1. Algorithms for Finding Distance-Edge-Colorings of Graphs
Takehiro Ito Akira Kato
Xiao Zhou Takao Nishizeki
Tohoku University

2. Distance between two edges
# of edges in a shortest path between two edges
Distance = 2

3. Distance between two edges
# of edges in a shortest path between two edges
Distance = 1

4. Distance between two edges
# of edges in a shortest path between two edges
Distance = 0

5. Distance-Edge-Coloring or l-Edge-Coloring
for a given bounded integer l,
any two edges within distance l have different colors
l = 1
Not 1-edge-coloring
1-edge-coloring
with six colors

6. Distance-Edge-Coloring or l-Edge-Coloring
for a given bounded integer l,
any two edges within distance l have different colors
0-edge-coloring
ordinary edge-coloring
l = 1
1-edge-coloring
strong edge-coloring
1-edge-coloring
with six colors

7. l-Edge-Coloring Problem
find an l-edge-coloring with min # of colors
The ordinary edge-coloring problem is NP-hard
The l-edge-coloring problem is NP-hard in general ?
restricted class
Partial k-trees

8. Class of Partial k-Trees
All Graphs
k=O(1)
Partial k-trees
(bounded tree-width)
k=2
Series-parallel
outerplanar graph
k=1
Trees
Forests

9. Related Results on Partial k-Trees
(ordinary edge-coloring problem)
Linear-time algorithm [Zhou et al. 1996]
l = 1
(strong edge-coloring problem)
Polynomial-time algorithm
[Salavatiour 2004] ?
l-edge-coloring problem

10. Our Results O( n a O(1) ) Partial k-Trees (k, l : constant)
Polynomial-time exact algorithm
O( n a O(1) )
partial k-tree has an l-edge-coloring with a colors?
n: # of vertices

11. Our Results O( n a O(1) ) 22(k+1)(l+1)+1 Partial k-Trees
(k, l : constant)
Polynomial-time exact algorithm
22(k+1)(l+1)+1
= O(1)
O( n a O(1) )
partial k-tree has an l-edge-coloring with a colors?
n: # of vertices

12. Our Results O( n a O(1) ) Partial k-Trees (k, l : constant)
Polynomial-time exact algorithm
O( n a O(1) )
partial k-tree has an l-edge-coloring with a colors?
min # of colors
a ≤ (# of edges)
a = O(1) O(n) time
n: # of vertices

13. Polynomial-time exact algorithm
Our Results
Partial k-Trees
Polynomial-time exact algorithm
Planar Graphs
Polynomial-time
2-approximation algorithm

14. Class of Partial k-Trees
All Graphs
Planar Graph
k=O(1)
Partial k-trees
(bounded tree-width)
k=2
Series-parallel
outerplanar graph
k=1
Trees
Forests

15. Polynomial-time exact algorithm
Our Results
Partial k-Trees
Polynomial-time exact algorithm
Planar Graphs
Polynomial-time
2-approximation algorithm

16. k-tree (recursive definition)
Partial k-Trees
k-tree (recursive definition) 1 2 3
k+1 k 4
(k+1)-clique
G : k-tree
k-clique k v
A graph obtained from G by adding a new vertex v and joining it with each of the k vertices is a k-tree.
A complete graph with k+1 vertices is a k-tree.

