A Method for Generating Colorings over Graph Automophism
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This document describes an algorithm for generating all non-equivalent colorings of a graph with certain symmetries. The algorithm uses family trees and blocks defined by the graph's automorphisms. It assigns numbers to blocks of vertices with the leading color, then decides the positions of the new color in each block to generate the child colorings recursively. This allows generating all colorings with polynomial delay by avoiding duplicate calculations.
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A Method for Generating Colorings over Graph Automophism
- 1. A METHOD FOR GENERATING COLORINGS OVER GRAPH AUTOMORPHISM Fei He, Hiroshi Nagamochi 2015.8.22. 1 The 12th International Symposium on Operations Research and its Applications in engineering, technology and management (ISORA 2015), Luoyang, August 21-24, 2015. 何 飛 プレゼン資料
- 2. 平成28年度 数理工学専攻説明会 第１回： 平成２８年５月７日(土) 第２回： 平成２８年５月３０日(月) 場所,プログラムの詳細は以下の専攻ＨＰをご覧ください． http://www.amp.i.kyoto-u.ac.jp 研究室見学できます．在学生から，入試勉強のしかた， 過去問の勉強方法などを聞くチャンスです． 京都大学大学院 情報学研究科 数理工学専攻 修士課程，博士課程の学生募集
- 3. Background Enumeration of isomers of chemical compounds Ex. N 10 Isomers Naphthalene N c l cl N cl N cl N cl N cl N cl N cl NclN cl Ncl Nitrogen Atom Chlorine Atom W 2
- 4. Background Enumeration of isomers of chemical compounds can be converted into graph coloring problems. Ex. N c l c l N 10 Isomers 10 Colorings Chemistry Problem Graph Coloring Problem N c l Color the vertices with the color of the atom attached to it. See the carbon atoms as vertices. 3
- 5. Assumption on Graphs -- Symmetries graph G automorphism φ1 automorphism φ2 automorphism φ3 Axial symmetries A combination of φ1 and φ2 We only deal with graphs of two axial symmetries. Ex. 180° 4
- 6. Assumption on Graphs --Blocks graph G A block: a set of vertices that can be mapped to each other by some automorphism. Ex. Block B1 180° Block B2 B1∨ B2 = W Blocks are equivalence classes defined by automorphisms. W 5
- 7. Equivalent Colorings Coloring c1 Coloring c2 φ2 equivalent There exists an automorphism φ1, φ2 or φ3 Coloring c1 Coloring c3 non-equivalent 6
- 8. Problem Statement A graph G=(V, E) of two axial symmetries, a subset W of vertices of V Input : Output : All non-equivalent colorings to W Ex. Input 1 Input 2 7
- 9. Output Output : All non-equivalent colorings to W Color IndexIn case of a Naphthalene ring:Ex. We want to realize the generation of colorings in polynomial delay. Avoid duplication of calculation Total number of possible colorings : > 460,000 … 8 Total number of non-equivalent colorings : 23,911
- 10. Idea of Algorithm -- Family Tree … … … … … … … * We put colorings with the same color index in one set. … The Family Tree 9
- 11. Parent-child Relationship … Color Priority: Constraints on #: # ≥# ≥# ≥# ≥# ≥# ≥ … … Color Index Parent Parent Child Child Child Parent # in Parent = # + #min in Child the leading color the leading color the leading color 10 min
- 12. Example of Family Tree # ≥ # ≥ # ≥ # ≥ # ≥ # Ex. |W| = 6 … 11
- 13. Generate Children Color Indices … min= … … … … # = 1 # = 2 # = #min # ≥ # ≥ … ≥ # ≥ #Color Priority: … The next color in the ordering 12 becomes the new min
- 14. Recursive Problem for Generating Colorings … … Recursive Problem 13 … Known Unknown
- 15. Recursive Problem Set A coloring cInput : Output : All colorings c’ that are the children of c c c’ Ex. How to generate colorings? 14 By using blocks
- 16. Summary A graph G=(V, E) of two axial symmetries, a subset W of vertices of V Input : Output : All non-equivalent colorings to W Can be realized in polynomial delay by using Family Trees Blocks defined after automorphisms graph G 15 colorings
- 17. 16 The End
- 18. Flow of the Algorithm Input: c Define Blocks Assign Numbers Output: c’ Assign Numbers Decide Positions Output: c’ Decide Positions Decide Positions Decide Positions Output: c’ Output: c’ … …… 17
- 19. Flow of the Algorithm Input: c Define Blocks Assign Numbers Check the symmetry of parent coloring c and define blocks on vertices with the leading color over c’s symmetries. Output: c’ Assign Numbers Decide Positions Output: c’ Decide Positions Decide Positions Decide Positions Output: c’ Output: c’ … …… 18
- 20. Define Blocks For a coloring c, its block is a collection of vertices with the leading color, and can be shifted to one another by one of the automorphisms of c. Ex. Naphthalene ring φ1 Block 1Block 2 Block 3 Blocks are equivalence classes defined by c’s automorphism. c 19
- 21. Define Blocks Define blocks: check the symmetry of c and define blocks on vertices with the leading color. Block 1 Block 2 Block 1 Block 2 Block 3 φ1, φ2, φ3 φ1 Block 1 Block 2 Block 3 Block 4 Block 5 ＿＿ Ex. c 20
- 22. Flow of the Algorithm Input: c Define Blocks Assign Numbers Assign the number of the new color to each block. Output: c’ Assign Numbers Decide Positions Output: c’ Decide Positions Decide Positions Decide Positions Output: c’ Output: c’ … …… 21
- 23. Assign Numbers Block 1 Block 2 Block 3 Change two into Block 1 Block 2 Block 3 2 0 0 1 1 0 1 0 1 0 2 0 0 1 1 0 0 2 Assign Numbers of to each block: c Ex. 22
- 24. Flow of the Algorithm Input: c Define Blocks Assign Numbers Output: c’ Assign Numbers Decide Positions Output: c’ Decide Positions Decide Positions Decide Positions Output: c’ Output: c’ … …… Decide the positions for the new color by the assignment of numbers in each block. 23
- 25. Decide Positions Block 1 Block 2 Block 3 Change two into Block 1 Block 2 Block 3 2 0 0 Assign Numbers of to each block: c c’ Block 1 Ex. 24
- 26. Decide Positions Block 1 Block 2 Block 3 Change two into Block 1 Block 2 Block 3 1 1 0 Assign Numbers of to each block: c c’ c’ Block 2 Block 1 Ex. 25
- 27. Decide Positions c c’ Block 1 c’ c’ Block 2 Block 1 Block 1 Block 2 Block 3 2 0 0 1 1 0 1 0 1 0 2 0 0 1 1 0 0 2 … Output Ex. 26
- 28. Flow of the Algorithm Input: c Define Blocks Assign Numbers Output: c’ Assign Numbers Decide Positions Output: c’ Decide Positions Decide Positions Decide Positions Output: c’ Output: c’ … …… Can be calculated automatically by our algorithm 27
- 29. Summary A graph G=(V, E) of two axial symmetries, a subset W of vertices of V Input : Output : All non-isomorphic colorings to W Can be realized in polynomial delay by using Family Trees Blocks defined after automorphisms Assignment of colors and decision on their positions.graph G 28