平面グラフ分枝幅と分枝分割

1. 平面グラフ分枝幅と分枝分割 明治大学理工学部情報科学科 玉木久夫 共同研究者： Qianping Gu (Simon Fraser大) 吉武由実（明治大学理工学研究科）

2. 内容 Ｐａｒｔ　１ • グラフの分枝幅と分枝分割：定義と背景 • 平面グラフの分枝分割アルゴリズム O(n4) 時間アルゴリズム（Seymour & Thomas 94) O(n3) 時間への改良（Gu & Tamaki 05) Ｐａｒｔ　２ • 平面グラフの分枝幅 　　ねずみ捕りゲームによる特徴づけとO(n2) 時間アルゴリズム （Seymour & Thomas 94) 特徴づけの理解：　実際的な改良へ向けて

3. Part 1

4. Branch-decomposition(分枝分割) A branch-decomposition of graph G : • Conceptual definition: • a recursive binary decomposition of E(G) f a e c g a e c g b f d d b G

5. Branch-decomposition A branch-decomposition of graph G : • Conceptual definition: • a recursive binary decomposition of E(G) f a c g a e c f e g d b b d G

6. Branch-decomposition A branch-decomposition of graph G : • Conceptual definition: • a recursive binary decomposition of E(G) f a c f g a e c d b e g b d G

7. Branch-decomposition A branch-decomposition of graph G : • Conceptual definition: • a recursive binary decomposition of E(G) f c a f g a e c d b e g b d G

8. Branch-decomposition A branch-decomposition of graph G : • Conceptual definition: • a recursive binary decomposition of E(G) f c a f g a e c d b b d e g G

9. Branch-decomposition A branch-decomposition of graph G : • Conceptual definition: • a recursive binary decomposition of E(G) f c a f g a e c d b g G b e d

10. Branch-decomposition A branch-decomposition of graph G : Formal definition: a ternary tree with leaf set E(G) f c a f g a e c d b g G b e d

11. Branch-decomposition A branch-decomposition of graph G : Formal definition: a ternary tree with leaf set E(G) … abstracts away the starting bipartition f c a f g a e c d b g G b e d

12. Branch-decomposition A branch-decomposition of graph G : Formal definition: a ternary tree with leaf set E(G) … abstracts away the starting bipartition… f c a f g a e c d b g G b e d

13. Branchwidth（分枝幅） The width of branch-decomposition T of G : The maximum cardinality of the vertex cuts of G associated with the edges of T. c f a f a g e c d b g G b e d

14. Branchwidth The width of branch-decomposition T of G : The maximum cardinality of the vertex cuts of G associated with the tree edges of T. c f a f a g e c d b 3 g G b e d

15. Branchwidth The width of branch-decomposition T of G : The maximum cardinality of the vertex cuts of G associated with the tree edges of T. f width = 4 c a 2 f g 3 a 2 e 2 c 4 d b 3 3 2 2 2 2 g G b e d

16. Branchwidth The branchwidth of G : The minimum width of all the branch-decompositions of G.

17. Background • Branch-decompositions are introduced by Robertson and Seymour (1991) in relation to tree-decompositions. vertex cutstree edges of a branch-decomposition. tree nodes of a tree-decomposition, • bw(G) ≦tw(G) + 1 ≦ (3/2) bw(G) • Many NP-hard combinatorial problems on graphs can be solved in 2O(bw(G))n time, via DP based on the decomposition. .

18. Known results（Seymour-Thomas 94) General graphs：　 NP-complete to decide whether bw(G) ≦ k for given G, k, if k is part of the input. Planar graphs：　 The decision problem: O(n2) time Constructing the corresponding decomposition: O(n4) time If k is fixed, then the decision and the construction can both be done in linear time on general graphs (Bodlaender & Thilikos 97).  O(n3) : This work

19. Rest of Part 1 • Carving decomposition • Seymour-Thomas algorithm for planar graphs • Key lemmas for improvement • Algorithm and analysis: some ideas

20. Carving decomposition（刻み分割） of G • A recursive binary decomposition of V(G) • Formally a ternary tree with leaf set V(G). • The width of carving decomposition T of G is the maximum cardinality of the edge cuts of G associated with tree edges of T. 5 3 5 4 1 1 5 4 2 3 2 G

21. Branch-decomposition vs carving-decomposition The problem of computing an optimal decomposition of planar graph G can be reduced to that of computing an optimal carving-decomposition of a related planar multi-graph M(G). (Seymour and Thomas 94).

22. Goal Tool: O(n2)-time Carving-width decision procedure (Seymour and Thomas 94) Given a planar multi-graph G with n vertices and O(n) edges, a minimum-width carving decomposition of G can be constructed in O(n3) time. Given a planar multi-graph G and a positive integer k, decides whether the carvingwidth of G exceeds k.

