Carving-width, tree-width and area-optimal planar graph drawing - - PowerPoint PPT Presentation

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Carving-width, tree-width and area-optimal planar graph drawing - - PowerPoint PPT Presentation

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Carving-width, tree-width and area-optimal planar graph drawing Therese Biedl University of Waterloo biedl@uwaterloo.ca May 5, 2014 Therese Biedl Carving-width, tree-width and graph drawing 1 / 19 Graph Drawing Given: A graph G = ( V , E

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Carving-width, tree-width and area-optimal planar graph drawing

Therese Biedl University of Waterloo biedl@uwaterloo.ca May 5, 2014

Therese Biedl Carving-width, tree-width and graph drawing 1 / 19

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Graph Drawing

Given: A graph G = (V , E).

Therese Biedl Carving-width, tree-width and graph drawing 2 / 19

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Graph Drawing

Given: A graph G = (V , E). Want: a pretty drawing (created by an algorithm.)

Therese Biedl Carving-width, tree-width and graph drawing 2 / 19

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Graph Drawing

Given: A graph G = (V , E). Want: a pretty drawing (created by an algorithm.) What is “pretty”?

Therese Biedl Carving-width, tree-width and graph drawing 2 / 19

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Graph Drawing

Given: A graph G = (V , E). Want: a pretty drawing (created by an algorithm.) What is “pretty”? This talk: Straight-line drawing, planar graph, no crossing.

Therese Biedl Carving-width, tree-width and graph drawing 2 / 19

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Previous work

Every planar graph has a planar straight-line drawing. (Wagner’36, F´ ary’48, Stein’50 )

Therese Biedl Carving-width, tree-width and graph drawing 3 / 19

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Previous work

Every planar graph has a planar straight-line drawing. (Wagner’36, F´ ary’48, Stein’50 ) Want small integer coordinates.

height 8 8 rows =

Therese Biedl Carving-width, tree-width and graph drawing 3 / 19

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Previous work

Every planar graph has a planar straight-line drawing. (Wagner’36, F´ ary’48, Stein’50 ) Want small integer coordinates.

height 8 8 rows =

Theorem (de Fraysseix, Pach, Pollack 1990; Schnyder 1990) Every planar graph can be drawn in an O(n) × O(n)-grid.

Therese Biedl Carving-width, tree-width and graph drawing 3 / 19

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Previous work

Theorem (de Fraysseix, Pach, Pollack 1990; Schnyder 1990) Every planar graph can be drawn in an O(n) × O(n)-grid. Theorem (de Fraysseix, Pach, Pollack 1988) Some planar graphs need an Ω(n) × Ω(n)-grid.

Therese Biedl Carving-width, tree-width and graph drawing 4 / 19

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Previous work

Theorem (de Fraysseix, Pach, Pollack 1990; Schnyder 1990) Every planar graph can be drawn in an O(n) × O(n)-grid. Theorem (de Fraysseix, Pach, Pollack 1988) Some planar graphs need an Ω(n) × Ω(n)-grid. Many planar graph drawing results since:

Drawing in an n × n-grid (Schnyder 1990) Drawing in a 2

3n × O(n)-grid (Chrobak, Nakano 1994)

Drawing in area 8

9n2 (Brandenburg 2008)

4-connected planar graphs in area 1

4n2 (Miura et al. 1999)

Trees in O(n log n) area (easy). Outer-planar graphs in O(n log n) area (B. 2002) Series-parallel graphs in O(n1.5) area (B. 2009) and many more like that....

Therese Biedl Carving-width, tree-width and graph drawing 4 / 19

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Area-optimal planar graph drawing?

Most graph drawing results have the form: Algorithm draws graph in class X with area f (n). Some graph in class X needs area Ω(f (n)).

Therese Biedl Carving-width, tree-width and graph drawing 5 / 19

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Area-optimal planar graph drawing?

Most graph drawing results have the form: Algorithm draws graph in class X with area f (n). Some graph in class X needs area Ω(f (n)). Very few graph drawing results of the form: Algorithm draws G with optimal area for G. (Or at least within constant factor.)

Therese Biedl Carving-width, tree-width and graph drawing 5 / 19

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Area-optimal planar graph drawing?

