2001 Fiscal Year Annual Research Report

グラフ論、離散最適化とその応用

Research Project

Project/Area Number	11640105
Research Institution	The University of Electro-Communications
Principal Investigator	安藤清電気通信大学, 電気通信学部, 教授 (20096944)
Co-Investigator(Kenkyū-buntansha)	石上嘉康電気通信大学, 電気通信学部, 助手 (50262374) 田村明久京都大学, 理学部, 助教授 (50217189) 浅野哲夫北陸先端科学技術大学院大学, 情報科学研究科, 教授 (90113133)
Keywords	グラフ論 / 離散最適化 / アルゴリズム / 計算幾何 / 離散幾何 / 組み合わせ幾何 / 組み合わせ論 / 位相幾何的グラフ論
Research Abstract	以下に本研究の本年度における実績の概要を示す。グラフ論、離散最適化、計算幾何学は一見無関係に思われる分野にも積極的に応用されている。計算幾何学分野における本研究では画像処理分野の濃淡画像を白黒2値画像で近似する問題(ディジタルハーフトーニング)を取り上げた。ここでは従来の手法と全く異なる切り口でこの問題に接近し、ディジタルハーフトーニングを数学的最適化問題として定義し、ある種の測度を基礎にした実数要素行列の整数要素行列への変換法を導入することにより、ある制約の下ではディジタルハーフトーニングを効率よく実行できるアルゴリズムの開発に成功した。 M-凸関数はマトロイド構造を有する離散的な非線形関数で離散最適化問題の効率的なアルゴリズムでは多くの場合基本的役割をはたしてきた。離散最適化分野における本研究での成果として離散凸解析における重要な基本問題の一つであるM-凸関数の最小化問題に対する新しい多項式時間アルゴリズムを提案した。グラフ論分野においては昨年の5-可縮臨界グラフの次数5の頂点の分布に関する研究に引き続き、6-可縮臨界グラフを調べ、6-可縮臨界グラフでは少なくともその1/7の頂点が次数6であることを示した。存在が保証される最大部分グラフの概形を確定し、その意味でErdos-Stoneの定理の最も強い形での拡張の一つであったBollobas-Kohayakawa予想を肯定的に解決した。さらにグラフを同型な部分グラフで敷き詰めることができるための次数条件を述べたAlon-Yusterの定理を拡張し、定数を超えるサイズの部分グラフの扱いを可能にする最良の結果を得た。また、Alon-Yusterの定理を必ずしも同型でない部分グラフで敷き詰めるための条件に拡張し、長年未解決であったE⊥-Zahar予想など多くの定理、予想を近似的に含む一般的な結果を得た。

Research Products
(22 results)

All Other

All Publications (22 results)

[Publications] T.Asano, D.Z.Cand, N.Katoh, T.Tokuyama: "Efficient algorithms for optimization-based image segmentation"Internat. J. Comput. Geom. Appl.. 11. 145-166 (2001)
[Publications] T.Asano, N.Katoh, T.Tokuyama: "A unified scheme for detecting fundamental curves in binary edge images"Comput. Geom.. 18. 73-93 (2001)
[Publications] T.Asano, T.Matsui, T.okuyama: "on the complexities of the optimal rounding problems of sequences and matrices"Lecture Notes in Comput.Sci.. 1851. 476-489 (2000)
[Publications] D.Nakamura, A.Tamura: "A revision of Minty's algorithm for finding a maximum weight stable set of a claw-free graph"J. Oper. Res. Soc.Japan. 44. 194-204 (2001)
[Publications] K.Murota, A.Tamura: "On circuit valuation of matroids"Adv. in Appl. Math.. 26. 192-225 (2001)
[Publications] A.Tamura: "Perfect (0, ± 1)-matrices and perfect bidirected graphs"Theoret.Comput. Sci.. 235. 339-356 (2000)
[Publications] Y.Ishigami: "Vertex-disjoint cycles of length at most four each of which contains a specified vertex"J.of Graph Theory. 37. 37-47 (2001)
[Publications] Y.Ishigami, H.Wang: "An extension of a theorem on cycles containing specified independent edges"Discrete Math.. (accepted).
[Publications] Y.Ishigami: "Proof of a conjecture of Bollobas and Kohayakawa on the Erdos-Stone theorem"J. of Comb. Theory(Ser. B). (accepted).
[Publications] Y.Ishigami: "A common extension of the Erdos-Stone theorem and the Alon-Yuster theorem for unbouded graphs"European Journal of Combinatorics. (accepted).
[Publications] T.Kojima, K.Ando: "Minimum length of cycles through specified vertices in graphs with wide-diameter at most d"Ars Combinatoria. 58. 245-256 (2001)
[Publications] K.Ando, A.Kaneko, K.Kawarabayashi: "Contractible edges and Bowtie in k-connected graphs"Ars Combinatoria. (accepted).
[Publications] K.Ando, Y.Egawi: "Maximum Number of Edges in a Critically k-connected Graph"Discrete Math.. (accepted).
[Publications] K.Ando, K.Kawarabayashi: "Some forbidden subgraph conditions for a graph to have a k-contractible edge"Discrete Math.. (accepted).
[Publications] K.Ando, M.Hagita, A.Kaneko, M.Kano, K.Kawarabayashi, A.Saito: "Cycles having the same modularity and removable edges in 2-connected graphs"Discrete Math.. (accepted).
[Publications] Y.Ishigami: "Proof of a conjecture of Bollobas and Kohayakawa on the Erdos-Stone theorem"J. of Comb. Theory(Ser. B). (accepted).
[Publications] Y.Ishigami: "A common extension of the Erdos-Stone theorem and the Alon-Yuster theorem for unbouded graphs"European Journal of Combinatorics. (accepted).
[Publications] T.Kojima, K.Ando: "Minimum length of cycles through specified vertices in graphs with wide-diameter at most d"Ars Combinatoria. 58. 245-256 (2001)
[Publications] K.Ando, A.Kaneko, K.Kawarabayashi: "Contractible edges and Bowtie in k-connected graphs"Ars Combinatoria. (accepted).
[Publications] K.Ando, Y.Egawi: "Maximum Number of Edges in a Critically k-connected Graph"Discrete Math.. (accepted).
[Publications] K.Ando, K.Kawarabayashi: "Some forbidden subgraph conditions for a graph to have a k-contractible edge"Discrete Math.. (accepted).
[Publications] K.Ando, M.Hagita, A.Kaneko, M.Kano, K.Kawarabayashi, A.Saito: "Cycles having the same modularity and removable edges in 2-connected graphs"Discrete Math.. (accepted).

