•
Very few problems are known to be W-hard (
•
Usual studied cases of
not FPT)
when parameterized by tree-width;
∃[σ, %]-Dominating
Set are FPT when
parameterized by tree-width;
→ •
i.e.
Is it always FPT?
We prove
∃[σ, %]-Dominating
[ ]
Set becomes W 1 -hard for
(many) other cases when parameterized by tree-width.
1/24
1
Some de nitions
2
FPT cases
3
W 1 -hardness
4
Conclusion
[ ]
2/24
1
Some de nitions
2
FPT cases
3
W 1 -hardness
4
Conclusion
[ ]
3/24
Parameterized complexity In computational complexity, general parameter:
• n
(size of the input).
4/24
Parameterized complexity In computational complexity, general parameter:
• n
(size of the input).
n
4/24
Parameterized complexity In computational complexity, general parameter:
• n
(size of the input).
n
In parameterized complexity, more speci c parameters:
• k
(size of an expected solution);
• tw •
(tree-width of the input graph);
...
4/24
Parameterized complexity In computational complexity, general parameter:
• n
(size of the input).
n
In parameterized complexity, more speci c parameters:
• k
(size of an expected solution);
• tw •
(tree-width of the input graph);
...
n
k 4/24
Parameterized complexity
De nition (FPT) P is FPT O f (k ) · poly(n) . A problem
parameterized by
k
if it can be solved in time
5/24
Parameterized complexity
De nition (FPT) P is FPT O f (k ) · poly(n) . A problem
Example: k -Vertex
parameterized by
k
if it can be solved in time
Cover parameterized by the size
k
of the
vertex cover.
5/24
Parameterized complexity
De nition (FPT) P is FPT O f (k ) · poly(n) . A problem
Example: k -Vertex
parameterized by
k
if it can be solved in time
Cover parameterized by the size
k
of the
vertex cover. A problem
P
which is not FPT is at least W[1]-hard.
5/24
Parameterized complexity
De nition (FPT) P is FPT O f (k ) · poly(n) . A problem
Example: k -Vertex
parameterized by
k
if it can be solved in time
Cover parameterized by the size
k
of the
vertex cover. A problem
P
which is not FPT is at least W[1]-hard.
Example: k -Independent
Set parameterized by the size
k
of the
independent set.
5/24
Parameterized complexity
De nition (FPT) P is FPT O f (k ) · poly(n) . A problem
Example: k -Vertex
parameterized by
k
if it can be solved in time
Cover parameterized by the size
k
of the
vertex cover. A problem
P
which is not FPT is at least W[1]-hard.
Example: k -Independent
Set parameterized by the size
k
of the
independent set.
Remark:
P is W[1]-hard, Q ≤fpt P .
To prove
and prove that
take a W[1]-hard problem
Q 5/24
Generalized domination
De nition
(dominating set)
D ⊆ V is a dominating set if, ∀v ∈ V : • v ∈ D ; or • ∃u (u ∈ D ∧ adj(u , v )).
6/24
Generalized domination
De nition
(dominating set)
D ⊆ V is a dominating set • v ∈ D ; or • ∃u (u ∈ D ∩ N (v )).
if,
∀v ∈ V :
6/24
Generalized domination
De nition
(dominating set)
D ⊆ V is a dominating set if, ∀v ∈ V : • v ∈ D ⇒ |D ∩ N (v )| ≥ 0; or • v ∈ / D ⇒ |D ∩ N (v )| ≥ 1.
6/24
Generalized domination
De nition
(dominating set)
D ⊆ V is a dominating set if, ∀v ∈ V : • v ∈ D ⇒ |D ∩ N (v )| ∈ {0, 1, 2, . . .}; • v ∈ / D ⇒ |D ∩ N (v )| ∈ {1, 2, 3, . . .}.
or
6/24
Generalized domination
De nition Let
([σ, %]-dominating set)
σ, % ⊆ N. D ⊆ V
is a
[σ, %]-dominating set
• v ∈ D ⇒ |D ∩ N (v )| ∈ σ ;
or
if,
∀v ∈ V :
• v ∈ / D ⇒ |D ∩ N (v )| ∈ %.
6/24
Generalized domination
De nition Let
([σ, %]-dominating set)
σ, % ⊆ N. D ⊆ V
is a
[σ, %]-dominating set
• v ∈ D ⇒ |D ∩ N (v )| ∈ σ ;
or
if,
∀v ∈ V :
• v ∈ / D ⇒ |D ∩ N (v )| ∈ %.
σ
and
%
• σ
constraints the neighborhood of vertices which are in
• %
x some constraints on the neighborhood of every vertex:
D.
constraints the neighborhood of vertices which are not in
D.
6/24
Generalized domination
De nition Let
([σ, %]-dominating set)
σ, % ⊆ N. D ⊆ V
is a
[σ, %]-dominating set
• v ∈ D ⇒ |D ∩ N (v )| ∈ σ ;
or
if,
∀v ∈ V :
• v ∈ / D ⇒ |D ∩ N (v )| ∈ %.
σ
and
%
• σ
constraints the neighborhood of vertices which are in
• %
x some constraints on the neighborhood of every vertex:
D.
constraints the neighborhood of vertices which are not in
Remark:
We usually suppose that 0
be a trivial
[σ, %]-dominating
set.
∈ / %,
as otherwise
D=∅
D.
would
6/24
Generalized domination
De nition
(selected vertex, satis ed vertex)
D ⊆ V be a [σ, %]-dominating u ∈ V is selected if u ∈ D .
Let
set.
7/24
Generalized domination
De nition
(selected vertex, satis ed vertex)
D ⊆ V be a [σ, %]-dominating u ∈ V is selected if u ∈ D . v ∈ V is satis ed if: • v ∈ D ⇒ |D ∩ N (v )| ∈ σ ; • v ∈ / D ⇒ |D ∩ N (v )| ∈ %.
Let
set.
7/24
1
Some de nitions
2
FPT cases
3
W 1 -hardness
4
Conclusion
[ ]
8/24
Known results
Theorem
[van Rooij, Bodlaender, Rossmanith, 2009]
Dominating Set can be solved in
O∗ (3tw )
time.
9/24
Known results
Theorem
[van Rooij, Bodlaender, Rossmanith, 2009]
Dominating Set can be solved in
and
time.
[σ, %]-Dominating % = {1, 2, 3, . . .}.