Problem DrawOptArea: Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A?

Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

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Area-optimal planar graph drawing?

Problem DrawOptArea: Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? NP-hard (Krug & Wagner 2007)

Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

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Area-optimal planar graph drawing?

Problem DrawOptArea: Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? NP-hard (Krug & Wagner 2007) Polynomial for Apollonian networks (Mondal et al., 2011) (planar 3-tree, maximal chordal planar, ...)

Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

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Area-optimal planar graph drawing?

Problem DrawOptArea: Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? NP-hard (Krug & Wagner 2007) Polynomial for Apollonian networks (Mondal et al., 2011) (planar 3-tree, maximal chordal planar, ...)

Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

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Area-optimal planar graph drawing?

Problem DrawOptArea: Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? NP-hard (Krug & Wagner 2007) Polynomial for Apollonian networks (Mondal et al., 2011) (planar 3-tree, maximal chordal planar, ...)

Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

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Area-optimal planar graph drawing?

Problem DrawOptArea: Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? NP-hard (Krug & Wagner 2007) Polynomial for Apollonian networks (Mondal et al., 2011) (planar 3-tree, maximal chordal planar, ...)

Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

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Area-optimal planar graph drawing?

Problem DrawOptArea: Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? NP-hard (Krug & Wagner 2007) Polynomial for Apollonian networks (Mondal et al., 2011) (planar 3-tree, maximal chordal planar, ...)

Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

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Area-optimal planar graph drawing?

Problem DrawOptArea: Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? NP-hard (Krug & Wagner 2007) Polynomial for Apollonian networks (Mondal et al., 2011) (planar 3-tree, maximal chordal planar, ...)

Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

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Area-optimal planar graph drawing?

Problem DrawOptArea: Given a planar graph G and a constant A, does G have a planar straight-line drawing of area at most A? NP-hard (Krug & Wagner 2007) Polynomial for Apollonian networks (Mondal et al., 2011) (planar 3-tree, maximal chordal planar, ...) Why are Apollonian networks easy? treewidth 3? faces are triangles? both?

Therese Biedl Carving-width, tree-width and graph drawing 6 / 19

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Treewidth

Definition Treewidth tw(G) = min{k: G has chordal super-graph with clique-size k} −1

Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

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Treewidth

Definition Treewidth tw(G) = min{k: G has chordal super-graph with clique-size k} −1

Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

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Treewidth

Definition Treewidth tw(G) = min{k: G has chordal super-graph with clique-size k} −1

Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

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Treewidth

Definition Treewidth tw(G) = min{k: G has chordal super-graph with clique-size k} −1

Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

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Treewidth

Definition Treewidth tw(G) = min{k: G has chordal super-graph with clique-size k} −1

Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

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Treewidth

Definition Treewidth tw(G) = min{k: G has chordal super-graph with clique-size k} −1

Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

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Treewidth

Definition Treewidth tw(G) = min{k: G has chordal super-graph with clique-size k} −1

Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

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Treewidth

Definition Treewidth tw(G) = min{k: G has chordal super-graph with clique-size k} −1

Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

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Treewidth

Definition Treewidth tw(G) = min{k: G has chordal super-graph with clique-size k} −1 G has bounded treewidth ⇒ many NP-hard problems become polynomial (FPT). Often a first step towards developing a PTAS.

Therese Biedl Carving-width, tree-width and graph drawing 7 / 19

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Planar graph drawing and treewidth

DrawOptArea is NP-hard. Polynomial if treewidth bounded?

Therese Biedl Carving-width, tree-width and graph drawing 8 / 19

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Planar graph drawing and treewidth

DrawOptArea is NP-hard. Polynomial if treewidth bounded? No!

(Reduction-graph in [KW07] for NP-hardness of DrawOptArea.)

Therese Biedl Carving-width, tree-width and graph drawing 8 / 19

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Planar graph drawing and treewidth

DrawOptArea is NP-hard. Polynomial if treewidth bounded? No!

(Reduction-graph in [KW07] for NP-hardness of DrawOptArea.)

Graph is drawn on 3 rows.

Therese Biedl Carving-width, tree-width and graph drawing 8 / 19

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Planar graph drawing and treewidth

DrawOptArea is NP-hard. Polynomial if treewidth bounded? No!

