complexity results for threedimensional orthogonal graph drawing

1
complexity results for three-dimensional
orthogonal graph drawing

• maurizio titto patrignani
• third university of rome
• graph drawing 2005

2
three-dimensional orthogonal drawings

• nodes are (distinct) points in 3D space
• edges are composed by sequences of axis-parallel
  segments

node
edge
bend
3
what we know

• existence
• only graphs of maximum degree six admit such
  drawings
• all graphs of maximum degree six admit such
  drawings
• volume
• ?(n3/2) rosenberg 1983
• ?(n3/2) eades, stirk, and whitesides 1996
• bends
• in the optimal O(n3/2) volume we can draw with up
  to 7 bends per edge in O(n3/2) time eades,
  symvonis, and whitesides 2000
• in O(n3) volume we can draw with up to 3 bends
  per edges in linear time papakostas and tollis
  1999eades, symvonis, and whitesides, 2000
• in O(n2) volume we can draw with 6 bends per edge
  in linear time and handle insertions/deletions in
  O(1) time closson, gartshore, johansen, and
  wismath. 2000

4
what we do not know

• is three bends per edge the lower bound?or
  rather does every graph admit a 2-bend drawing?
• can we extend to 3D the topology-shape-metrics
  approach?

5
2-bend drawing problem

• an algorithm to produce 2-bends drawings could be
  particularly effective for information
  visualization
• is such algorithm does not exists then the
  algorithms we have are the best possible
• the K7 graph that was thought to require 3 bends
  turned out to admit a 2-bend drawing wood 97
• problem 46 of the open problem project demaine,
  mitchell, and orourke

6
topology-shape-metrics approach in 2D
V1,2,3,4,5,6 E(1,4),(1,5),(1,6),
(2,4),(2,5),(2,6), (3,4),(3,5),(3,6)
6
1
2
5
3
6
4
5
2
1
6
5
2
1
3
3
4
4
7
topology-shape-metrics approach in 3D
V1,2,3,4,5 E(1,2),(1,3),(1,4),
(2,3),(2,4),(2,5), (3,4),(3,5)
1
1
3
4
5
3
4
2
2
5
8
simple and not simple shape graphs
not simple shape graph (always intersects)
simple shape graph (admitting non-intersecting
metrics)
9
simplicity testing in 3D

• known results
• simplicity test for cycles di battista, liotta,
  lubiw, and whitesides, 01
• simplicity test for paths (with additional
  constraints) di battista, liotta, lubiw, and
  whitesides, 02
• the above two characterizations are not easy to
  extend to simple graphs (theta graphs) di
  giacomo, liotta, and patrignani, 04
• simplicity testing is an open problem in the
  general case
• problem 20 of brandenburg, eppstein, goodrich,
  kobourov, liotta, and mutzel. 03

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two open problems

• existence of a 2-bend drawing
• simplicity testing

can complexity considerations give us some
insight?
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what we show

• given a 6-degree graph we prove that
• simplicity testing is NP-complete
• if you fix edge shapes (with a maximum of 2
  bends per edge) finding the metrics corresponding
  to a non intersecting drawing is NP-complete
• 2-bend routing is NP-complete
• if you fix node positions finding a routing
  without intersections with a maximum of two bends
  per edge is NP-complete

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how we prove the statements

• reductions from the 3SAT problem
• instance a set of clauses c1, c2, , cm each
  containing three literals from a set of boolean
  variables v1, v2, , vn
• question can truth values be assigned to the
  variables so that each clause contains at least
  one true literal?

example of 3SAT instance (v1 ? v3 ? v4) ? (v1
? v2 ? v5) ? (v2 ? v3 ? v5)
c3
c1
c2
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we consider a generic target problem

• structure of the target problem
• instance a graph G and a set of constraints S
  expressed with respect to its nodes and edges
• question does G admit a 3D orthogonal drawing
  satisfying S?

14
the 3SAT reduction framework
variable gadgets
joint gadgets
clause gadgets
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use of the 3SAT framework

• theorem
• if these four statements are true
• there is at least one non intersecting drawing
  for each truth assignment satisfying the 3SAT
  instance
• in any non intersecting drawing if a variable
  gadget is true, the corresponding joint gadget is
  true and vice versa
• in any non intersecting drawing of a clause
  gadget one of the literals is true
• the construction can be done in polynomial time
• then the target problem is NP-hard

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simplicity testing problem

• instance a graph G and a shape for each edge
• question does G admit a 3D orthogonal drawing
  where the edges have the prescribed shape?

17
variable gadget
true variable
false variable
18
variable gadget propagating truth values
false variable
19
joint-gadget
T
F
T
F
20
joint-gadget
T
F
T
F
F
T
T
F
21
clause gadget
from the joint gadget
from the variable gadget
from the joint gadget
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all literals false ? intersecting clause gadget
F
T
F
T
F
F
T
T
F
F
F
F
F
T
F
T
F
T
F
T
T
T
T
T
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2-bend routing problem

• instance a graph G and the coordinates for its
  nodes
• question does G admit a 2-bend orthogonal
  drawing where the nodes have such coordinates?

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variable gadget
25
variable gadget propagating truth values
to clause gadget c1
variable gadget
to clause gadget c2
to clause gadget c3
26
joint gadget
from the variable gadget
27
joint gadget
from the variable gadget
28
joint gadget
to the clause gadget
from the variable gadget
29
clause gadget
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conclusionssimplicity testing is NP-hard

• any characterization of simple orthogonal shapes
  involves a hard computation
• even if we were able to find simple orthogonal
  shapes the compaction step would be NP-hard
• open problems
• are there classes of graphs such that the
  compaction step is polynomial?
• given a generic graph, are there families of
  shapes such that the metrics can always be
  computed in polynomial time?

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conclusions 2-bend routing is NP-hard

• yet another problem where two bends per edge
  implies NP-hardness
• two bends per edge fixed shape ? NP-hardness
• two bends per edge fixed positions ?
  NP-hardness
• two bends per edge diagonal layout ?
  NP-hardness
• wood, 2004
• open problem
• what is the problem of finding a 2-bend drawing
  of a graph?

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thank you!