23. Bottom-up construction of a carving-decomposition Start from singleton sets of vertices. 1 2 3 6 4 5 7

24. Bottom-up construction of a carving-dec. Merge two vertex sets into one, at a time. 1 2 3 6 4 5 7

25. Bottom-up construction of a carving-dec. Merge two vertex sets into one, at a time. 1 2 1 2 3 3 6 6 4 4 5 5 7 7

26. Bottom-up construction of a carving-dec. Merge two vertex sets into one, at a time. 1 2 1 2 3 3 6 4 6 4 5 5 7 7

27. Bottom-up construction of a carving-dec. Merge two vertex sets into one, at a time. 1 2 1 2 3 3 6 4 6 4 5 5 7 7

28. Bottom-up construction of a carving-dec. Merge two vertex sets into one, at a time. 1 2 1 2 3 3 6 4 6 4 5 5 7 7

29. Bottom-up construction of a carving-dec. Merge two vertex sets into one, at a time. 1 2 1 2 3 3 6 4 6 4 5 5 7 7

30. Bottom-up construction of a carving-dec. Merge two vertex sets into one, at a time. 1 2 1 2 3 3 6 4 6 4 5 5 7 7

31. Bond carving Bond carving of G: a carving decomposition of G in which every cut bipartitions V(G) into two connected sets, i.e., every cut is a dual cycle Lemma (Seymour and Thomas 94)  In the bottom up process, we can only merge adjacent vertex sets For every planar multi-graph G, the optimal carvingwidth can be achieved by a bond carving.

32. How to guide the bottom-up construction We have a contracted multi-graph at each step. 1 2 3 6 4 5 7

33. How to guide the bottom-up construction We have a contracted multi-graph at each step. 1 2 3 6 5 7

34. How to guide the bottom-up construction We have a contracted multi-graph at each step. Use the width decision procedure to ensure that the carvingwidth does not exceed the original width at any step. 1 2 3 6 5 We say that two vertex sets X and Y are mergeable if merging them into one does not cause the carvingwidth to exceed the original optimal width 7

35. A carving-decomposition algorithm • Decide the carvingwidth k of G. • M  the set of all singleton vertex sets of G. • While |M| > 1 do Find a mergeable pair X , Y of vertex sets in M and replace them by XUY. At each iteration, the O(n2)-time decision procedure is called O(n) times for mergeability testing.  O(n4) time in total for n iterations

36. Our refinement Reduce the number of calls to the decision procedure througout the execution from O(n2) to O(n). The answers to all the other mergeability tests are deduced from previous test results in O(n) time each.

37. Key lemma Let X, Y, W, Z be in the current set M of vertex sets, such that • |dG(X U Y)| ≦ k, where k is the carving width of G • X and Y are not mergeable, • No edge of G between W and Z. 4. X and W are mergeable and so are Y and Z. Let M’ be obtained from M by merging these two pairs Then, XUW and YUZare not mergeable in M’. W Z Y X

38. Key lemma Let X, Y, W, Z be in the current set M of vertex sets, such that • |dG(X U Y)| ≦ k, • X and Y are not mergeable, • No edge of G between W and Z. 4. X and W are mergeable and so are Y and Z. Let M’ be obtained from M by merging these two pairs Then, XUW and YUZare not mergeable in M’. ≦ k W Z Y X

39. Key lemma Let X, Y, W, Z be in the current set M of vertex sets, such that • |dG(X U Y)| ≦ k, • X and Y are not mergeable, • No edge of G between W and Z. 4. X and W are mergeable and so are Y and Z. Let M’ be obtained from M by merging these two pairs Then, XUW and YUZare not mergeable in M’. W Z Y X

40. Key lemma Let X, Y, W, Z be in the current set M of vertex sets, such that • |dG(X U Y)| ≦ k, • X and Y are not mergeable, • No edge of G between W and Z. 4. X and W are mergeable and so are Y and Z. Let M’ be obtained from M by merging these two pairs Then, XUW and YUZare not mergeable in M’. W Z Y X

41. Key lemma Let X, Y, W, Z be in the current set M of vertex sets, such that • |dG(X U Y)| ≦ k, • X and Y are not mergeable, • No edge of G between W and Z. 4. X and W are mergeable and so are Y and Z. Let M’ be obtained from M by merging these two pairs Then, XUW and YUZare not mergeable in M’. W Z Y X

42. Key lemma Let X, Y, W, Z be in the current set M of vertex sets, such that • |dG(X U Y)| ≦ k, • X and Y are not mergeable, • No edge of G between W and Z. 4. X and W are mergeable and so are Y and Z. Let M’ be obtained from M by merging these two pairs Then, XUW and YUZare not mergeable in M’. ? W Z Y X

43. Key lemma Let X, Y, W, Z be in the current set M of vertex sets, such that • |dG(X U Y)| ≦ k, • X and Y are not mergeable, • No edge of G between W and Z. 4. X and W are mergeable and so are Y and Z. Let M’ be obtained from M by merging these two pairs Then, XUW and YUZare not mergeable in M’. W Z Y X

44. Proof of the key lemma We assume that XUW and YUZare mergeable and show that X and Y would then be mergeable A Assume we have Z W X Y

45. Proof of the key lemma We assume that XUW and YUZare mergeable and show that X and Y would then be mergeable A A A Assume we have or then, we have W Z Z X Y Z W W X Y X Y

46. Proof of the key lemma We only need to consider the red cuts below. (Blue cuts are ok by the assumption of the lemma) A A A W Z Z X Y Z W W X Y X Y

47. Proof of the key lemma (completed) cut1 + cut2 = cut3 + cut4 ≦ 2k So, either cut1 ≦ k or cut2 ≦ k Note there are no edges between W and Z by assumption 3 1 A W Z 2 X ∪Y 4

48. Finished? Only one expensive test between a pair, as long as the set of edges between them does not change? Y X

49. Finished? Only one expensive test between a pair, as long as the set of edges between them does not change? Z1 W1 Y X

50. Finished? Only one expensive test between a pair, as long as the set of edges between them does not change? Z2 W2 Z1 W1 Y X