17. 1 2 4 3
Tree-decomposition
node 1 4 2 3
4-clique
3-tree

18. Tree-decomposition 1 2 4 3 1 4 5 1 2 3 5 2 3
3-clique
3-tree

19. Tree-decomposition 1 2 4 3 1 4 6 6 4 3 2 1 2 3 5 2 3 5
3-tree

20. Tree-decomposition 3-tree 1 4 7 2 3 6 5 1 2 4 3 6 4 3 2 1 2 3 5 5 1 2

21. Tree-decomposition 3-tree 1 4 7 2 3 6 8 5 1 2 4 3 6 4 3 2 1 2 3 5 5 1

22. Tree-decomposition 3-tree 1 4 7 2 3 6 8 5 9 1 2 4 3 6 4 3 2 1 2 3 5 5

23. Tree-decomposition 3-tree 1 4 7 2 3 10 6 8 5 9 1 2 4 3 6 4 3 2 1 2 3 5

24. Tree-decomposition Partial k-Tree 3-tree a subgraph of k-tree 9 5 8 710 3 5 8 2 9 1 7 6 4 1 2 4 3 9 5 8 7 6 10 4 1 3 2 1 4 6 4 3 2 1 2 3 5 7 2 3 6 8 5 1 2 7 2 8 3 5 5 9 10 10 3 5 8 2 8 5 9
Partial k-Tree
a subgraph of k-tree
3-tree

25. Tree-decomposition Partial 3-tree 1 4 7 2 3 6 8 5 9 10 1 4 2 3 1 2 3 5

26. Algorithm for Partial k-Trees
Dynamic Programming 10 3 5 8 2 9 6 4 1 7
leaves
root

27. Corresponding Subgraphs
Algorithm for Partial k-Trees
What should we store in the table for the problem ?
DP table 9 5 8 10 3 2 7 1 2 8 3 5 1 7
DP table
DP table
Nodes of
Tree-decomposition
Corresponding Subgraphs

28. Corresponding Subgraphs
Algorithm for Partial k-Trees
Tree-Decomposition
Corresponding Subgraphs

29. Corresponding Subgraph
Algorithm for Partial k-Trees
Tree-Decomposition
Corresponding Subgraph

30. Corresponding Subgraph
Algorithm for Partial k-Trees
store all colors which are assigned to edges with (distance from a vertex) ≦ l
Tree-Decomposition
Corresponding Subgraph

31. Algorithm for Partial k-Trees
store all colors which are assigned to edges with (distance from a vertex) ≦ l v2 v1
vk+1 v1 v2 …
vk+1
color 1 l

32. Algorithm for Partial k-Trees
store all colors which are assigned to edges with (distance from a vertex) ≦ l v2 v1
vk+1 v1 v2 …
vk+1
color 1 l …

33. Algorithm for Partial k-Trees
store all colors which are assigned to edges with (distance from a vertex) ≦ l v2 v1
vk+1 v1 v2 …
vk+1
color 1 l … …

34. Algorithm for Partial k-Trees
store all colors which are assigned to edges with (distance from a vertex) ≦ l v2 v1
vk+1 v1 v2 …
vk+1
color 1 l … … … … … … …

35. Algorithm for Partial k-Trees
store all colors which are assigned to edges with (distance from a vertex) ≦ l v2 v1
vk+1 v1 v2 …
vk+1
color 1 l … … … … … … …

36. Algorithm for Partial k-Trees
store all colors which are assigned to edges with (distance from a vertex) ≦ l v2 v1
vk+1 v1 v2 …
vk+1
colors 1 l … … … …

37. Algorithm for Partial k-Trees
store all colors which are assigned to edges with (distance from a vertex) ≦ l v2 v1
vk+1 v1 v2 …
vk+1
colors 1 l … … … … … … … … … … … … …

38. v1 v2 vk+1 v1 v2 vk+1 exponential size l … 1 vk+1 v2 v1 l … 1 vk+1 v2
colors l … 1
vk+1 v2 v1
colors l … 1
vk+1 v2 v1 v1 v2
vk+1 v1 v2
vk+1
exponential
size

39. v1 v2 vk+1 v1 v2 vk+1 Permutation same color pattern l … 1 vk+1 v2 v1
colors l … 1
vk+1 v2 v1
colors l … 1
vk+1 v2 v1 v1 v2
vk+1 v1 v2
vk+1
Permutation
same color pattern

40. v1 v2 vk+1 v1 v2 vk+1 same color pattern l … 1 vk+1 v2 v1 l … 1 vk+1
colors l … 1
vk+1 v2 v1
colors l … 1
vk+1 v2 v1 v1 v2
vk+1 v1 v2
vk+1
same color pattern

41. v1 v2 vk+1 v1 v2 vk+1 Permutation same color pattern l … 1 vk+1 v2 v1
colors l … 1
vk+1 v2 v1
colors l … 1
vk+1 v2 v1 v1 v2
vk+1 v1 v2
vk+1
Permutation
same color pattern

