Degree-constrained orientations of embedded graphs Yann Disser - - PowerPoint PPT Presentation

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

Graph orientation

1 1 2 2

1 3 2 Problem Given graph G and α : V → N0, is there an orientation s.t. every vertex v has in-degree α(v)?

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

Graph orientation

1 1 2 2

1 3 2 Problem Given graph G and α : V → N0, is there an orientation s.t. every vertex v has in-degree α(v)?

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

Graph orientation

1 1 2 2

1 3 2 Problem Given graph G and α : V → N0, is there an orientation s.t. every vertex v has in-degree α(v)?

◮ applications in graph drawing,

evacuation, data structures, theoretical insights ...

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

Graph orientation

1 1 2 2

1 3 2 Problem Given graph G and α : V → N0, is there an orientation s.t. every vertex v has in-degree α(v)?

◮ applications in graph drawing,

evacuation, data structures, theoretical insights ...

◮ solvable in poly-time, even for

general upper and lower bounds

[Hakimi 1965, Frank & Gyárfás 1976]

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

Graph orientation

1 1 2 2

1 3 2 Problem Given graph G and α : V → N0, is there an orientation s.t. every vertex v has in-degree α(v)?

◮ applications in graph drawing,

evacuation, data structures, theoretical insights ...

◮ solvable in poly-time, even for

general upper and lower bounds

[Hakimi 1965, Frank & Gyárfás 1976]

Question What 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 3 2 Problem Given graph G and α : V → N0, is there an orientation s.t. every vertex v has in-degree α(v)?

◮ applications in graph drawing,

evacuation, data structures, theoretical insights ...

◮ solvable in poly-time, even for

general upper and lower bounds

[Hakimi 1965, Frank & Gyárfás 1976]

Question What 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 3 2 Problem Given graph G and α : V → N0, is there an orientation s.t. every vertex v has in-degree α(v)?

◮ applications in graph drawing,

evacuation, data structures, theoretical insights ...

◮ solvable in poly-time, even for

general upper and lower bounds

[Hakimi 1965, Frank & Gyárfás 1976]

Question What 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 3 2 [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 ∗?

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 3 2

1 Uniqueness for planar embeddings

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

Directed cuts and rigid edges

Observation Let S ⊆ V. If

v∈S α(v) = |E[S]|,

then all edges in δ(S) must be

• riented away from S.

S

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

Directed cuts and rigid edges

Observation Let S ⊆ V. If

v∈S α(v) = |E[S]|,

then all edges in δ(S) must be

• riented away from S.

S

+1

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

Directed cuts and rigid edges

Observation Let S ⊆ V. If

v∈S α(v) = |E[S]|,

then all edges in δ(S) must be

• riented away from S.

S

+1

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

Directed cuts and rigid edges

Observation Let S ⊆ V. If

v∈S α(v) = |E[S]|,

then all edges in δ(S) must be

• riented away from S.

S

+|E[S]|

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

Directed cuts and rigid edges

Observation Let S ⊆ V. If

v∈S α(v) = |E[S]|,

then all edges in δ(S) must be

• riented away from S.

S

+|E[S]|

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

Directed cuts and rigid edges

Observation Let S ⊆ V. If

v∈S α(v) = |E[S]|,

then all edges in δ(S) must be

• riented away from S.

S

+|E[S]|

Definiton An 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}.

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

Directed cuts and rigid edges

Observation Let S ⊆ V. If

v∈S α(v) = |E[S]|,

then all edges in δ(S) must be

• riented away from S.

S

+|E[S]|

Definiton An 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}.

Lemma If D feasible orientation, then e ∈ R iff e on directed cut w.r.t. D.

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

Directed cuts and rigid edges

Observation Let S ⊆ V. If

v∈S α(v) = |E[S]|,

then all edges in δ(S) must be

• riented away from S.

S

+|E[S]|

Definiton An 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}.

Lemma If D feasible orientation, then e ∈ R iff e on directed cut w.r.t. D.

◮ Same argumentation in dual graph gives set R∗.

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

Uniqueness of solution in planar embeddings

Theorem If G is a plane graph and there is a globally feasible orientation D, then E = R ˙ ∪ R∗. Thus, D is the unique solution.

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

Uniqueness of solution in planar embeddings

Theorem If G is a plane graph and there is a globally feasible orientation D, then E = R ˙ ∪ R∗. Thus, D is the unique solution. Proof.

◮ e either on directed cycle or directed cut of GD

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

Uniqueness of solution in planar embeddings

Theorem If G is a plane graph and there is a globally feasible orientation D, then E = R ˙ ∪ R∗. Thus, D is the unique solution. Proof.

◮ e either on directed cycle or directed cut of GD ◮ e on di-cut of GD ⇔ e ∈ R

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

Uniqueness of solution in planar embeddings

Theorem If G is a plane graph and there is a globally feasible orientation D, then E = R ˙ ∪ R∗. Thus, D is the unique solution. Proof.

◮ e either on directed cycle or directed cut of GD ◮ e on di-cut of GD ⇔ e ∈ R ◮ e on di-cycle of GD ⇔ e on di-cut of G∗ D ⇔ e ∈ R∗

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

Uniqueness of solution in planar embeddings

Theorem If G is a plane graph and there is a globally feasible orientation D, then E = R ˙ ∪ R∗. Thus, D is the unique solution. Proof.

