グラフ同型性判定問題に対する幅パラメータ固定アルゴリズムの研究

2015 Fiscal Year Annual Research Report

• Project/Area Number: 25730003
• Research Institution: Japan Advanced Institute of Science and Technology
• Principal Investigator: 大舘 陽太 北陸先端科学技術大学院大学, 情報科学研究科, 助教 (80610196)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: アルゴリズム理論 / グラフ同型性判定問題

Outline of Annual Research Achievements

この年度では，ある一つの固定されたグラフを誘導マイナーとして含まないグラフクラスに対する同型性判定問題の研究を行った．結果として，計算量二分法を得た．以下に詳しく説明する．
グラフの辺の縮約とは，その辺の両端点を同一視する操作である．あるグラフHが別のグラフGの「誘導マイナー」であるとは，Gに対して頂点削除と辺の縮約を繰り返してHに同型なグラフが得られることを言う．あるグラフクラスCが，禁止誘導マイナーHを持つとは，Cに含まれるどのグラフGもHを誘導マイナーとして持たないことを言う．
本研究では，グラフクラスCがある定数サイズのHを禁止誘導マイナーとして持つ場合の，Cに対するグラフ同型性判定問題の計算量を調べた．結果として，以下の完全な計算量二分法を得た: 1. H が，完全グラフ，ジェム，ハウスのある変種のいずれかのグラフである場合は，多項式時間で解ける．2. H がそれ以外のグラフである場合はGI完全，つまり，一般の場合と同じ難しさである．
証明は，Hを禁止誘導マイナーとして持つグラフの分類と特徴付けに基づくが，その分類・特徴付け自体，アルゴリズム理論的に非常に有用である．例えば，本結果の副産物として，クリーク幅と呼ばれるグラフの複雑度を示す値についても，その値がグラフクラス全体で定数であるかそれともいくらでも大きくなりえるかの特徴付けを与えた．
グラフ同型性判定問題に対して，禁止構造による計算量の解析は今までにもいくつか行われてきた．例えば，あるグラフクラスが禁止マイナーHを持つ場合は，Hがどんなグラフでも多項式時間で解けることが知られている．禁止誘導マイナーに関してもある程度の研究が行われていたが，完全な計算量二分法は知られていなかった．今回の結果で，グラフ同型性判定問題と禁止構造に関する知見を深めることができた．

Research Products

• [Journal Article] A polynomial-time approximation scheme for the geometric unique coverage problem on unit squares (2016)
  • Author(s): Takehiro Ito, Shin-ichi Nakano, Yoshio Okamoto, Yota Otachi, Ryuhei Uehara, Takeaki Uno, and Yushi Uno
  • Journal Title: Computational Geometry: Theory and Applications Volume: 51 Pages: 25-39
  • DOI: 10.1016/j.comgeo.2015.10.004
  • Peer Reviewed
• [Journal Article] On the treewidth of toroidal grids (2016)
  • Author(s): Masashi Kiyomi, Yoshio Okamoto, and Yota Otachi
  • Journal Title: Discrete Applied Mathematics Volume: 198 Pages: 303-306
  • DOI: 10.1016/j.dam.2015.06.027
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Polynomial-time algorithms for Subgraph Isomorphism in small graph classes of perfect graphs (2016)
  • Author(s): Matsuo Konagaya, Yota Otachi, and Ryuhei Uehara
  • Journal Title: Discrete Applied Mathematics Volume: 199 Pages: 37-45
  • DOI: 10.1016/j.dam.2015.01.040
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Thin strip graphs (2016)
  • Author(s): Takashi Hayashi, Akitoshi Kawamura, Yota Otachi, Hidehiro Shinohara, and Koichi Yamazaki
  • Journal Title: Discrete Applied Mathematics Volume: - Pages: -
  • DOI: 10.1016/j.dam.2015.01.018
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Ferrers dimension of grid intersection graphs (2016)
  • Author(s): Steven Chaplick, Pavol Hell, Yota Otachi, Toshiki Saitoh, and Ryuhei Uehara
  • Journal Title: Discrete Applied Mathematics Volume: - Pages: -
  • DOI: 10.1016/j.dam.2015.05.035
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Extending partial representations of proper and unit interval graphs (2016)
  • Author(s): Pavel Klavik, Jan Kratochvil, Yota Otachi, Ignaz Rutter, Toshiki Saitoh, Maria Saumell, and Tomas Vyskocil
  • Journal Title: Algorithmica Volume: - Pages: -
  • DOI: 10.1007/s00453-016-0133-z
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Finding a chain graph in a bipartite permutation graph (2016)
  • Author(s): Masashi Kiyomi and Yota Otachi
  • Journal Title: Information Processing Letters Volume: - Pages: -
  • DOI: 10.1016/j.ipl.2016.04.006
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Extending partial representations of subclasses of chordal graphs (2015)
  • Author(s): Pavel Klavik, Jan Kratochvil, Yota Otachi, and Toshiki Saitoh
  • Journal Title: Theoretical Computer Science Volume: 576 Pages: 85-101
  • DOI: 10.1016/j.tcs.2015.02.007
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Completely independent spanning trees in (partial) k-trees (2015)
  • Author(s): Masayoshi Matsushita, Yota Otachi, and Toru Araki,
  • Journal Title: Discussiones Mathematicae Graph Theory Volume: 35 Pages: 427-437
  • DOI: 10.7151/dmgt.1806
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Linear-time algorithm for sliding tokens on trees (2015)
  • Author(s): Erik D. Demaine, Martin L. Demaine, Eli Fox-Epstein, Duc A. Hoang, Takehiro Ito, Hirotaka Ono, Yota Otachi, Ryuhei Uehara, and Takeshi Yamada
  • Journal Title: Theoretical Computer Science Volume: 600 Pages: 132-142
  • DOI: 10.1016/j.tcs.2015.07.037
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Presentation] Sliding token on bipartite permutation graphs (2015)
  • Author(s): Eli Fox-Epstein, Duc A. Hoang, Yota Otachi, and Ryuhei Uehara
  • Organizer: 26th International Symposium on Algorithms and Computation (ISAAC 2015)
  • Place of Presentation: Nagoya Marriott Associa Hotel, Nagoya, Japan
  • Year and Date: 2015-12-09 – 2015-12-11
  • Int'l Joint Research
• [Presentation] Swapping colored tokens on graphs (2015)
  • Author(s): Katsuhisa Yamanaka, Takashi Horiyama, David Kirkpatrick, Yota Otachi, Toshiki Saitoh, Ryuhei Uehara, and Yushi Uno
  • Organizer: 14th International Symposium on Algorithms and Data Structures (WADS 2015)
  • Place of Presentation: Victoria, BC, Canada
  • Year and Date: 2015-08-05 – 2015-08-07
  • Int'l Joint Research
• [Presentation] Competitive diffusion on weighted graphs (2015)
  • Author(s): Takehiro Ito, Yota Otachi, Toshiki Saitoh, Hisayuki Satoh, Akira Suzuki, Kei Uchizawa, Ryuhei Uehara, Katsuhisa Yamanaka, and Xiao Zhou
  • Organizer: 14th International Symposium on Algorithms and Data Structures (WADS 2015)
  • Place of Presentation: Victoria, BC, Canada
  • Year and Date: 2015-08-05 – 2015-08-07
  • Int'l Joint Research
• [Presentation] Induced minor free graphs: Isomorphism and clique-width (2015)
  • Author(s): Remy Belmonte, Yota Otachi, and Pascal Schweitzer
  • Organizer: 41st International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2015)
  • Place of Presentation: Munich, Germany
  • Year and Date: 2015-07-17 – 2015-07-19
  • Int'l Joint Research
• [Presentation] Reconfiguration of cliques in a graph (2015)
  • Author(s): Takehiro Ito, Hirotaka Ono, Yota Otachi
  • Organizer: 12th Annual Conference on Theory and Applications of Models of Computation (TAMC 2015)
  • Place of Presentation: Singapore
  • Year and Date: 2015-05-18 – 2015-05-20
  • Int'l Joint Research