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 Problem Given graph G and α : V → N 0 , is 1 there an orientation s.t. every vertex v has in-degree α ( v ) ? 2 3 2 1 1 2 0 Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Graph orientation Problem Given graph G and α : V → N 0 , is 1 there an orientation s.t. every vertex v has in-degree α ( v ) ? 2 3 2 1 1 2 0 Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Graph orientation Problem Given graph G and α : V → N 0 , is 1 there an orientation s.t. every vertex v has in-degree α ( v ) ? 2 3 ◮ applications in graph drawing, 2 evacuation, data structures, 1 1 2 theoretical insights ... 0 Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Graph orientation Problem Given graph G and α : V → N 0 , is 1 there an orientation s.t. every vertex v has in-degree α ( v ) ? 2 3 ◮ applications in graph drawing, 2 evacuation, data structures, 1 1 2 theoretical insights ... ◮ solvable in poly-time, even for 0 general upper and lower bounds [Hakimi 1965, Frank & Gyárfás 1976] Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Graph orientation Problem Given graph G and α : V → N 0 , is 1 there an orientation s.t. every vertex v has in-degree α ( v ) ? 2 3 ◮ applications in graph drawing, 2 evacuation, data structures, 1 1 2 theoretical insights ... ◮ solvable in poly-time, even for 0 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 Problem Given graph G and α : V → N 0 , is 1 there an orientation s.t. every vertex v has in-degree α ( v ) ? 2 3 ◮ applications in graph drawing, 2 evacuation, data structures, 1 1 2 theoretical insights ... ◮ solvable in poly-time, even for 0 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 Problem Given graph G and α : V → N 0 , is 1 there an orientation s.t. every vertex v has in-degree α ( v ) ? 2 3 ◮ applications in graph drawing, 2 evacuation, data structures, 1 1 2 theoretical insights ... ◮ solvable in poly-time, even for 0 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 2 3 Uniqueness for planar embeddings 1 2 1 1 2 Bound for general embeddings 2 3 Hardness for interval version [ 4 , 6 ] Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Problem definition Primal-dual orientation problem Input: embedded graph G = ( V , E ) , α : V → N 0 , α ∗ : V ∗ → N 0 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 → N 0 , α ∗ : V ∗ → N 0 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 3 2 Uniqueness for planar embeddings 1 2 1 1 2 Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Directed cuts and rigid edges S Observation Let S ⊆ V . If � v ∈ S α ( v ) = | E [ S ] | , then all edges in δ ( S ) must be oriented away from S . Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Directed cuts and rigid edges S Observation + 1 Let S ⊆ V . If � v ∈ S α ( v ) = | E [ S ] | , then all edges in δ ( S ) must be oriented away from S . Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Directed cuts and rigid edges S Observation Let S ⊆ V . If � v ∈ S α ( v ) = | E [ S ] | , then all edges in δ ( S ) must be + 1 oriented away from S . Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Directed cuts and rigid edges S Observation Let S ⊆ V . If � v ∈ S α ( v ) = | E [ S ] | , + | E [ S ] | then all edges in δ ( S ) must be oriented away from S . Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Directed cuts and rigid edges S Observation Let S ⊆ V . If � v ∈ S α ( v ) = | E [ S ] | , + | E [ S ] | then all edges in δ ( S ) must be oriented away from S . Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Directed cuts and rigid edges S Observation Let S ⊆ V . If � v ∈ S α ( v ) = | E [ S ] | , + | E [ S ] | then all edges in δ ( S ) must be oriented away from 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 S Observation Let S ⊆ V . If � v ∈ S α ( v ) = | E [ S ] | , + | E [ S ] | then all edges in δ ( S ) must be oriented away from 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 S Observation Let S ⊆ V . If � v ∈ S α ( v ) = | E [ S ] | , + | E [ S ] | then all edges in δ ( S ) must be oriented away from 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 ∪ R ∗ . Thus, D is the unique solution. D , then 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 ∪ R ∗ . Thus, D is the unique solution. D , then E = R ˙ Proof. ◮ e either on directed cycle or directed cut of G D 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 ∪ R ∗ . Thus, D is the unique solution. D , then E = R ˙ Proof. ◮ e either on directed cycle or directed cut of G D ◮ 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 ∪ R ∗ . Thus, D is the unique solution. D , then E = R ˙ Proof. ◮ e either on directed cycle or directed cut of G D ◮ e on di-cut of G D ⇔ e ∈ R ◮ e on di-cycle of G D ⇔ 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 ∪ R ∗ . Thus, D is the unique solution. D , then E = R ˙ Proof. ◮ e either on directed cycle or directed cut of G D ◮ e on di-cut of G D ⇔ e ∈ R ◮ e on di-cycle of G D ⇔ e on di-cut of G ∗ D ⇔ e ∈ R ∗ Corollary We can find D in time O ( | E | 3 / 2 ) by computing a feasible orientation in G and G ∗ and combining their rigid parts. Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs

Outline Bound for general embeddings 2 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 ? � � x ( e ) + | δ − x ( e ) − D ( v ) | = α ( v ) ∀ v ∈ V e ∈ δ + D ( v ) e ∈ δ − D ( v ) x ( e ) + | δ − � � α ∗ ( f ) ∀ f ∈ V ∗ x ( e ) − D ( f ) | = e ∈ δ + D ( f ) e ∈ δ − D ( f ) Yann Disser, Jannik Matuschke Degree-constrained orientations of embedded graphs