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The Analysis of Computational Complexity of Discrete Problems

Research Project

Project/Area Number 13640139
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionNihon University

Principal Investigator

TODA Seinosuke  Nihon University, Dept. Computer Science and System Analysis, Professor, 文理学部, 教授 (90172163)

Co-Investigator(Kenkyū-buntansha) CHEN Zhi-zhong  Tokyo Denki University, Dept. Mathematical Science, Associate Professor, 理工学部, 助教授 (00242933)
TANI Sei-ichi  Nihon University, Dept. Computer Science and System Analysis, Associate Professor, 文理学部, 助教授 (70266708)
YAKU Takeo  Nihon University, Dept. Computer Science and System Analysis, Professor, 文理学部, 教授 (90102821)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2003: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2002: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2001: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsComputational Complexity / graph theory / graph isomorphism / self avoiding walk / polynomial time algorithm / #P complete / chordal graph / grid graph / 2次元格子グラフ / 超立方体グラフ / self-avoiding walk / 数え上げ問題 / 結び目 / 自明性判定問題 / ブレイド / 共役問題 / 絡み目 / 同型写真像 / 多項式時間 / マッチング / 近似アルゴリズム / アルゴリズム / 多次元格子 / 超立方体 / 数え上げ / 多項式
Research Abstract

In this research project, we mainly investigate the computational complexity of discrete problems. In particular, we dealt with graph Isomorphism problem and the problem of counting self avoiding walks in graphs. At this point, exploring the. precise complexity of the problems has remained to be important open questions in computational complexity theory while many researches were done so far. We currently believe that investigating their computational complexity may give us a new insight on the structure of computations. In this research project, we obtained several results mentioned as follows. Related to the graph isomorphism problem, we first showed that the problem of counting graph isomorphisms among partial k-trees was computable in polynomial time with developing a dynamic programming algorithm. In this algorithm, we had to compute the permanent of bipartite graphs, which is the number of perfect matching in bipartite graphs. In usual, such a computation has appeared to be hard. But, in our case, we found that bipartite graphs concerned had a strong symmetry, and then we succeeded to design an efficient algorithm for computing the permanent. We further showed that the graph isomorphism problem on the class of chordal bipartite graphs and on the class of strongly chordal graphs remained to be GI -complete.. These results refine the previous knowledge on the complexity. of the problem. We also showed that the problems of counting self-avoiding walks both in two-dimensional grid graphs and in hypercube graphs were complete for #P. This is a first result concerned on the complexity of the problem. We further showed that the problem was #EXP-complete in case that an input graph was given in a succinct representation form. We further designed a linear-time algorithm for 7-coloring 1 -planar graphs, and we study a possibility of developing software systems with using graph grammar theory.

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (23 results)

All Other

All Publications (23 results)

  • [Publications] 戸田誠之助: "グラフ同型性判定問題の計算量"電子情報通信学会論文誌. J85-D-I. 100-115 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 名古屋孝幸, 谷聖一, 戸田誠之助: "グラフ同型写像数え上げ問題に対するアルゴリズムについて"電子情報通信学会論文誌. J85-D-I. 424-435 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Liskeiwicz, M.Ogihara, S.Toda: "The Complexity of Counting Self-Avoiding Walks in Two-Dimensional Grid graphs and in Hypercube Graphs"Theoretical Computer Science. 304. 129-156 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Z-Z.Chen, M.Kouno: "A Linear-Time Algorithm for 7-Coloring 1-Planar Graphs"Lecture Notes in Computer Science. 2747. 348-357 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 戸田誠之助: "グラフ同型性判定問題"冨山房. 129 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Toda: "Computational Complexity of Graph Isomorphism Problem"IEICE Japanese Transactions on Information and Systems. Vol.J85-D-I, No2. 100-115 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Nagoya, S.Tani, S.Toda: "A Polynomial-Time Algorithm for Counting Graph Isomorphisms among Partial k-trees"IEICE Japanese Transactions on Information and Systems. Vol.J85-D-I, No2. 100-115 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Liskeiwicz, M.Ogihara, S.Toda: "The Complexity of Counting Self-Avoiding Walks in Two-Dimensional Grid Graphs and in Hypercube Graphs"Theoretical Computer Science. Vol.304,No.1-3. 129-156 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Z-Z.Chen, M.Kouno: "A Linear-Time Algorithm for 7-Coloring 1-Planar Graphs"Lecuture Notes in Computer Science. Vol.2747. 348-357 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Liskiewicz, M.Ogihara, S.Toda: "The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes"Theoretical Computer Science. Vol.304 No.1-3. 129-156 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 原正雄, 谷聖一, 山本慎: "結び目の非自明性判定問題の計算量について"京都大学数理解析研究所講究録. 1325. 227-232 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Matsuba, S.Tani: "On Computational Complexity of the Conjugacy Problems for Braids"電子情報通信学会信学技報. CPMP-2003-88. 17-23 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] 名古屋, 谷, 戸田: "グラフ同型写像の数え上げ問題に対するアルゴリズムについて"電子情報通信学会論文誌. Vol.J85-D-1 No.5. 424-435 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 原, 谷, 山本: "Anborescent絡み目に対するジョーンズ多項式計算アルゴリズム"情報技術レターズ. Vol.1. 16-17 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Motohashi, K.Tsuchido, T.Yaku: "Attribute graph for tables and their algorithms"ソフトウェア工学の基礎(近代科学社). 9. 183-186 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Z.Z.Chen et al.: "Better Approximation Algorithms for NMR Spectral Peak Assignment"Lecture Notes in Computer Science. Vol.2452. 82-96 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 戸田誠之助: "グラフ同型性判定問題の計算量"電子情報通信学会論文誌 D-I. D-I,J85-D-I. 100-115 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] M. Ogihara, S. Toda: "The Complexity of Computing the Number of Self-Avoiding Walks in Two-Dimensional Grids and in Hypercube Graphs"Lecture Notes in Computer Science. 2136. 585-597 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] G.-H. Lin, Z.-Z. Chen, T. Jiang, J. Wen: "The Longest Common Subsequence Problem for Sequences with Nested Arc Annotations"Lecture Notes in Computer Science. 2076. 444-455 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T. Arita et. al.: "Syntactic Tabular Form Processing by Precedence Attribute Graph Grammars"Proc. IASTED Applied Informatics. 637-642 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K. Yamazaki, S. Tani, T. Nishino: "A Characterization of k-th Powers Pn, k of Paths in terms of k-Trees"Int. J. of Found. of Comput. Sci.. Vol.12,No.4. 435-443 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 原正雄, 谷聖一, 山本慎: "Arborescent linkのJones多項式の計算アルゴリズムについて"京都大学数理解析研究所講究録. (掲載予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] 戸田誠之助: "グラフ同型性判定問題"冨山房. 129 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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