(Reduction-graph in [KW07] for NP-hardness of DrawOptArea.)

Graph is drawn on 3 rows. Therefore [FLW03]: pathwidth ≤ 3.

Therese Biedl Carving-width, tree-width and graph drawing 8 / 19

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Planar graph drawing and treewidth

DrawOptArea is NP-hard. Polynomial if treewidth bounded? No!

(Reduction-graph in [KW07] for NP-hardness of DrawOptArea.)

Graph is drawn on 3 rows. Therefore [FLW03]: pathwidth ≤ 3. Well-known: then treewidth ≤ 3.

Therese Biedl Carving-width, tree-width and graph drawing 8 / 19

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Planar graph drawing and treewidth

(Reduction-graph in [KW07] for NP-hardness of DrawOptArea.)

But this graph was disconnected. And it has many possible planar embeddings.

Therese Biedl Carving-width, tree-width and graph drawing 9 / 19

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Planar graph drawing and treewidth

(Reduction-graph in [KW07] for NP-hardness of DrawOptArea.)

But this graph was disconnected. And it has many possible planar embeddings. Theorem (B. 2014) DrawOptArea is NP-hard even for a 3-connected planar graph with treewidth at most 8.

Therese Biedl Carving-width, tree-width and graph drawing 9 / 19

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Planar graph drawing and treewidth

Theorem (B. 2014) DrawOptArea is NP-hard even for a 3-connected planar graph with treewidth at most 8.

Therese Biedl Carving-width, tree-width and graph drawing 10 / 19

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Planar graph drawing and triangulated graphs

Back to: Why is DrawOptArea easy on Apollonian networks? constant treewidth? faces are triangles? both?

Therese Biedl Carving-width, tree-width and graph drawing 11 / 19

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Planar graph drawing and triangulated graphs

Back to: Why is DrawOptArea easy on Apollonian networks? constant treewidth no, still NP-hard faces are triangles? both?

Therese Biedl Carving-width, tree-width and graph drawing 11 / 19

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Planar graph drawing and triangulated graphs

Back to: Why is DrawOptArea easy on Apollonian networks? constant treewidth no, still NP-hard faces are triangles? both? Definition (Planar triangulated graph) Planar graph where all faces (including

  • uter-face) are triangles.

Therese Biedl Carving-width, tree-width and graph drawing 11 / 19

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Planar graph drawing and triangulated graphs

Theorem (B. 2014) DrawOptArea is NP-hard even for planar graphs that are one edge away from being triangulated.

Therese Biedl Carving-width, tree-width and graph drawing 12 / 19

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Planar graph drawing and triangulated graphs

Theorem (B. 2014) DrawOptArea is NP-hard even for planar graphs that are one edge away from being triangulated.

Therese Biedl Carving-width, tree-width and graph drawing 12 / 19

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Planar graph drawing and triangulated graphs

Theorem (B. 2014) DrawOptArea is NP-hard even for planar graphs that are one edge away from being triangulated.

Therese Biedl Carving-width, tree-width and graph drawing 12 / 19

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Planar graph drawing and triangulated graphs

Theorem (B. 2014) DrawOptArea is NP-hard even for planar graphs that are one edge away from being triangulated. NP-hard for triangulated? Conjectured yes.

Therese Biedl Carving-width, tree-width and graph drawing 12 / 19

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Planar graph drawing and carving width

Back to: Why is DrawOptArea easy on Apollonian networks? constant treewidth? no, still NP-hard small face-degrees? both?

Therese Biedl Carving-width, tree-width and graph drawing 13 / 19

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Planar graph drawing and carving width

Back to: Why is DrawOptArea easy on Apollonian networks? constant treewidth? no, still NP-hard small face-degrees? no, still NP-hard both?

Therese Biedl Carving-width, tree-width and graph drawing 13 / 19

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Planar graph drawing and carving width

Back to: Why is DrawOptArea easy on Apollonian networks? constant treewidth? no, still NP-hard small face-degrees? no, still NP-hard both? Theorem (based on B., Vatshelle 2012) If G is a plane graph with bounded treewidth and bounded face-degrees, then DrawOptArea is polynomial.