Recall that Dominating Set is
σ = {0, 1, 2, . . .}
O∗ (3tw )
Set with
9/24
Known results
Theorem
[van Rooij, Bodlaender, Rossmanith, 2009]
Dominating Set can be solved in
O∗ (3tw )
time.
[σ, %]-Dominating % = {1, 2, 3, . . .}.
Recall that Dominating Set is
σ = {0, 1, 2, . . .}
Theorem
and
[van Rooij, Bodlaender, Rossmanith, 2009]
∃[σ, %]-Dominating
Set can be solved in
needed to represents
σ
are nite or co nite, where and
s
O∗ s tw
Set with
time if
σ
and
%
is the minimum number of states
%.
9/24
More cases
Using Courcelle's theorem:
If
σ
and
%
are nite or co nite, then
∃[σ, %]-Dominating
Set is
expressible in MSOL2 . Hence it is FPT when parameterized by tree-width.
10/24
More cases
Using Courcelle's theorem:
If
σ
and
%
are ultimately periodic, then
∃[σ, %]-Dominating
Set is
expressible in CMSOL. Hence it is FPT when parameterized by tree-width.
10/24
More cases
Using Courcelle's theorem:
If
σ
and
%
are ultimately periodic, then
∃[σ, %]-Dominating
Set is
expressible in CMSOL. Hence it is FPT when parameterized by tree-width. But the hidden constant is very huge.
10/24
More cases
Using Courcelle's theorem:
If
σ
and
%
are ultimately periodic, then
∃[σ, %]-Dominating
Set is
expressible in CMSOL. Hence it is FPT when parameterized by tree-width. But the hidden constant is very huge.
Theorem
[C., 20082010]
∃[σ, %]-Dominating
Set can be solved in
are ultimately periodic, where states needed to represents
σ
s
O∗ s tw
time if
σ
and
%
is (almost) the minimum number of
and
%.
10/24
More cases
Theorem
[C., 20082010]
∃[σ, %]-Dominating
Set can be solved in
are ultimately periodic, where states needed to represents
σ
s
O∗ s tw
time if
σ
and
%
is (almost) the minimum number of
and
%.
11/24
More cases
Theorem
[C., 20082010]
∃[σ, %]-Dominating
Set can be solved in
are ultimately periodic, where states needed to represents
σ
s
O∗ s tw
time if
σ
and
%
is (almost) the minimum number of
and
%.
Ideas of the proof:
11/24
More cases
Theorem
[C., 20082010]
∃[σ, %]-Dominating
Set can be solved in
are ultimately periodic, where states needed to represents
σ
s
O∗ s tw
time if
σ
and
%
is (almost) the minimum number of
and
%.
Ideas of the proof:
•
Represent
σ
and
%
with nite unary-language automata;
11/24
More cases
Theorem
[C., 20082010]
∃[σ, %]-Dominating
Set can be solved in
are ultimately periodic, where states needed to represents
σ
s
O∗ s tw
time if
σ
and
%
is (almost) the minimum number of
and
%.
Ideas of the proof:
•
•
Represent
σ
and
%
with nite unary-language automata;
Apply dynamic programming on a (nice) tree-decomposition of the input graph;
11/24
More cases
Theorem
[C., 20082010]
∃[σ, %]-Dominating
Set can be solved in
are ultimately periodic, where states needed to represents
σ
s
O∗ s tw
time if
σ
and
%
is (almost) the minimum number of
and
%.
Ideas of the proof:
•
Represent
σ
and
%
with nite unary-language automata;
•
Apply dynamic programming on a (nice) tree-decomposition of
•
Encode the number of selected neighbors of each vertex using
the input graph;
the corresponding state in one of the automata;
11/24
More cases
Theorem
[C., 20082010]
∃[σ, %]-Dominating
Set can be solved in
are ultimately periodic, where states needed to represents
σ
s
O∗ s tw
time if
σ
and
%
is (almost) the minimum number of
and
%.
Ideas of the proof:
•
Represent
σ
and
%
with nite unary-language automata;
•
Apply dynamic programming on a (nice) tree-decomposition of
•
Encode the number of selected neighbors of each vertex using
•
Use fast subset convolution to fasten the join operation.
the input graph;
the corresponding state in one of the automata;
11/24
1
Some de nitions
2
FPT cases
3
W 1 -hardness
4
Conclusion
[ ]
12/24
Some
Theorem If
σ
W[ ]
1 -hard cases
[C., 2010]
contains arbitrary large gaps between two consecutive elements
% is co nite (and an additional technical constraint on σ ), then ∃[σ, %]-Dominating Set is W[1]-hard parameterized by tree-width. and
13/24
Some
Theorem If
σ
W[ ]
1 -hard cases
[C., 2010]
contains arbitrary large gaps between two consecutive elements
% is co nite (and an additional technical constraint on σ ), then ∃[σ, %]-Dominating Set is W[1]-hard parameterized by tree-width. and
σ and %, we will reduce k -Capacitated ∃[σ, %]-Dominating Set.
Given
Dominating Set to
13/24
Some
Theorem If
σ
W[ ]
1 -hard cases
[C., 2010]
contains arbitrary large gaps between two consecutive elements
% is co nite (and an additional technical constraint on σ ), then ∃[σ, %]-Dominating Set is W[1]-hard parameterized by tree-width. and
σ and %, we will reduce k -Capacitated ∃[σ, %]-Dominating Set.
Given
k -Capacitated
Dominating Set to
[ ]
Dominating Set is W 1 -hard when parameterized
by the tree-width of the input graph and the size
k
of the expected
solution.
13/24
Capacitated domination
De nition Let
(capacitated dominating set)
G = (V , E ),
cap
: V → N.
