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Degree-constrained orientations ofembedded graphs
Yann Disser Jannik Matuschke
The Combinatorial Optimization WorkshopAussois, January 9, 2013
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Graph orientation
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1 2
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1
32
0
ProblemGiven graph G and α : V → N0, isthere an orientation s.t. every vertex vhas in-degree α(v)?
I applications in graph drawing,evacuation, data structures,theoretical insights ...
I solvable in poly-time, even forgeneral upper and lower bounds[Hakimi 1965, Frank & Gyárfás 1976]
QuestionWhat if we have degree-constraints in primal and dual graph?
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Graph orientation
1
1 2
2
1
32
0
ProblemGiven graph G and α : V → N0, isthere an orientation s.t. every vertex vhas in-degree α(v)?
I applications in graph drawing,evacuation, data structures,theoretical insights ...
I solvable in poly-time, even forgeneral upper and lower bounds[Hakimi 1965, Frank & Gyárfás 1976]
QuestionWhat if we have degree-constraints in primal and dual graph?
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Graph orientation
1
1 2
2
1
32
0
ProblemGiven graph G and α : V → N0, isthere an orientation s.t. every vertex vhas in-degree α(v)?
I applications in graph drawing,evacuation, data structures,theoretical insights ...
I solvable in poly-time, even forgeneral upper and lower bounds[Hakimi 1965, Frank & Gyárfás 1976]
QuestionWhat if we have degree-constraints in primal and dual graph?
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Graph orientation
1
1 2
2
1
32
0
ProblemGiven graph G and α : V → N0, isthere an orientation s.t. every vertex vhas in-degree α(v)?
I applications in graph drawing,evacuation, data structures,theoretical insights ...
I solvable in poly-time, even forgeneral upper and lower bounds[Hakimi 1965, Frank & Gyárfás 1976]
QuestionWhat if we have degree-constraints in primal and dual graph?
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Graph orientation
1
1 2
2
1
32
0
ProblemGiven graph G and α : V → N0, isthere an orientation s.t. every vertex vhas in-degree α(v)?
I applications in graph drawing,evacuation, data structures,theoretical insights ...
I solvable in poly-time, even forgeneral upper and lower bounds[Hakimi 1965, Frank & Gyárfás 1976]
QuestionWhat if we have degree-constraints in primal and dual graph?
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Graph orientation
1
1 2
2
1
32
0
ProblemGiven graph G and α : V → N0, isthere an orientation s.t. every vertex vhas in-degree α(v)?
I applications in graph drawing,evacuation, data structures,theoretical insights ...
I solvable in poly-time, even forgeneral upper and lower bounds[Hakimi 1965, Frank & Gyárfás 1976]
QuestionWhat if we have degree-constraints in primal and dual graph?
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Graph orientation
1
1 2
2
1
32
0
ProblemGiven graph G and α : V → N0, isthere an orientation s.t. every vertex vhas in-degree α(v)?
I applications in graph drawing,evacuation, data structures,theoretical insights ...
I solvable in poly-time, even forgeneral upper and lower bounds[Hakimi 1965, Frank & Gyárfás 1976]
QuestionWhat if we have degree-constraints in primal and dual graph?
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Outline
1
1 2
2
1
32
[4, 6]
1 Uniqueness for planar embeddings
2 Bound for general embeddings
3 Hardness for interval version
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Problem definition
Primal-dual orientation problem
Input: embedded graph G = (V ,E),α : V → N0, α∗ : V ∗ → N0
Task: Is there orientation D, s.t.|δ−D (v)| = α(v) for all v ∈ V and|δ−D (f )| = α∗(f ) for all f ∈ V ∗?
Existence of primal and dual solution not sufficient
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Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Problem definition
Primal-dual orientation problem
Input: embedded graph G = (V ,E),α : V → N0, α∗ : V ∗ → N0
Task: Is there orientation D, s.t.|δ−D (v)| = α(v) for all v ∈ V and|δ−D (f )| = α∗(f ) for all f ∈ V ∗?
Existence of primal and dual solution not sufficient
1
1
1
1
1
1
1
1
1
1
1
1
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Outline
1
1 2
2
1
32 1 Uniqueness for planar embeddings
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Directed cuts and rigid edges
ObservationLet S ⊆ V . If
∑v∈S α(v) = |E [S]|,
then all edges in δ(S) must beoriented away from S.
S
DefinitonAn edge is called rigid if e ∈ δ(S) for some S ⊆ V with∑
v∈S α(v) = |E [S]|. R := {e ∈ E : e is rigid}.