42. colors l … 1
vk+1 v2 v1
# of colors 1 2 v1 v2
vk+1

43. Algorithm for Partial k-Trees2 l … 1
vk+1
other sets of { , } # v2 v1
2O(1)
(k+1)(l+1) = O(1)
a color pattern

44. color pattern: 2O(1) rows {0, 1, …, a}
Algorithm for Partial k-Trees 2 l … 1
vk+1
other sets of { , } # v2 v1
2O(1)
color pattern: 2O(1) rows {0, 1, …, a}

45. Algorithm for Partial k-Trees2 l … 1
vk+1
other sets of { , } # v2 v1 2 l … 1
vk+1 3
other sets of { , } # v2 v1 1 l …
vk+1 2
other sets of { , } 4 # v2 v1 1 l …
vk+1 2
other sets of { , } 3 # v2 v1
O( a O(1) )
2O(1)

46. Corresponding Subgraphs
Algorithm for Partial k-Trees
time
O( a O(1) )
DP table 9 5 8 10 3 2 7 1 2 8 3 5 1 7
size
O( a O(1) )
DP table
size
O( a O(1) )
DP table
Nodes of
Tree-decomposition
Corresponding Subgraphs

47. Algorithm for Partial k-Trees10 3 5 8 2 9 6 4 1 7
leaves
root
# of nodes
= O(n)
time
O( a O(1) )
time
O( n a O(1) )

48. Polynomial-time exact algorithm
Our Results
Partial k-Trees
Polynomial-time exact algorithm
Planar Graphs
Polynomial-time
2-approximation algorithm

49. Planar Graphs
planar graph
non-planar graph

50. 2-Approx. Algorithm for Planar Graphs
Planar graph can be partitioned into layers

51. 2-Approx. Algorithm for Planar Graphs
Planar graph can be partitioned into layers
layer 1

52. 2-Approx. Algorithm for Planar Graphs
Planar graph can be partitioned into layers
layer 2

53. 2-Approx. Algorithm for Planar Graphs
Planar graph can be partitioned into layers
layer 3

54. 2-Approx. Algorithm for Planar Graphs
Planar graph can be partitioned into layers

55. colorings of components can be found separately
2-Approx. Algorithm for Planar Graphs
given planar graph G
(l+2) layers G’
G’’
colorings of components can be found separately

56. 2-Approx. Algorithm for Planar Graphs
(l+2)-outerplanar graph
(l+2)-outerplanar graph G’
G’’

57. 2-Approx. Algorithm for Planar Graphs
(l+2)-outerplanar graph
(l+2)-outerplanar graph G’
G’’

58. 2-Approx. Algorithm for Planar Graphs
(l+2)-outerplanar graphs partial (3l+5)-trees
Poly.-time exact Algorithm
(l+2)-outerplanar graph
(l+2)-outerplanar graph G’
G’’
OPT(G’) colors
OPT(G’’) colors

59. APPRO = OPT(G’) + OPT(G’’)
2-Approx. Algorithm for Planar Graphs
given planar graph G G’
G’’
APPRO = OPT(G’) + OPT(G’’)
OPT(G’) ≤ OPT(G)
OPT(G’) colors
OPT(G’’) ≤ OPT(G)
OPT(G’’) colors

60. 2-Approx. Algorithm for Planar Graphs
given planar graph G G’
G’’
APPRO = OPT(G’) + OPT(G’’)
≤ OPT(G) + OPT(G)
APPRO = OPT(G’) + OPT(G’’)
≤ OPT(G) + OPT(G)
= 2 OPT(G)
OPT(G’) ≤ OPT(G)
OPT(G’) colors
OPT(G’’) colors
OPT(G’’) ≤ OPT(G)

61. Conclusions Partial k-Trees Planar Graphs
Polynomial-time exact algorithm
Polynomial-time
2-approximation algorithm
It is easy to modify all algorithms so that
it actually finds an l-edge-coloring.

62. END