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NewDevelopments of Discrete-System Algorithmics Based on Complexes

Research Project

Project/Area Number 10205204
Research Category

Grant-in-Aid for Scientific Research on Priority Areas (B)

Allocation TypeSingle-year Grants
Research InstitutionThe University of Tokyo

Principal Investigator

IMAI Hiroshi  The University Tokyo, Assistant professor, 理学系研究科, 助教授 (80183010)

Co-Investigator(Kenkyū-buntansha) INABA Mari  The University Tokyo, Lecturer, 理学系研究科, 講師 (60282711)
ASAI Kenichi  The University Tokyo, Assistant, 理学系研究科, 助手 (10262156)
Project Period (FY) 1998 – 2000
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥10,900,000 (Direct Cost: ¥10,900,000)
Fiscal Year 2000: ¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 1999: ¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 1998: ¥4,600,000 (Direct Cost: ¥4,600,000)
Keywordssimplicial complex / discrete system / invariant polynomials / Tutte polynomial / Binary Decision Diagram (BDD) / Groebner bases / quantun information / マトロイド / Grobner基底 / 計算幾何・計算代数 / 最適化 / 情報幾何 / Voronoi図
Research Abstract

Simplicial complexes arise in various discrete systems, such as graphs, networks, matroids, convex polytopes, triangulations, etc. There have been known discrete invariant polynomials featuring a given simplicial complex, such as network reliability, chromatic polynomial, invariants of statistical physics and knots/links. We have developed efficient algorithms to compute these invariant polynomials based on the Binary Decision Diagram (BDD), and have shown that moderate-size problems can be solved in practice. Since most of these computation problems are known to be #P-hard, this is a quite remarkable result. We have also presented algebraic approaches to these problems, based on the Groebner bases in computational algebra and also triangulations in computational geometry. This unified approach was analyzed from the viewpoint of combinatorial complexity. We have further investigated new search methods for combinatorial search, geometric and probabilistic proximity structures, full text database search algorithms, etc. Finally, quantum computing and quantum information theory are treated as a natural extension of probabilistic computation and classical information theory, and their discrete structures have been revealed. In so doing, quantum computation simulators have been implemented and used. Some submodularity property of the quantum entropy is also shown.

Report

(4 results)
  • 2001 Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • Research Products

    (32 results)

All Other

All Publications (32 results)