14/24
Capacitated domination
De nition
(capacitated dominating set)
G = (V , E ), cap : V → N. (S , dom) is a capacitated dominating set
Let
of
G,
14/24
Capacitated domination
De nition
(capacitated dominating set)
G = (V , E ), cap : V → N. (S , dom) is a capacitated dominating set ∀v ∈ V , dom(v ) ⊆ N (v ), if: Let
of
G,
with
S ⊆V
and
14/24
Capacitated domination
De nition
(capacitated dominating set)
G = (V , E ), cap : V → N. (S , dom) is a capacitated dominating set of G , with S ⊆ V ∀v ∈ V , dom(v ) ⊆ N (v ), if: • S is a dominating set of G (in the classical sense); Let
and
14/24
Capacitated domination
De nition
(capacitated dominating set)
G = (V , E ), cap : V → N. (S , dom) is a capacitated dominating set of G , with S ⊆ V ∀v ∈ V , dom(v ) ⊆ N (v ), if: • S is a dominating set of G (in the classical sense); • ∀v ∈ S , |dom(v )| ≤ cap(v ); Let
and
14/24
Capacitated domination
De nition
(capacitated dominating set)
G = (V , E ), cap : V → N. (S , dom) is a capacitated dominating set of G , with S ⊆ V ∀v ∈ V , dom(v ) ⊆ N (v ), if: • S is a dominating set of G (in the classical sense); • ∀v ∈ S , |dom(v )| ≤ cap(v ); • ∀v ∈ / S , dom(v ) = ∅. Let
and
14/24
Capacitated domination
De nition
(capacitated dominating set)
G = (V , E ), cap : V → N. (S , dom) is a capacitated dominating set of G , with S ⊆ V ∀v ∈ V , dom(v ) ⊆ N (v ), if: • S is a dominating set of G (in the classical sense); • ∀v ∈ S , |dom(v )| ≤ cap(v ); • ∀v ∈ / S , dom(v ) = ∅. Let
k -Capacitated
Dominating Set: search
S ⊆V
such that
and
|S | ≤ k .
14/24
Ideas of the reduction
2
3
4
1
1
2
3
1
1
2
Input: A graph with capacities
on vertices.
15/24
Ideas of the reduction
2
3
4
1
1
2
3
1
1
2
Input: A graph with capacities
on vertices.
15/24
Ideas of the reduction
2
3
4
1
1
2
3
1
1
2
Input: A graph with capacities
on vertices.
15/24
Ideas of the reduction
2
3
4
1
1
2
3
1
2
1
3
2
Input: A graph with capacities
on vertices.
4
1
1
→
2
3
1
1
2
Transformation: The
incidence graph.
15/24
Ideas of the reduction
2
3
4
1
1
2
3
1
2
1
3
2
Input: A graph with capacities
on vertices.
4
1
1
→
2
3
1
1
2
Transformation: The
incidence graph.
15/24
Ideas of the reduction
2
3
4
1
1
2
3
1
2
1
3
2
Input: A graph with capacities
on vertices.
4
1
1
→
2
3
1
1
2
Transformation: The
incidence graph.
15/24
Some functions on
σ
σ contains arbitrary large gaps between two consecutive elements
16/24
Some functions on
σ
σ contains arbitrary large gaps between two consecutive elements We de ne some functions on
σ:
16/24
Some functions on
σ
σ contains arbitrary large gaps between two consecutive elements We de ne some functions on
• Γ− (x , q ),
σ:
p∈σ p ≥ q;
the minimum element
length at least
x
before
p,
and
s.t. there is a gap of
16/24
Some functions on
σ
σ contains arbitrary large gaps between two consecutive elements We de ne some functions on
• Γ− (x , q ),
σ:
p∈σ p ≥ q; • Γ+ (x , q ), the minimum element p ∈ σ length at least x after p , and p ≥ q ; the minimum element
length at least
x
before
p,
s.t. there is a gap of
and
s.t. there is a gap of
16/24
Some functions on
σ
σ contains arbitrary large gaps between two consecutive elements We de ne some functions on
• Γ− (x , q ),
σ:
p ∈ σ s.t. there p ≥ q; • Γ+ (x , q ), the minimum element p ∈ σ s.t. there length at least x after p , and p ≥ q ; • Γ0 (q ), the minimum element p ∈ σ s.t. p ≥ q . the minimum element
length at least
x
before
p,
is a gap of
and
is a gap of
16/24
Some functions on
σ
σ contains arbitrary large gaps between two consecutive elements We de ne some functions on
• Γ− (x , q ),
σ:
p ∈ σ s.t. there p ≥ q; • Γ+ (x , q ), the minimum element p ∈ σ s.t. there length at least x after p , and p ≥ q ; • Γ0 (q ), the minimum element p ∈ σ s.t. p ≥ q . the minimum element
length at least
x
before
p,
is a gap of
and
is a gap of
Technical constraint: We suppose that there exists a polynomial
pσ
such that a gap of length
t
exists at distance
pσ (t )
in
σ.
16/24
The reduction Let
G
be an instance of
2
3
k -Capacitated
4
1
1
2
3
2
Dominating Set.
1
1
17/24
The reduction Let
G
k -Capacitated Dominating Set. I (G ), incidence graph of G , and add some gadgets:
be an instance of
We start with
2
3
4
1
1
2
3
2
1
1
17/24
The reduction
k -Capacitated Dominating Set. We start with I (G ), incidence graph of G , and add some gadgets: domination (D): forces G to have a dominating set; Let
G
be an instance of
2
3
4
1
1
2
3
2
1
1
17/24
The reduction
k -Capacitated Dominating Set. We start with I (G ), incidence graph of G , and add some gadgets: domination (D): forces G to have a dominating set; capacity (C ): encodes the capacity function cap, and allows Let
G
be an instance of
any selected vertex to be satis ed;
2
3
4
1
1
2
3
2
1
1
17/24
The reduction
k -Capacitated Dominating Set. We start with I (G ), incidence graph of G , and add some gadgets: domination (D): forces G to have a dominating set; capacity (C ): encodes the capacity function cap, and allows Let
G
be an instance of
any selected vertex to be satis ed;
edge-selection (E ):
encodes the domination function dom;
2
3
4
1
1
2
3
2
1
1
17/24
The reduction
k -Capacitated Dominating Set. We start with I (G ), incidence graph of G , and add some gadgets: domination (D): forces G to have a dominating set; capacity (C ): encodes the capacity function cap, and allows Let
G
be an instance of
any selected vertex to be satis ed;
edge-selection (E ): satis ability (S ):
encodes the domination function dom; allows any non-selected vertex to be satis ed;
2
3
4
1
1
2
3
2
1
1
17/24
The reduction
k -Capacitated Dominating Set. We start with I (G ), incidence graph of G , and add some gadgets: domination (D): forces G to have a dominating set; capacity (C ): encodes the capacity function cap, and allows Let
G
be an instance of
any selected vertex to be satis ed;
edge-selection (E ): satis ability (S ): limitation (L):
encodes the domination function dom; allows any non-selected vertex to be satis ed; encodes the parameter
2
3
4
1
1
2
3
2
k; 1
1
17/24
The reduction
k -Capacitated Dominating Set. We start with I (G ), incidence graph of G , and add some gadgets: domination (D): forces G to have a dominating set; capacity (C ): encodes the capacity function cap, and allows Let
G
be an instance of
any selected vertex to be satis ed;
edge-selection (E ): satis ability (S ): limitation (L): force (F ):
encodes the domination function dom; allows any non-selected vertex to be satis ed; encodes the parameter
k;
forces a given vertex to be selected.