LemmaIf D feasible orientation, then e ∈ R iff e on directed cut w.r.t. D.
I Same argumentation in dual graph gives set R∗.
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Directed cuts and rigid edges
ObservationLet S ⊆ V . If
∑v∈S α(v) = |E [S]|,
then all edges in δ(S) must beoriented away from S.
S+1
DefinitonAn edge is called rigid if e ∈ δ(S) for some S ⊆ V with∑
v∈S α(v) = |E [S]|. R := {e ∈ E : e is rigid}.
LemmaIf D feasible orientation, then e ∈ R iff e on directed cut w.r.t. D.
I Same argumentation in dual graph gives set R∗.
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Directed cuts and rigid edges
ObservationLet S ⊆ V . If
∑v∈S α(v) = |E [S]|,
then all edges in δ(S) must beoriented away from S.
S
+1
DefinitonAn edge is called rigid if e ∈ δ(S) for some S ⊆ V with∑
v∈S α(v) = |E [S]|. R := {e ∈ E : e is rigid}.
LemmaIf D feasible orientation, then e ∈ R iff e on directed cut w.r.t. D.
I Same argumentation in dual graph gives set R∗.
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Directed cuts and rigid edges
ObservationLet S ⊆ V . If
∑v∈S α(v) = |E [S]|,
then all edges in δ(S) must beoriented away from S.
S
+|E [S]|
DefinitonAn edge is called rigid if e ∈ δ(S) for some S ⊆ V with∑
v∈S α(v) = |E [S]|. R := {e ∈ E : e is rigid}.
LemmaIf D feasible orientation, then e ∈ R iff e on directed cut w.r.t. D.
I Same argumentation in dual graph gives set R∗.
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Directed cuts and rigid edges
ObservationLet S ⊆ V . If
∑v∈S α(v) = |E [S]|,
then all edges in δ(S) must beoriented away from S.
S
+|E [S]|
DefinitonAn edge is called rigid if e ∈ δ(S) for some S ⊆ V with∑
v∈S α(v) = |E [S]|. R := {e ∈ E : e is rigid}.
LemmaIf D feasible orientation, then e ∈ R iff e on directed cut w.r.t. D.
I Same argumentation in dual graph gives set R∗.
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Directed cuts and rigid edges
ObservationLet S ⊆ V . If
∑v∈S α(v) = |E [S]|,
then all edges in δ(S) must beoriented away from S.
S
+|E [S]|
DefinitonAn edge is called rigid if e ∈ δ(S) for some S ⊆ V with∑
v∈S α(v) = |E [S]|. R := {e ∈ E : e is rigid}.
LemmaIf D feasible orientation, then e ∈ R iff e on directed cut w.r.t. D.
I Same argumentation in dual graph gives set R∗.
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Directed cuts and rigid edges
ObservationLet S ⊆ V . If
∑v∈S α(v) = |E [S]|,
then all edges in δ(S) must beoriented away from S.
S
+|E [S]|
DefinitonAn edge is called rigid if e ∈ δ(S) for some S ⊆ V with∑
v∈S α(v) = |E [S]|. R := {e ∈ E : e is rigid}.
LemmaIf D feasible orientation, then e ∈ R iff e on directed cut w.r.t. D.
I Same argumentation in dual graph gives set R∗.
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Directed cuts and rigid edges
ObservationLet S ⊆ V . If
∑v∈S α(v) = |E [S]|,
then all edges in δ(S) must beoriented away from S.
S
+|E [S]|
DefinitonAn edge is called rigid if e ∈ δ(S) for some S ⊆ V with∑
v∈S α(v) = |E [S]|. R := {e ∈ E : e is rigid}.
LemmaIf D feasible orientation, then e ∈ R iff e on directed cut w.r.t. D.
I Same argumentation in dual graph gives set R∗.
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Uniqueness of solution in planar embeddings
TheoremIf G is a plane graph and there is a globally feasible orientationD, then E = R ∪̇R∗. Thus, D is the unique solution.
Proof.I e either on directed cycle or directed cut of GD
I e on di-cut of GD ⇔ e ∈ RI e on di-cycle of GD ⇔ e on di-cut of G∗D ⇔ e ∈ R∗
Corollary
We can find D in time O(|E |3/2) by computing a feasibleorientation in G and G∗ and combining their rigid parts.
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Uniqueness of solution in planar embeddings
TheoremIf G is a plane graph and there is a globally feasible orientationD, then E = R ∪̇R∗. Thus, D is the unique solution.