Therese Biedl Carving-width, tree-width and graph drawing 13 / 19

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Planar graph drawing and carving width

Theorem (B. 2014, based on B., Vatshelle 2012) If G is a plane graph with bounded treewidth and bounded face-degrees, then DrawOptArea is polynomial. Look at dual graph:

Therese Biedl Carving-width, tree-width and graph drawing 14 / 19

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Planar graph drawing and carving width

Theorem (B. 2014, based on B., Vatshelle 2012) If G is a plane graph with bounded treewidth and bounded face-degrees, then DrawOptArea is polynomial. Look at dual graph: bounded face degrees ⇔ bounded maximum degree in dual bounded treewidth ⇔ bounded treewidth in dual ⇒ dual has bounded carving width

Therese Biedl Carving-width, tree-width and graph drawing 14 / 19

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Carving-width

Definition (Carving decomposition) Recursive vertex-partitioning (always split into two groups)

Therese Biedl Carving-width, tree-width and graph drawing 15 / 19

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Carving-width

Definition (Carving decomposition) Recursive vertex-partitioning (always split into two groups)

3 edges in cut

Definition Width of carving decomposition: maximum # edges in cut at arc. Carving width: Smallest possible width of carving decomposition.

Therese Biedl Carving-width, tree-width and graph drawing 15 / 19

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Carving-width of dual

Definition (Carving decomposition of dual) Recursive face-partitioning (always split into two groups)

Therese Biedl Carving-width, tree-width and graph drawing 16 / 19

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Carving-width of dual

Definition (Carving decomposition of dual) Recursive face-partitioning (always split into two groups)

5 edges in boundary

Carving width of dual: maximum # edges in boundary.

Therese Biedl Carving-width, tree-width and graph drawing 16 / 19

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Planar graph drawing and carving width

Theorem (B. 2014, based on B., Vatshelle 2012) If G is a plane graph whose dual has bounded carving width, then DrawOptArea is polynomial. Idea: Dynamic programming.

5 edges in boundary

Therese Biedl Carving-width, tree-width and graph drawing 17 / 19

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Planar graph drawing and carving width

Theorem (B. 2014, based on B., Vatshelle 2012) If G is a plane graph whose dual has bounded carving width, then DrawOptArea is polynomial. Idea: Dynamic programming. Ga: graph “below” arc a. Ba: boundary-vertices of Ga πa: mapping from Ba to points in W × H-grid.

5 edges in boundary

M(a, π) = TRUE if we can draw Ga planar with Ba at π(Ba)

Therese Biedl Carving-width, tree-width and graph drawing 17 / 19

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Planar graph drawing and carving width

Theorem (B. 2014, based on B., Vatshelle 2012) If G is a plane graph whose dual has bounded carving width, then DrawOptArea is polynomial. Idea: Dynamic programming. Ga: graph “below” arc a. Ba: boundary-vertices of Ga πa: mapping from Ba to points in W × H-grid.

5 edges in boundary

M(a, π) = TRUE if we can draw Ga planar with Ba at π(Ba) Compute M (for all W · H ≤ A) bottom-up in decomposition. G has drawing ⇔ TRUE at root for some π, W , H. Run-time O∗(A

3 2 cw(G ∗)). Therese Biedl Carving-width, tree-width and graph drawing 17 / 19

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Area-optimal planar graph drawing

Planar embedding Fixed Free constant treewidth NPC [KW07] constant face-degrees constant vertex-degrees constant treewidth and face-degrees constant treewidth and vertex-degrees

Therese Biedl Carving-width, tree-width and graph drawing 18 / 19

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Area-optimal planar graph drawing

Planar embedding Fixed Free constant treewidth NPC [KW07] NPC constant face-degrees constant vertex-degrees NPC NPC constant treewidth and face-degrees constant treewidth and vertex-degrees NPC NPC

Therese Biedl Carving-width, tree-width and graph drawing 18 / 19

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Area-optimal planar graph drawing

Planar embedding Fixed Free constant treewidth NPC [KW07] NPC constant face-degrees NPC NPC constant vertex-degrees NPC NPC constant treewidth and face-degrees constant treewidth and vertex-degrees NPC NPC