◮ e either on directed cycle or directed cut of GD ◮ e on di-cut of GD ⇔ e ∈ R ◮ 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 feasible

• rientation 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 ∗

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 ∗

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 ∗ Observation

◮ rank of the system is |V| − 1 + |V ∗| − 1

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 ∗ Observation

◮ rank of the system is |V| − 1 + |V ∗| − 1 ◮ 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

Theorem

◮ There are at most 22g feasible orientations. ◮ All orientations can be found in time O(22g|E|2 + |E|3).

Remark The 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 and

α∗(f) ≤ |δ−

D (f)| ≤ β∗(f) for all f ∈ V ∗?

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 and

α∗(f) ≤ |δ−

D (f)| ≤ β∗(f) for all f ∈ V ∗?

Theorem This 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. Example C1 = x1 ∨ ¬x2 ∨ ¬x3 C2 = x2 ∨ x3 ∨ ¬x4

x1 x2 x3 x4 C1 C2

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. Example C1 = x1 ∨ ¬x2 ∨ ¬x3 C2 = x2 ∨ x3 ∨ ¬x4

x1 x2 x3 x4 C1 C2

Variable gadget

2

1 T

2

F

2

1 T

2

F

2 2 2 T 2

1 F

2

T

2

1 F C1 C2 Cd−1 Cd

[0, 2 deg(xi)]

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. Example C1 = x1 ∨ ¬x2 ∨ ¬x3 C2 = x2 ∨ x3 ∨ ¬x4

x1 x2 x3 x4 C1 C2

Variable gadget

2

1 T

2

F

2

1 T

2

F

2 2 2 T 2

1 F

2

T

2

1 F C1 C2 Cd−1 Cd

[0, 2 deg(xi)]

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. Example C1 = x1 ∨ ¬x2 ∨ ¬x3 C2 = x2 ∨ x3 ∨ ¬x4

x1 x2 x3 x4 C1 C2

Variable gadget

2

1 T

2

F

2

1 T

2

F

2 2 2 T 2

1 F

2

T

2

1 F C1 C2 Cd−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. Example C1 = x1 ∨ ¬x2 ∨ ¬x3 C2 = x2 ∨ x3 ∨ ¬x4

x1 x2 x3 x4 C1 C2

Variable gadget

2

1 T

2

F

2

1 T

2

F

2 2 2 T 2

1 F

2

T

2

1 F C1 C2 Cd−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. Example C1 = x1 ∨ ¬x2 ∨ ¬x3 C2 = x2 ∨ x3 ∨ ¬x4

x1 x2 x3 x4 C1 C2

Variable gadget

2

1 T

2

F

2

1 T

2

F

2 2 2 T 2

1 F

2

T

2

1 F C1 C2 Cd−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. Example C1 = x1 ∨ ¬x2 ∨ ¬x3 C2 = x2 ∨ x3 ∨ ¬x4

x1 x2 x3 x4 C1 C2

Clause gadget

2 1 1 2 1 1 2 1 1

F T F T F T x2 x3 x1

[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. Example C1 = x1 ∨ ¬x2 ∨ ¬x3 C2 = x2 ∨ x3 ∨ ¬x4

x1 x2 x3 x4 C1 C2

Clause gadget

2 1 1 2 1 1 2 1 1

F T F T F T x2 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. Example C1 = x1 ∨ ¬x2 ∨ ¬x3 C2 = x2 ∨ x3 ∨ ¬x4

x1 x2 x3 x4 C1 C2

Edge gadget

2

F

2

T 1

1

F

1

T [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. Example C1 = x1 ∨ ¬x2 ∨ ¬x3 C2 = x2 ∨ x3 ∨ ¬x4

x1 x2 x3 x4 C1 C2

Edge gadget

2

F

2

T 1

1

F

1

T [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. Example C1 = x1 ∨ ¬x2 ∨ ¬x3 C2 = x2 ∨ x3 ∨ ¬x4

x1 x2 x3 x4 C1 C2

Edge gadget

2

F

2

T 1

1

F

1

T [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. Example C1 = x1 ∨ ¬x2 ∨ ¬x3 C2 = x2 ∨ x3 ∨ ¬x4

x1 x2 x3 x4 C1 C2

Edge gadget (negated)

2

F

2

T 1

1

F

1

T

2 2

1 [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. Example C1 = x1 ∨ ¬x2 ∨ ¬x3 C2 = x2 ∨ x3 ∨ ¬x4

x1 x2 x3 x4 C1 C2

Edge gadget (negated)

2

F

2

T 1

1

F

1

T

2 2

1 [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. Example C1 = x1 ∨ ¬x2 ∨ ¬x3 C2 = x2 ∨ x3 ∨ ¬x4

x1 x2 x3 x4 C1 C2

Edge gadget (negated)

2

F

2

T 1

1

F

1

T

2 2

1 [0, 4] [0, 4]

variable clause

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

Summary

The primal-dual orientation problem ...

◮ has at most 22g solutions if degrees are fixed numbers

(enumeration in O(22g|E|2 + |E|3)).

◮ is NP-hard if only upper and lower bounds are given, even

for planar embeddings.

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

Summary

The primal-dual orientation problem ...

◮ has at most 22g solutions if degrees are fixed numbers

(enumeration in O(22g|E|2 + |E|3)).

◮ is NP-hard if only upper and lower bounds are given, even

for planar embeddings. Open question

◮ Can we find a feasible orientation in time poly(g, |E|)? Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Summary

The primal-dual orientation problem ...

◮ has at most 22g solutions if degrees are fixed numbers

(enumeration in O(22g|E|2 + |E|3)).

◮ is NP-hard if only upper and lower bounds are given, even

for planar embeddings. Open question

◮ Can we find a feasible orientation in time poly(g, |E|)?

Thank you!

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