Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained...

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Degree-constrained orientations of embedded graphs Yann Disser Jannik Matuschke The Combinatorial Optimization Workshop Aussois, January 9, 2013 Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Transcript of Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained...

Page 1: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 2: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

Graph orientation

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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

Page 3: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

Graph orientation

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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

Page 4: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

Graph orientation

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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

Page 5: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

Graph orientation

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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

Page 6: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

Graph orientation

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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

Page 7: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

Graph orientation

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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

Page 8: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

Graph orientation

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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

Page 9: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

Outline

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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

Page 10: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 11: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 12: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

Outline

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32 1 Uniqueness for planar embeddings

Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Page 13: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 14: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 15: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 16: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 17: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 18: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 19: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 20: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 21: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 22: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 23: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 24: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 25: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 26: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

Outline

2 Bound for general embeddings

Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Page 27: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 28: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 29: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 30: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 31: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 32: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

Outline

[4, 6] 3 Hardness for interval version

Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Page 33: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 34: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 35: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 36: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 37: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 38: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 39: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 40: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 41: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 42: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 43: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 44: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 45: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 46: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 47: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 48: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 49: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 50: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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

Page 51: Degree-constrained orientations of embedded · PDF file... Jannik Matuschke Degree-constrained orientations of embedded ... Existence of primal and dual solution not sufficient 1

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