2
3
4
1
1
2
3
2
1
1
17/24
The reduction
k -Capacitated Dominating Set. We start with I (G ), incidence graph of G , and add some gadgets: domination (D): forces G to have a dominating set; capacity (C ): encodes the capacity function cap, and allows Let
G
be an instance of
any selected vertex to be satis ed;
edge-selection (E ): satis ability (S ): limitation (L): force (F ):
encodes the domination function dom; allows any non-selected vertex to be satis ed; encodes the parameter
forces a given vertex to be selected.
D S
D C
k;
E
S
D C
E
L
S
D C
S
C
17/24
The gadgets Gadget
force (F ):
forces a given vertex to be selected.
← neighbor in H clique with min σ vertices →
← forced vertex
clique with ασ,ρ vertices → clique with βσ,ρ vertices →
18/24
The gadgets Gadget
force (F ):
forces a given vertex to be selected.
← neighbor in H clique with min σ vertices →
← forced vertex
clique with ασ,ρ vertices → clique with βσ,ρ vertices →
Gadget
domination (D):
forces
G
to have a dominating set.
edge-vertex in I(G) →
← original-vertex v in I(G)
neighbor of v in G → clique with min σ vertices → Γ+ (1, min σ) − min σ forced vertices →
← forced vertices q0 − 2 18/24
The gadgets Gadget
edge-selection (E ):
encodes the domination function dom.
← neighbor in I(G) ← edge-vertex in I(G) ← Γ− (1, q0 + 1) − 1 forced vertices
19/24
The gadgets Gadget
edge-selection (E ):
encodes the domination function dom.
← neighbor in I(G) ← edge-vertex in I(G) ← Γ− (1, q0 + 1) − 1 forced vertices
Gadget
satis ability (S ):
allows any non-selected vertex to be
satis ed.
Γ0 (q0 ) forced vertices
← q0 choosable vertices ← original-vertex in I(G) 19/24
The gadgets
Gadget
capacity (C ):
encodes the capacity function cap, and allows
any selected vertex to be satis ed.
Γ0 (q0 + 1) − 1 forced vertices
← cap(v) vertices ← original-vertex v in I(G) Γ+ degG (v) + q0 , cap(v) − cap(v) − 1 forced vertices
20/24
The gadgets
Gadget
limitation (L):
encodes the parameter
k.
Γ+ |V (G)|, k − k forced vertices original-vertices in I(G)
← central forced vertex c ← k choosable vertices
Γ0 (q0 ) − 1 forced vertices
21/24
Correctness
Correctness of the reduction:
•
Each gadget has small tree-width → tw (H ) = f tw (G )
(at most min σ
+ 1).
22/24
Correctness
Correctness of the reduction:
• •
Each gadget has small tree-width → tw (H ) = f tw (G )
(at most min σ
The number of added vertices depends only on
→ |V (H )| = g |V (G )|
+ 1).
k ≤ n, σ
and
%.
22/24
Correctness
Correctness of the reduction:
• •
Each gadget has small tree-width → tw (H ) = f tw (G )
(at most min σ
The number of added vertices depends only on
→ |V (H )| = g |V (G )|
• H
admits a
[σ, %]-dominating
set i
G
+ 1).
k ≤ n, σ
admits a
and
%.
k -capacitated
dominating set.
22/24
Correctness
Correctness of the reduction:
• •
Each gadget has small tree-width → tw (H ) = f tw (G )
(at most min σ
The number of added vertices depends only on
→ |V (H )| = g |V (G )|
• H
admits a
[σ, %]-dominating
set i
G
+ 1).
k ≤ n, σ
admits a
and
%.
k -capacitated
dominating set.
Theorem If
σ
[C., 2010]
contains arbitrary large gaps between two consecutive elements
% is co nite (and an additional technical constraint), then ∃[σ, %]-Dominating Set is W[1]-hard parameterized by tree-width. and
22/24
1
Some de nitions
2
FPT cases
3
W 1 -hardness
4
Conclusion
[ ]
23/24
Conclusion
∃[σ, %]-Dominating Set is FPT parameterized when σ and % are ultimately periodic.
by tree-width,
24/24
Conclusion
∃[σ, %]-Dominating Set is FPT parameterized when σ and % are ultimately periodic.
by tree-width,
∃[σ, %]-Dominating Set is W[1]-hard parameterized by tree-width, when σ contains arbitrary large gaps between two consecutive elements and % is co nite (and an additional constraint on σ ).
24/24
Conclusion
∃[σ, %]-Dominating Set is FPT parameterized when σ and % are ultimately periodic.
by tree-width,
∃[σ, %]-Dominating Set is W[1]-hard parameterized by tree-width, when σ contains arbitrary large gaps between two consecutive elements and % is co nite (and an additional constraint on σ ). And now?
• •
•
W
[t ]-completeness ;
Other cases of
[σ, %] (e.g.
recursive with bounded gaps);
...
24/24
Conclusion
∃[σ, %]-Dominating Set is FPT parameterized when σ and % are ultimately periodic.
by tree-width,
∃[σ, %]-Dominating Set is W[1]-hard parameterized by tree-width, when σ contains arbitrary large gaps between two consecutive elements and % is co nite (and an additional constraint on σ ). And now?
• •
•
W
[t ]-completeness ;
Other cases of
[σ, %] (e.g.
recursive with bounded gaps);
...
And voilà!