  • [Publications] 関根京子, 今井浩, 今井桂子: "Jones多項式の計算"日本応用数理学会論文誌. Vol.8,No.3. 341-354 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Imai, H., Inaba, M.: "Geometric Clustering by Divergence and Its Underlying Discrete Proximity Structures"IEICE Transactions on Information and Systems. Vol.E83-D,No.2. 27-35 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Imai, H.: "Computing the Invariant Polynomials of Graphs,Networks and Matroids"IEICE Transactions on Information and Systems. Vol.E83-D,No.3. 330-343 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Inaba, M., Katoh, N., Imai, H.: "Clustering Algorithms by Voronoi Diagrams and Randomization"IEICE Transactions on Information and Systems. Vol.E83-D,No.6. 1199-1206 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Ishizeki, T., Imai, H.: "Bases of Acyclic Directed Graphs and Minimum Cost Flow Problems"Proceedings of the 2nd Japanese-Hungarian Symposium on Discrete Mathematics and Its Application Budapest. 82-91 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Gruska, J., Imai, H.: "Power,Puzzles and Properties of Entanglement Proceedings of the 3rd International Conference on Machines,Computations and Universality"Lecture Notes in Computer Science. Vol.2055. 25-68 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 浅野孝夫, 今井浩: "オーム社"計算とアルゴリズム. (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Imai, H., Inaba, M: "Geometric Clustering by Divergence and Its Underlying Discrete Proximity Structures"IEICE Transactions on Information and Systems. Vol.E83-D, No.2. 27-35 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Imai, H: "Clustering Algorithms by Voronoi Diagrams and Randomization"IEICE Transactions on Information and Systems. Vol.E83-D, No.3. 330-343 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Inaba, M., Katoh, N., Imai, H: "Clustering Algorithms by Voronoi Diagrams and Randomization"IEICE Transactions on Information and Systems. Vol.E83-D, No.6. 1199-1206 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Ishizeki, T., Imai, H: "Bases of Acyclic Directed Graphs and Minimum Cost Flow Problems"Proceedings of the 2nd Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications Budapest. 82-91 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Ishizeki and H.Imai: "Grobner Bases of Acyclic Tournament Graphs and Hypergeometric Systems on the Group of Unipotent Matrices"京都大学数理解析研究所講究録. 1171. 7-20 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Ishizeki and H.Imai: "Grobner Bases for Toric Ideals of Acyclic Directed Graphs and Their Applications to Minimum Cost Flow Problem"統計数理研究所共同研究レポート. 135. 152-165 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Imai and M.Inaba: "Geometric Clustering by Divergence and Its Underlying Discrete Proximity Structures"IEICE Trans.Information and Systems. E83-D,1. 27-35 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Inaba,N.Katoh and H.Imai: "Variance-Based k-Clustering Algorithms by Voronoi Diagrams and Randomization"IEICE Trans.Information and Systems. E83-D,6. 1199-1206 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Imai,M.Hachimori,M.Hamada,H.Kobayashi and K.Matsumoto: "Optimization in Quantum Computation and Information"Proceedings of the 2nd Japanese-Hungarian Symposium on Discrete Mathematics. (4月出版予定). (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Ishizeki and H.Imai: "Grobner Bases of Acyclic Directed Graphs and Minimum Cost Flow Problems"Proceedings of the 2nd Japanese-Hungarian Symposium on Discrete Mathematics. (4月出版予定). (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H. Imai, K. Sekine and K. Imai: "Computational Investigations of All-Terminal Network Reliability via BDDs"IEICE Trans. Fundamentals. E82-A,5. 714-721 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 今井浩: "離散システム不変多項式と計算位相・幾何"日本数学会1999年度秋季総合分科会応用数学分科会特別講演. 44-56 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 今井浩、関根京子: "グラフのTutte多項式計算システム"京都大学数理解析研究所講究録. 1120. 120-129 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 石関隆幸、今井浩: "トーナメントグラフのトーリックイデアルのGrobner基底について"情報処理学会SIGAL. (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 徳永裕己、今井浩: "量子計算機シミュレーションシステム"京都大学数理解析研究所講究録. 1120. 110-119 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H. Imai: "Computing the Invariant Polynomials of Graphs, Networks and Matroids"IEICE Trans. Information and Systems. E83-D,3(印刷中). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] T. Asano, H. Imai, D. T. Lee, S. Nakano, T. Tokuyama, eds.: "Computing and Combinatorics"Soriger-Verlag. 494 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 浅野孝夫、今井浩: "計算とアルゴリズム"オーム社. 295 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Hayase and H.Imai: "OBDDs of a Monotone Function and of ITS Prime Implicarts" Theory of Computing Systems. vol.31. 579-591 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 関根京子.今井浩.今井桂子: "Jones多項式の計算" 日本応用数理学会論文誌. vol.8.no.3. 341-354 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Ikeda and H.Imai: "Enhauced A^* Algorithms for Multiple Alignments : Optimal Alignments for Several Sequences and \K\-Opt Aporaximate Alignments for Large Cases" Theoretical Computer Science. vol.210, No.2. 341-374 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Inaba and H.Imai: "Geomatric Clustering in Feature Space" Lecture Notes in Artificial Intelligence. vol.1532. 421-422 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Keiko Imai and Hiroshi Imai: "Dynamic Weighted Voronei Diagrams and Weighted Minimax Matching of Two Corresponding Prints sets" Optimizatiom Methods and Software. vol.10. 261-274 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Imai, K.Sekine and K.Imai: "Computational Investigations of All-Tarminal Network Reliability via BDDs," IEICE Trams Fundamantals. vol.E82・A No.5. (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 今井 浩: "ネットワーク信頼度計算の周辺-組合せ数え上げの新展開離散構造とアルゴリズムV(藤重 悟編)" 近代科学社, 50 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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