Proof.I e either on directed cycle or directed cut of GD
I e on di-cut of GD ⇔ e ∈ RI e on di-cycle of GD ⇔ e on di-cut of G∗D ⇔ e ∈ R∗
Corollary
We can find D in time O(|E |3/2) by computing a feasibleorientation in G and G∗ and combining their rigid parts.
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Uniqueness of solution in planar embeddings
TheoremIf G is a plane graph and there is a globally feasible orientationD, then E = R ∪̇R∗. Thus, D is the unique solution.
Proof.I e either on directed cycle or directed cut of GD
I e on di-cut of GD ⇔ e ∈ R
I e on di-cycle of GD ⇔ e on di-cut of G∗D ⇔ e ∈ R∗
Corollary
We can find D in time O(|E |3/2) by computing a feasibleorientation in G and G∗ and combining their rigid parts.
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Uniqueness of solution in planar embeddings
TheoremIf G is a plane graph and there is a globally feasible orientationD, then E = R ∪̇R∗. Thus, D is the unique solution.
Proof.I e either on directed cycle or directed cut of GD
I e on di-cut of GD ⇔ e ∈ RI e on di-cycle of GD ⇔ e on di-cut of G∗D ⇔ e ∈ R∗
Corollary
We can find D in time O(|E |3/2) by computing a feasibleorientation in G and G∗ and combining their rigid parts.
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Uniqueness of solution in planar embeddings
TheoremIf G is a plane graph and there is a globally feasible orientationD, then E = R ∪̇R∗. Thus, D is the unique solution.
Proof.I e either on directed cycle or directed cut of GD
I e on di-cut of GD ⇔ e ∈ RI e on di-cycle of GD ⇔ e on di-cut of G∗D ⇔ e ∈ R∗
Corollary
We can find D in time O(|E |3/2) by computing a feasibleorientation in G and G∗ and combining their rigid parts.
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Outline
2 Bound for general embeddings
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Linear algebra for general embeddings
Linear algebra formulation
D: arbitrary orientation x(e) ∈ {0,1}: reverse edge e?∑e∈δ+D (v)
x(e)−∑
e∈δ−D (v)
x(e) + |δ−D (v)| = α(v) ∀v ∈ V∑e∈δ+D (f )
x(e)−∑
e∈δ−D (f )
x(e) + |δ−D (f )| = α∗(f ) ∀f ∈ V ∗
ObservationI rank of the system is |V | − 1 + |V ∗| − 1I all solutions in space of dimension|E | − |V | − |V ∗| − 2 = 2g (Euler’s formula)
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Linear algebra for general embeddings
Linear algebra formulation
D: arbitrary orientation x(e) ∈ {0,1}: reverse edge e?∑e∈δ+D (v)
x(e)−∑
e∈δ−D (v)
x(e) = α(v)− |δ−D (v)| ∀v ∈ V∑e∈δ+D (f )
x(e)−∑
e∈δ−D (f )
x(e) = α∗(f )− |δ−D (f )| ∀f ∈ V ∗
ObservationI rank of the system is |V | − 1 + |V ∗| − 1I all solutions in space of dimension|E | − |V | − |V ∗| − 2 = 2g (Euler’s formula)
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Linear algebra for general embeddings
Linear algebra formulation
D: arbitrary orientation x(e) ∈ {0,1}: reverse edge e?∑e∈δ+D (v)
x(e)−∑
e∈δ−D (v)
x(e) = α(v)− |δ−D (v)| ∀v ∈ V∑e∈δ+D (f )
x(e)−∑
e∈δ−D (f )
x(e) = α∗(f )− |δ−D (f )| ∀f ∈ V ∗
ObservationI rank of the system is |V | − 1 + |V ∗| − 1
I all solutions in space of dimension|E | − |V | − |V ∗| − 2 = 2g (Euler’s formula)
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Linear algebra for general embeddings
Linear algebra formulation
D: arbitrary orientation x(e) ∈ {0,1}: reverse edge e?∑e∈δ+D (v)
x(e)−∑
e∈δ−D (v)
x(e) = α(v)− |δ−D (v)| ∀v ∈ V∑e∈δ+D (f )
x(e)−∑
e∈δ−D (f )
x(e) = α∗(f )− |δ−D (f )| ∀f ∈ V ∗
ObservationI rank of the system is |V | − 1 + |V ∗| − 1I all solutions in space of dimension|E | − |V | − |V ∗| − 2 = 2g (Euler’s formula)
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Bound on the number of solutions
TheoremI There are at most 22g feasible orientations.I All orientations can be found in time O(22g |E |2 + |E |3).