24/24
Mathieu Chapelle LIFO, University of Orléans, France
JGA 2010
CIRM, Luminy, France 9th November 2010
In brief
•
Very few problems are known to be W-hard (
i.e.
not FPT)
when parameterized by tree-width;
1/24
In brief
•
Very few problems are known to be W-hard (
•
Usual studied cases of
i.e.
not FPT)
when parameterized by tree-width;
∃[σ, %]-Dominating
Set are FPT when
parameterized by tree-width;
1/24
In brief
•
Very few problems are known to be W-hard (
•
Usual studied cases of
i.e.
not FPT)
when parameterized by tree-width;
∃[σ, %]-Dominating
Set are FPT when
parameterized by tree-width;
→
Is it always FPT?
1/24
In brief
•
Very few problems are known to be W-hard (
•
Usual studied cases of
not FPT)
when parameterized by tree-width;
∃[σ, %]-Dominating
Set are FPT when
parameterized by tree-width;
→ •
i.e.
Is it always FPT?
We prove
∃[σ, %]-Dominating
[ ]
Set becomes W 1 -hard for
(many) other cases when parameterized by tree-width.
1/24
1
Some de nitions
2
FPT cases
3
W 1 -hardness
4
Conclusion
[ ]
2/24
1
Some de nitions
2
FPT cases
3
W 1 -hardness
4
Conclusion
[ ]
3/24
Parameterized complexity In computational complexity, general parameter:
• n
(size of the input).
4/24
Parameterized complexity In computational complexity, general parameter:
• n
(size of the input).
n
4/24
Parameterized complexity In computational complexity, general parameter:
• n
(size of the input).
n
In parameterized complexity, more speci c parameters:
• k
(size of an expected solution);
• tw •
(tree-width of the input graph);
...
4/24
Parameterized complexity In computational complexity, general parameter:
• n
(size of the input).
n
In parameterized complexity, more speci c parameters:
• k
(size of an expected solution);
• tw •
(tree-width of the input graph);
...
n
k 4/24
Parameterized complexity
De nition (FPT) P is FPT O f (k ) · poly(n) . A problem
parameterized by
k
if it can be solved in time
5/24
Parameterized complexity
De nition (FPT) P is FPT O f (k ) · poly(n) . A problem
Example: k -Vertex
parameterized by
k
if it can be solved in time
Cover parameterized by the size
k
of the
vertex cover.
5/24
Parameterized complexity
De nition (FPT) P is FPT O f (k ) · poly(n) . A problem
Example: k -Vertex
parameterized by
k
if it can be solved in time
Cover parameterized by the size
k
of the
vertex cover. A problem
P
which is not FPT is at least W[1]-hard.
5/24
Parameterized complexity
De nition (FPT) P is FPT O f (k ) · poly(n) . A problem
Example: k -Vertex
parameterized by
k
if it can be solved in time
Cover parameterized by the size
k
of the
vertex cover. A problem
P
which is not FPT is at least W[1]-hard.
Example: k -Independent
Set parameterized by the size
k
of the
independent set.
5/24
Parameterized complexity
De nition (FPT) P is FPT O f (k ) · poly(n) . A problem
Example: k -Vertex
parameterized by
k
if it can be solved in time
Cover parameterized by the size
k
of the
vertex cover. A problem
P
which is not FPT is at least W[1]-hard.
Example: k -Independent
Set parameterized by the size
k
of the
independent set.
Remark:
P is W[1]-hard, Q ≤fpt P .
To prove
and prove that
take a W[1]-hard problem
Q 5/24
Generalized domination
De nition
(dominating set)
D ⊆ V is a dominating set if, ∀v ∈ V : • v ∈ D ; or • ∃u (u ∈ D ∧ adj(u , v )).
6/24
Generalized domination
De nition
(dominating set)
D ⊆ V is a dominating set • v ∈ D ; or • ∃u (u ∈ D ∩ N (v )).
if,
∀v ∈ V :
6/24
Generalized domination
De nition
(dominating set)
D ⊆ V is a dominating set if, ∀v ∈ V : • v ∈ D ⇒ |D ∩ N (v )| ≥ 0; or • v ∈ / D ⇒ |D ∩ N (v )| ≥ 1.
6/24
Generalized domination
De nition
(dominating set)
D ⊆ V is a dominating set if, ∀v ∈ V : • v ∈ D ⇒ |D ∩ N (v )| ∈ {0, 1, 2, . . .}; • v ∈ / D ⇒ |D ∩ N (v )| ∈ {1, 2, 3, . . .}.
or
6/24
Generalized domination
De nition Let
([σ, %]-dominating set)
σ, % ⊆ N. D ⊆ V
is a
[σ, %]-dominating set
• v ∈ D ⇒ |D ∩ N (v )| ∈ σ ;
or
if,
∀v ∈ V :
• v ∈ / D ⇒ |D ∩ N (v )| ∈ %.
6/24
Generalized domination
De nition Let
([σ, %]-dominating set)
σ, % ⊆ N. D ⊆ V
is a
[σ, %]-dominating set
• v ∈ D ⇒ |D ∩ N (v )| ∈ σ ;
or
if,
∀v ∈ V :
• v ∈ / D ⇒ |D ∩ N (v )| ∈ %.
σ
and
%
• σ
constraints the neighborhood of vertices which are in
• %
x some constraints on the neighborhood of every vertex:
D.
constraints the neighborhood of vertices which are not in
D.
6/24
Generalized domination
De nition Let
([σ, %]-dominating set)
σ, % ⊆ N. D ⊆ V
is a
[σ, %]-dominating set
• v ∈ D ⇒ |D ∩ N (v )| ∈ σ ;
or
if,
∀v ∈ V :
• v ∈ / D ⇒ |D ∩ N (v )| ∈ %.
σ
and
%
• σ
constraints the neighborhood of vertices which are in
• %
x some constraints on the neighborhood of every vertex:
D.
constraints the neighborhood of vertices which are not in
Remark:
We usually suppose that 0
be a trivial
[σ, %]-dominating
set.
∈ / %,
as otherwise
D=∅
D.
would
6/24
Generalized domination
De nition
(selected vertex, satis ed vertex)
D ⊆ V be a [σ, %]-dominating u ∈ V is selected if u ∈ D .
Let
set.
7/24
Generalized domination
De nition
(selected vertex, satis ed vertex)
D ⊆ V be a [σ, %]-dominating u ∈ V is selected if u ∈ D . v ∈ V is satis ed if: • v ∈ D ⇒ |D ∩ N (v )| ∈ σ ; • v ∈ / D ⇒ |D ∩ N (v )| ∈ %.
Let
set.
7/24
1
Some de nitions
2
FPT cases
3
W 1 -hardness
4
Conclusion
[ ]
8/24
Known results
Theorem
[van Rooij, Bodlaender, Rossmanith, 2009]
Dominating Set can be solved in
O∗ (3tw )
time.
9/24
Known results
Theorem
[van Rooij, Bodlaender, Rossmanith, 2009]
Dominating Set can be solved in
and
time.
[σ, %]-Dominating % = {1, 2, 3, . . .}.
Recall that Dominating Set is
σ = {0, 1, 2, . . .}
O∗ (3tw )
Set with
9/24
Known results
Theorem
[van Rooij, Bodlaender, Rossmanith, 2009]
Dominating Set can be solved in
O∗ (3tw )
time.
[σ, %]-Dominating % = {1, 2, 3, . . .}.
Recall that Dominating Set is
σ = {0, 1, 2, . . .}
Theorem
and
[van Rooij, Bodlaender, Rossmanith, 2009]
∃[σ, %]-Dominating
Set can be solved in
needed to represents
σ
are nite or co nite, where and
s
O∗ s tw
Set with
time if
σ
and
%
is the minimum number of states
%.
9/24
More cases
Using Courcelle's theorem:
If
σ
and
%
are nite or co nite, then
∃[σ, %]-Dominating
Set is
expressible in MSOL2 . Hence it is FPT when parameterized by tree-width.
10/24
More cases
Using Courcelle's theorem:
If
σ
and
%
are ultimately periodic, then
∃[σ, %]-Dominating
Set is
expressible in CMSOL. Hence it is FPT when parameterized by tree-width.
10/24
More cases
Using Courcelle's theorem:
If
σ
and
%
are ultimately periodic, then
∃[σ, %]-Dominating
Set is
expressible in CMSOL. Hence it is FPT when parameterized by tree-width. But the hidden constant is very huge.
10/24
More cases
Using Courcelle's theorem:
If
σ
and
%
are ultimately periodic, then
∃[σ, %]-Dominating
Set is
expressible in CMSOL. Hence it is FPT when parameterized by tree-width. But the hidden constant is very huge.
Theorem
[C., 20082010]
∃[σ, %]-Dominating
Set can be solved in
are ultimately periodic, where states needed to represents
σ
s
O∗ s tw
time if
σ
and
%
is (almost) the minimum number of
and
%.
10/24
More cases
Theorem
[C., 20082010]
∃[σ, %]-Dominating
Set can be solved in
are ultimately periodic, where states needed to represents
σ
s
O∗ s tw
time if
σ
and
%
is (almost) the minimum number of
and
%.
11/24
More cases
Theorem
[C., 20082010]
∃[σ, %]-Dominating
Set can be solved in
are ultimately periodic, where states needed to represents
σ
s
O∗ s tw
time if
σ
and
%
is (almost) the minimum number of
and
%.
Ideas of the proof:
11/24
More cases
Theorem
[C., 20082010]
∃[σ, %]-Dominating
Set can be solved in
are ultimately periodic, where states needed to represents
σ
s
O∗ s tw
time if
σ
and
%
is (almost) the minimum number of
and
%.
Ideas of the proof:
•
Represent
σ
and
%
with nite unary-language automata;
11/24
More cases
Theorem
[C., 20082010]
∃[σ, %]-Dominating
Set can be solved in
are ultimately periodic, where states needed to represents
σ
s
O∗ s tw
time if
σ
and
%
is (almost) the minimum number of
and
%.
Ideas of the proof:
•
•
Represent
σ
and
%
with nite unary-language automata;
Apply dynamic programming on a (nice) tree-decomposition of the input graph;
11/24
More cases
Theorem
[C., 20082010]
∃[σ, %]-Dominating
Set can be solved in
are ultimately periodic, where states needed to represents
σ
s
O∗ s tw
time if
σ
and
%
is (almost) the minimum number of
and
%.
Ideas of the proof:
•
Represent
σ
and
%
with nite unary-language automata;
•
Apply dynamic programming on a (nice) tree-decomposition of
•
Encode the number of selected neighbors of each vertex using
the input graph;
the corresponding state in one of the automata;
11/24
More cases
Theorem
[C., 20082010]
∃[σ, %]-Dominating
Set can be solved in
are ultimately periodic, where states needed to represents
σ
s
O∗ s tw
time if
σ
and
%
is (almost) the minimum number of
and
%.
Ideas of the proof:
•
Represent
σ
and
%
with nite unary-language automata;
•
Apply dynamic programming on a (nice) tree-decomposition of
•
Encode the number of selected neighbors of each vertex using
•
Use fast subset convolution to fasten the join operation.
the input graph;
the corresponding state in one of the automata;
11/24
1
Some de nitions
2
FPT cases
3
W 1 -hardness
4
Conclusion
[ ]
12/24
Some
Theorem If
σ
W[ ]
1 -hard cases
[C., 2010]
contains arbitrary large gaps between two consecutive elements
% is co nite (and an additional technical constraint on σ ), then ∃[σ, %]-Dominating Set is W[1]-hard parameterized by tree-width. and
13/24
Some
Theorem If
σ
W[ ]
1 -hard cases
[C., 2010]
contains arbitrary large gaps between two consecutive elements
% is co nite (and an additional technical constraint on σ ), then ∃[σ, %]-Dominating Set is W[1]-hard parameterized by tree-width. and
σ and %, we will reduce k -Capacitated ∃[σ, %]-Dominating Set.
Given
Dominating Set to
13/24
Some
Theorem If
σ
W[ ]
1 -hard cases
[C., 2010]
contains arbitrary large gaps between two consecutive elements
% is co nite (and an additional technical constraint on σ ), then ∃[σ, %]-Dominating Set is W[1]-hard parameterized by tree-width. and
σ and %, we will reduce k -Capacitated ∃[σ, %]-Dominating Set.
Given
k -Capacitated
Dominating Set to
[ ]
Dominating Set is W 1 -hard when parameterized
by the tree-width of the input graph and the size
k
of the expected
solution.
13/24
Capacitated domination
De nition Let
(capacitated dominating set)
G = (V , E ),
cap
: V → N.
14/24
Capacitated domination
De nition
(capacitated dominating set)
G = (V , E ), cap : V → N. (S , dom) is a capacitated dominating set
Let
of
G,
14/24
Capacitated domination
De nition
(capacitated dominating set)
G = (V , E ), cap : V → N. (S , dom) is a capacitated dominating set ∀v ∈ V , dom(v ) ⊆ N (v ), if: Let
of
G,
with
S ⊆V
and
14/24
Capacitated domination
De nition
(capacitated dominating set)
G = (V , E ), cap : V → N. (S , dom) is a capacitated dominating set of G , with S ⊆ V ∀v ∈ V , dom(v ) ⊆ N (v ), if: • S is a dominating set of G (in the classical sense); Let
and
14/24
Capacitated domination
De nition
(capacitated dominating set)
G = (V , E ), cap : V → N. (S , dom) is a capacitated dominating set of G , with S ⊆ V ∀v ∈ V , dom(v ) ⊆ N (v ), if: • S is a dominating set of G (in the classical sense); • ∀v ∈ S , |dom(v )| ≤ cap(v ); Let
and
14/24
Capacitated domination
De nition
(capacitated dominating set)
G = (V , E ), cap : V → N. (S , dom) is a capacitated dominating set of G , with S ⊆ V ∀v ∈ V , dom(v ) ⊆ N (v ), if: • S is a dominating set of G (in the classical sense); • ∀v ∈ S , |dom(v )| ≤ cap(v ); • ∀v ∈ / S , dom(v ) = ∅. Let
and
14/24
Capacitated domination
De nition
(capacitated dominating set)
G = (V , E ), cap : V → N. (S , dom) is a capacitated dominating set of G , with S ⊆ V ∀v ∈ V , dom(v ) ⊆ N (v ), if: • S is a dominating set of G (in the classical sense); • ∀v ∈ S , |dom(v )| ≤ cap(v ); • ∀v ∈ / S , dom(v ) = ∅. Let
k -Capacitated
Dominating Set: search
S ⊆V
such that
and
|S | ≤ k .
14/24
Ideas of the reduction
2
3
4
1
1
2
3
1
1
2
Input: A graph with capacities
on vertices.
15/24
Ideas of the reduction
2
3
4
1
1
2
3
1
1
2
Input: A graph with capacities
on vertices.
15/24
Ideas of the reduction
2
3
4
1
1
2
3
1
1
2
Input: A graph with capacities
on vertices.
15/24
Ideas of the reduction
2
3
4
1
1
2
3
1
2
1
3
2
Input: A graph with capacities
on vertices.
4
1
1
→
2
3
1
1
2
Transformation: The
incidence graph.
15/24
Ideas of the reduction
2
3
4
1
1
2
3
1
2
1
3
2
Input: A graph with capacities
on vertices.
4
1
1
→
2
3
1
1
2
Transformation: The
incidence graph.
15/24
Ideas of the reduction
2
3
4
1
1
2
3
1
2
1
3
2
Input: A graph with capacities
on vertices.
4
1
1
→
2
3
1
1
2
Transformation: The
incidence graph.
15/24
Some functions on
σ
σ contains arbitrary large gaps between two consecutive elements
16/24
Some functions on
σ
σ contains arbitrary large gaps between two consecutive elements We de ne some functions on
σ:
16/24
Some functions on
σ
σ contains arbitrary large gaps between two consecutive elements We de ne some functions on
• Γ− (x , q ),
σ:
p∈σ p ≥ q;
the minimum element
length at least
x
before
p,
and
s.t. there is a gap of
16/24
Some functions on
σ
σ contains arbitrary large gaps between two consecutive elements We de ne some functions on
• Γ− (x , q ),
σ:
p∈σ p ≥ q; • Γ+ (x , q ), the minimum element p ∈ σ length at least x after p , and p ≥ q ; the minimum element
length at least
x
before
p,
s.t. there is a gap of
and
s.t. there is a gap of
16/24
Some functions on
σ
σ contains arbitrary large gaps between two consecutive elements We de ne some functions on
• Γ− (x , q ),
σ:
p ∈ σ s.t. there p ≥ q; • Γ+ (x , q ), the minimum element p ∈ σ s.t. there length at least x after p , and p ≥ q ; • Γ0 (q ), the minimum element p ∈ σ s.t. p ≥ q . the minimum element
length at least
x
before
p,
is a gap of
and
is a gap of
16/24
Some functions on
σ
σ contains arbitrary large gaps between two consecutive elements We de ne some functions on
• Γ− (x , q ),
σ:
p ∈ σ s.t. there p ≥ q; • Γ+ (x , q ), the minimum element p ∈ σ s.t. there length at least x after p , and p ≥ q ; • Γ0 (q ), the minimum element p ∈ σ s.t. p ≥ q . the minimum element
length at least
x
before
p,
is a gap of
and
is a gap of
Technical constraint: We suppose that there exists a polynomial
pσ
such that a gap of length
t
exists at distance
pσ (t )
in
σ.
16/24
The reduction Let
G
be an instance of
2
3
k -Capacitated
4
1
1
2
3
2
Dominating Set.
1
1
17/24
The reduction Let
G
k -Capacitated Dominating Set. I (G ), incidence graph of G , and add some gadgets:
be an instance of
We start with
2
3
4
1
1
2
3
2
1
1
17/24
The reduction
k -Capacitated Dominating Set. We start with I (G ), incidence graph of G , and add some gadgets: domination (D): forces G to have a dominating set; Let
G
be an instance of
2
3
4
1
1
2
3
2
1
1
17/24
The reduction
k -Capacitated Dominating Set. We start with I (G ), incidence graph of G , and add some gadgets: domination (D): forces G to have a dominating set; capacity (C ): encodes the capacity function cap, and allows Let
G
be an instance of
any selected vertex to be satis ed;
2
3
4
1
1
2
3
2
1
1
17/24
The reduction
k -Capacitated Dominating Set. We start with I (G ), incidence graph of G , and add some gadgets: domination (D): forces G to have a dominating set; capacity (C ): encodes the capacity function cap, and allows Let
G
be an instance of
any selected vertex to be satis ed;
edge-selection (E ):
encodes the domination function dom;
2
3
4
1
1
2
3
2
1
1
17/24
The reduction
k -Capacitated Dominating Set. We start with I (G ), incidence graph of G , and add some gadgets: domination (D): forces G to have a dominating set; capacity (C ): encodes the capacity function cap, and allows Let
G
be an instance of
any selected vertex to be satis ed;
edge-selection (E ): satis ability (S ):
encodes the domination function dom; allows any non-selected vertex to be satis ed;
2
3
4
1
1
2
3
2
1
1
17/24
The reduction
k -Capacitated Dominating Set. We start with I (G ), incidence graph of G , and add some gadgets: domination (D): forces G to have a dominating set; capacity (C ): encodes the capacity function cap, and allows Let
G
be an instance of
any selected vertex to be satis ed;
edge-selection (E ): satis ability (S ): limitation (L):
encodes the domination function dom; allows any non-selected vertex to be satis ed; encodes the parameter
2
3
4
1
1
2
3
2
k; 1
1
17/24
The reduction
k -Capacitated Dominating Set. We start with I (G ), incidence graph of G , and add some gadgets: domination (D): forces G to have a dominating set; capacity (C ): encodes the capacity function cap, and allows Let
G
be an instance of
any selected vertex to be satis ed;
edge-selection (E ): satis ability (S ): limitation (L): force (F ):
encodes the domination function dom; allows any non-selected vertex to be satis ed; encodes the parameter
k;
forces a given vertex to be selected.
2
3
4
1
1
2
3
2
1
1
17/24
The reduction
k -Capacitated Dominating Set. We start with I (G ), incidence graph of G , and add some gadgets: domination (D): forces G to have a dominating set; capacity (C ): encodes the capacity function cap, and allows Let
G
be an instance of
any selected vertex to be satis ed;
edge-selection (E ): satis ability (S ): limitation (L): force (F ):
encodes the domination function dom; allows any non-selected vertex to be satis ed; encodes the parameter
forces a given vertex to be selected.
D S
D C
k;
E
S
D C
E
L
S
D C
S
C
17/24
The gadgets Gadget
force (F ):
forces a given vertex to be selected.
← neighbor in H clique with min σ vertices →
← forced vertex
clique with ασ,ρ vertices → clique with βσ,ρ vertices →
18/24
The gadgets Gadget
force (F ):
forces a given vertex to be selected.
← neighbor in H clique with min σ vertices →
← forced vertex
clique with ασ,ρ vertices → clique with βσ,ρ vertices →
Gadget
domination (D):
forces
G
to have a dominating set.
edge-vertex in I(G) →
← original-vertex v in I(G)
neighbor of v in G → clique with min σ vertices → Γ+ (1, min σ) − min σ forced vertices →
← forced vertices q0 − 2 18/24
The gadgets Gadget
edge-selection (E ):
encodes the domination function dom.
← neighbor in I(G) ← edge-vertex in I(G) ← Γ− (1, q0 + 1) − 1 forced vertices
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The gadgets Gadget
edge-selection (E ):
encodes the domination function dom.
← neighbor in I(G) ← edge-vertex in I(G) ← Γ− (1, q0 + 1) − 1 forced vertices
Gadget
satis ability (S ):
allows any non-selected vertex to be
satis ed.
Γ0 (q0 ) forced vertices
← q0 choosable vertices ← original-vertex in I(G) 19/24
The gadgets
Gadget
capacity (C ):
encodes the capacity function cap, and allows
any selected vertex to be satis ed.
Γ0 (q0 + 1) − 1 forced vertices
← cap(v) vertices ← original-vertex v in I(G) Γ+ degG (v) + q0 , cap(v) − cap(v) − 1 forced vertices
20/24
The gadgets
Gadget
limitation (L):
encodes the parameter
k.
Γ+ |V (G)|, k − k forced vertices original-vertices in I(G)
← central forced vertex c ← k choosable vertices
Γ0 (q0 ) − 1 forced vertices
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Correctness
Correctness of the reduction:
•
Each gadget has small tree-width → tw (H ) = f tw (G )
(at most min σ
+ 1).
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Correctness
Correctness of the reduction:
• •
Each gadget has small tree-width → tw (H ) = f tw (G )
(at most min σ
The number of added vertices depends only on
→ |V (H )| = g |V (G )|
+ 1).
k ≤ n, σ
and
%.
22/24
Correctness
Correctness of the reduction:
• •
Each gadget has small tree-width → tw (H ) = f tw (G )
(at most min σ
The number of added vertices depends only on
→ |V (H )| = g |V (G )|
• H
admits a
[σ, %]-dominating
set i
G
+ 1).
k ≤ n, σ
admits a
and
%.
k -capacitated
dominating set.
22/24
Correctness
Correctness of the reduction:
• •
Each gadget has small tree-width → tw (H ) = f tw (G )
(at most min σ
The number of added vertices depends only on
→ |V (H )| = g |V (G )|
• H
admits a
[σ, %]-dominating
set i
G
+ 1).
k ≤ n, σ
admits a
and
%.
k -capacitated
dominating set.
Theorem If
σ
[C., 2010]
contains arbitrary large gaps between two consecutive elements
% is co nite (and an additional technical constraint), then ∃[σ, %]-Dominating Set is W[1]-hard parameterized by tree-width. and
22/24
1
Some de nitions
2
FPT cases
3
W 1 -hardness
4
Conclusion
[ ]
23/24
Conclusion
∃[σ, %]-Dominating Set is FPT parameterized when σ and % are ultimately periodic.
by tree-width,
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Conclusion
∃[σ, %]-Dominating Set is FPT parameterized when σ and % are ultimately periodic.
by tree-width,
∃[σ, %]-Dominating Set is W[1]-hard parameterized by tree-width, when σ contains arbitrary large gaps between two consecutive elements and % is co nite (and an additional constraint on σ ).
24/24
Conclusion
∃[σ, %]-Dominating Set is FPT parameterized when σ and % are ultimately periodic.
by tree-width,
∃[σ, %]-Dominating Set is W[1]-hard parameterized by tree-width, when σ contains arbitrary large gaps between two consecutive elements and % is co nite (and an additional constraint on σ ). And now?
• •
•
W
[t ]-completeness ;
Other cases of
[σ, %] (e.g.
recursive with bounded gaps);
...
24/24
Conclusion
∃[σ, %]-Dominating Set is FPT parameterized when σ and % are ultimately periodic.
by tree-width,
∃[σ, %]-Dominating Set is W[1]-hard parameterized by tree-width, when σ contains arbitrary large gaps between two consecutive elements and % is co nite (and an additional constraint on σ ). And now?
• •
•
W
[t ]-completeness ;
Other cases of
[σ, %] (e.g.
recursive with bounded gaps);
...
And voilà!
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