2001 Fiscal Year Annual Research Report

グラフ論、離散最適化とその応用

Principal Investigator

安藤 清 電気通信大学, 電気通信学部, 教授 (20096944)

Research Products

[Publications] T.Asano, D.Z.Cand, N.Katoh, T.Tokuyama: "Efficient algorithms for optimization-based image segmentation"Internat. J. Comput. Geom. Appl.. 11. 145-166 (2001)

[Publications] T.Asano, N.Katoh, T.Tokuyama: "A unified scheme for detecting fundamental curves in binary edge images"Comput. Geom.. 18. 73-93 (2001)

[Publications] T.Asano, T.Matsui, T.okuyama: "on the complexities of the optimal rounding problems of sequences and matrices"Lecture Notes in Comput.Sci.. 1851. 476-489 (2000)

[Publications] D.Nakamura, A.Tamura: "A revision of Minty's algorithm for finding a maximum weight stable set of a claw-free graph"J. Oper. Res. Soc.Japan. 44. 194-204 (2001)

[Publications] K.Murota, A.Tamura: "On circuit valuation of matroids"Adv. in Appl. Math.. 26. 192-225 (2001)

[Publications] A.Tamura: "Perfect (0, ± 1)-matrices and perfect bidirected graphs"Theoret.Comput. Sci.. 235. 339-356 (2000)

[Publications] Y.Ishigami: "Vertex-disjoint cycles of length at most four each of which contains a specified vertex"J.of Graph Theory. 37. 37-47 (2001)

[Publications] Y.Ishigami, H.Wang: "An extension of a theorem on cycles containing specified independent edges"Discrete Math.. (accepted).

[Publications] Y.Ishigami: "Proof of a conjecture of Bollobas and Kohayakawa on the Erdos-Stone theorem"J. of Comb. Theory(Ser. B). (accepted).

[Publications] Y.Ishigami: "A common extension of the Erdos-Stone theorem and the Alon-Yuster theorem for unbouded graphs"European Journal of Combinatorics. (accepted).

[Publications] T.Kojima, K.Ando: "Minimum length of cycles through specified vertices in graphs with wide-diameter at most d"Ars Combinatoria. 58. 245-256 (2001)

[Publications] K.Ando, A.Kaneko, K.Kawarabayashi: "Contractible edges and Bowtie in k-connected graphs"Ars Combinatoria. (accepted).

[Publications] K.Ando, Y.Egawi: "Maximum Number of Edges in a Critically k-connected Graph"Discrete Math.. (accepted).

[Publications] K.Ando, K.Kawarabayashi: "Some forbidden subgraph conditions for a graph to have a k-contractible edge"Discrete Math.. (accepted).

[Publications] K.Ando, M.Hagita, A.Kaneko, M.Kano, K.Kawarabayashi, A.Saito: "Cycles having the same modularity and removable edges in 2-connected graphs"Discrete Math.. (accepted).

[Publications] Y.Ishigami: "Proof of a conjecture of Bollobas and Kohayakawa on the Erdos-Stone theorem"J. of Comb. Theory(Ser. B). (accepted).

[Publications] Y.Ishigami: "A common extension of the Erdos-Stone theorem and the Alon-Yuster theorem for unbouded graphs"European Journal of Combinatorics. (accepted).

[Publications] T.Kojima, K.Ando: "Minimum length of cycles through specified vertices in graphs with wide-diameter at most d"Ars Combinatoria. 58. 245-256 (2001)

[Publications] K.Ando, A.Kaneko, K.Kawarabayashi: "Contractible edges and Bowtie in k-connected graphs"Ars Combinatoria. (accepted).

[Publications] K.Ando, Y.Egawi: "Maximum Number of Edges in a Critically k-connected Graph"Discrete Math.. (accepted).

[Publications] K.Ando, K.Kawarabayashi: "Some forbidden subgraph conditions for a graph to have a k-contractible edge"Discrete Math.. (accepted).

[Publications] K.Ando, M.Hagita, A.Kaneko, M.Kano, K.Kawarabayashi, A.Saito: "Cycles having the same modularity and removable edges in 2-connected graphs"Discrete Math.. (accepted).

安藤清電気通信大学, 電気通信学部, 教授 (20096944)