RemarkThe bound on the number of orientations is tight.
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Outline
[4, 6] 3 Hardness for interval version
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Orientations with upper and lower bounds
Bounded primal-dual orientation problem
Input: embedded graph G = (V ,E),α, β : V → N0, α∗, β∗ : V ∗ → N0
Task: Is there orientation D, s.t.α(v) ≤ |δ−D (v)| ≤ β(v) for all v ∈ V andα∗(f ) ≤ |δ−D (f )| ≤ β∗(f ) for all f ∈ V ∗?
TheoremThis problem is NP-hard, even for planar embeddings.
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Orientations with upper and lower bounds
Bounded primal-dual orientation problem
Input: embedded graph G = (V ,E),α, β : V → N0, α∗, β∗ : V ∗ → N0
Task: Is there orientation D, s.t.α(v) ≤ |δ−D (v)| ≤ β(v) for all v ∈ V andα∗(f ) ≤ |δ−D (f )| ≤ β∗(f ) for all f ∈ V ∗?
TheoremThis problem is NP-hard, even for planar embeddings.
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Reduction from PLANAR 3-SAT
PLANAR 3-SAT
Instance of 3-SAT s.t. the induced bipartite graph is planar.
ExampleC1 = x1 ∨ ¬x2 ∨ ¬x3
C2 = x2 ∨ x3 ∨ ¬x4
x1 x2 x3 x4
C1 C2
Variable gadget
21T
2F
21
T
2F
22
2 T
21
F
2T
21 F
C1
C2Cd−1
Cd
[0,2 deg(xi)]
�: true: false
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Reduction from PLANAR 3-SAT
PLANAR 3-SAT
Instance of 3-SAT s.t. the induced bipartite graph is planar.
ExampleC1 = x1 ∨ ¬x2 ∨ ¬x3
C2 = x2 ∨ x3 ∨ ¬x4
x1 x2 x3 x4
C1 C2
Variable gadget
21T
2F
21
T
2F
22
2 T
21
F
2T
21 F
C1
C2Cd−1
Cd
[0,2 deg(xi)]
�: true: false
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Reduction from PLANAR 3-SAT
PLANAR 3-SAT
Instance of 3-SAT s.t. the induced bipartite graph is planar.
ExampleC1 = x1 ∨ ¬x2 ∨ ¬x3
C2 = x2 ∨ x3 ∨ ¬x4
x1 x2 x3 x4
C1 C2
Variable gadget
21T
2F
21
T
2F
22
2 T
21
F
2T
21 F
C1
C2Cd−1
Cd
[0,2 deg(xi)]
�: true: false
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Reduction from PLANAR 3-SAT
PLANAR 3-SAT
Instance of 3-SAT s.t. the induced bipartite graph is planar.
ExampleC1 = x1 ∨ ¬x2 ∨ ¬x3
C2 = x2 ∨ x3 ∨ ¬x4
x1 x2 x3 x4
C1 C2
Variable gadget
21T
2F
21
T
2F
22
2 T
21
F
2T
21 F
C1
C2Cd−1
Cd
[0,2 deg(xi)]
�: true: false
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Reduction from PLANAR 3-SAT
PLANAR 3-SAT
Instance of 3-SAT s.t. the induced bipartite graph is planar.
ExampleC1 = x1 ∨ ¬x2 ∨ ¬x3
C2 = x2 ∨ x3 ∨ ¬x4
x1 x2 x3 x4
C1 C2
Variable gadget
21T
2F
21
T
2F
22
2 T
21
F
2T
21 F
C1
C2Cd−1
Cd
[0,2 deg(xi)]
�: true: false
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Reduction from PLANAR 3-SAT
PLANAR 3-SAT
Instance of 3-SAT s.t. the induced bipartite graph is planar.
ExampleC1 = x1 ∨ ¬x2 ∨ ¬x3
C2 = x2 ∨ x3 ∨ ¬x4
x1 x2 x3 x4
C1 C2
Variable gadget
21T
2F
21
T
2F
22
2 T
21
F
2T
21 F
C1
C2Cd−1
Cd
[0,2 deg(xi)]
�: true: false
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Reduction from PLANAR 3-SAT
PLANAR 3-SAT
Instance of 3-SAT s.t. the induced bipartite graph is planar.