Therese Biedl Carving-width, tree-width and graph drawing 18 / 19

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Area-optimal planar graph drawing

Planar embedding Fixed Free constant treewidth NPC [KW07] NPC constant face-degrees NPC NPC constant vertex-degrees NPC NPC constant treewidth and face-degrees P constant treewidth and vertex-degrees NPC NPC

Therese Biedl Carving-width, tree-width and graph drawing 18 / 19

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Area-optimal planar graph drawing

Planar embedding Fixed Free constant treewidth NPC [KW07] NPC constant face-degrees NPC NPC constant vertex-degrees NPC NPC constant treewidth and face-degrees P NPC constant treewidth and vertex-degrees NPC NPC

Therese Biedl Carving-width, tree-width and graph drawing 18 / 19

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Area-optimal planar graph drawing

Convex drawing No Yes Planar embedding Fixed Free Fixed Free constant treewidth NPC [KW07] NPC constant face-degrees NPC NPC constant vertex-degrees NPC NPC constant treewidth and face-degrees P NPC constant treewidth and vertex-degrees NPC NPC DrawConvexOptArea: Only consider drawings where all faces (including outer-face) are convex.

Therese Biedl Carving-width, tree-width and graph drawing 18 / 19

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Area-optimal planar graph drawing

Convex drawing No Yes Planar embedding Fixed Free Fixed Free constant treewidth NPC [KW07] NPC constant face-degrees NPC NPC NPC NPC constant vertex-degrees NPC NPC NPC NPC constant treewidth and face-degrees P NPC constant treewidth and vertex-degrees NPC NPC DrawConvexOptArea: Only consider drawings where all faces (including outer-face) are convex.

Therese Biedl Carving-width, tree-width and graph drawing 18 / 19

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Area-optimal planar graph drawing

Convex drawing No Yes Planar embedding Fixed Free Fixed Free constant treewidth NPC [KW07] NPC constant face-degrees NPC NPC NPC NPC constant vertex-degrees NPC NPC NPC NPC constant treewidth and face-degrees P NPC constant treewidth and vertex-degrees NPC NPC P DrawConvexOptArea: Only consider drawings where all faces (including outer-face) are convex. Theorem (based on B.,Vatshelle 2012) If G is a plane graph that has bounded carving width, then DrawConvexOptArea is polynomial.

Therese Biedl Carving-width, tree-width and graph drawing 18 / 19

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Area-optimal planar graph drawing

Convex drawing No Yes Planar embedding Fixed Free Fixed Free constant treewidth NPC [KW07] NPC P P constant face-degrees NPC NPC NPC NPC constant vertex-degrees NPC NPC NPC NPC constant treewidth and face-degrees P NPC P P constant treewidth and vertex-degrees NPC NPC P P DrawConvexOptArea: Only consider drawings where all faces (including outer-face) are convex. Theorem (based on B.,Vatshelle 2012 B. 2014) If G is a planeplanar graph that has bounded carving width treewidth, then DrawConvexOptArea is polynomial.

Therese Biedl Carving-width, tree-width and graph drawing 18 / 19

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Some open problems

Is DrawOptArea polynomial for other graph classes?

Therese Biedl Carving-width, tree-width and graph drawing 19 / 19

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Some open problems

Is DrawOptArea polynomial for other graph classes? Is DrawConvexOptArea FPT in the treewidth?

Therese Biedl Carving-width, tree-width and graph drawing 19 / 19

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Some open problems

Is DrawOptArea polynomial for other graph classes? Is DrawConvexOptArea FPT in the treewidth? Approximation algorithms, or even a PTAS?

Therese Biedl Carving-width, tree-width and graph drawing 19 / 19

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Some open problems

Is DrawOptArea polynomial for other graph classes? Is DrawConvexOptArea FPT in the treewidth? Approximation algorithms, or even a PTAS?

References:

  • T. Biedl, M. Vatshelle, The point-set embeddability problem for plane

graphs, SoCG 2012, to appear in IJCGA.

  • T. Biedl, On area-optimal planar drawings, to appear at ICALP 2014.

Therese Biedl Carving-width, tree-width and graph drawing 19 / 19