ExampleC1 = x1 ∨ ¬x2 ∨ ¬x3
C2 = x2 ∨ x3 ∨ ¬x4
x1 x2 x3 x4
C1 C2
Clause gadget
21
1
2
11
2
1
1F
T
FT
F
Tx2
x3
x1
false
false
true
[4,6]
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Reduction from PLANAR 3-SAT
PLANAR 3-SAT
Instance of 3-SAT s.t. the induced bipartite graph is planar.
ExampleC1 = x1 ∨ ¬x2 ∨ ¬x3
C2 = x2 ∨ x3 ∨ ¬x4
x1 x2 x3 x4
C1 C2
Clause gadget
21
1
2
11
2
1
1F
T
FT
F
Tx2
x3
x1false
false
true
[4,6]
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Reduction from PLANAR 3-SAT
PLANAR 3-SAT
Instance of 3-SAT s.t. the induced bipartite graph is planar.
ExampleC1 = x1 ∨ ¬x2 ∨ ¬x3
C2 = x2 ∨ x3 ∨ ¬x4
x1 x2 x3 x4
C1 C2
Edge gadget
(negated)
2F
2T
1
1F
1T
[0, 4]
2 21
[0, 4]
[0, 4]
variable
clause
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Reduction from PLANAR 3-SAT
PLANAR 3-SAT
Instance of 3-SAT s.t. the induced bipartite graph is planar.
ExampleC1 = x1 ∨ ¬x2 ∨ ¬x3
C2 = x2 ∨ x3 ∨ ¬x4
x1 x2 x3 x4
C1 C2
Edge gadget
(negated)
2F
2T
1
1F
1T
[0, 4]
2 21
[0, 4]
[0, 4]
variable
clause
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Reduction from PLANAR 3-SAT
PLANAR 3-SAT
Instance of 3-SAT s.t. the induced bipartite graph is planar.
ExampleC1 = x1 ∨ ¬x2 ∨ ¬x3
C2 = x2 ∨ x3 ∨ ¬x4
x1 x2 x3 x4
C1 C2
Edge gadget
(negated)
2F
2T
1
1F
1T
[0, 4]
2 21
[0, 4]
[0, 4]
variable
clause
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Reduction from PLANAR 3-SAT
PLANAR 3-SAT
Instance of 3-SAT s.t. the induced bipartite graph is planar.
ExampleC1 = x1 ∨ ¬x2 ∨ ¬x3
C2 = x2 ∨ x3 ∨ ¬x4
x1 x2 x3 x4
C1 C2
Edge gadget (negated)
2F
2T
1
1F
1T
[0, 4]
2 21
[0, 4]
[0, 4]
variable
clause
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Reduction from PLANAR 3-SAT
PLANAR 3-SAT
Instance of 3-SAT s.t. the induced bipartite graph is planar.
ExampleC1 = x1 ∨ ¬x2 ∨ ¬x3
C2 = x2 ∨ x3 ∨ ¬x4
x1 x2 x3 x4
C1 C2
Edge gadget (negated)
2F
2T
1
1F
1T
[0, 4]
2 21
[0, 4]
[0, 4]
variable
clause
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Reduction from PLANAR 3-SAT
PLANAR 3-SAT
Instance of 3-SAT s.t. the induced bipartite graph is planar.
ExampleC1 = x1 ∨ ¬x2 ∨ ¬x3
C2 = x2 ∨ x3 ∨ ¬x4
x1 x2 x3 x4
C1 C2
Edge gadget (negated)
2F
2T
1
1F
1T
[0, 4]
2 21
[0, 4]
[0, 4]
variable
clause
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Summary
The primal-dual orientation problem ...I has at most 22g solutions if degrees are fixed numbers
(enumeration in O(22g |E |2 + |E |3)).I is NP-hard if only upper and lower bounds are given, even
for planar embeddings.
Open questionI Can we find a feasible orientation in time poly(g, |E |)?
Thank you!
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Summary
The primal-dual orientation problem ...I has at most 22g solutions if degrees are fixed numbers
(enumeration in O(22g |E |2 + |E |3)).I is NP-hard if only upper and lower bounds are given, even
for planar embeddings.
Open questionI Can we find a feasible orientation in time poly(g, |E |)?
Thank you!
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs
Summary
The primal-dual orientation problem ...I has at most 22g solutions if degrees are fixed numbers
(enumeration in O(22g |E |2 + |E |3)).I is NP-hard if only upper and lower bounds are given, even
for planar embeddings.
Open questionI Can we find a feasible orientation in time poly(g, |E |)?
Thank you!
Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs