| Project/Area Number |
15340007
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Osaka University |
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Principal Investigator |
HIBI Takayuki Osaka University, Graduate School of Information Science and Technology, Professor, 大学院・情報科学研究科, 教授 (80181113)
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| Co-Investigator(Kenkyū-buntansha) |
SAITO Mutsumi Hokkaido University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (70215565)
OHSUGI Hidefumi Rikkyo University, School of Science, Associate Professor, 理学部, 助教授 (80350289)
MATSUI Yasuko Tokai University, School of Science, Assistant Professor, 理学部, 講師 (10264582)
TAKAYAMA Yukihide Ritsumeikan University, School of Information Science, Professor, 理工学部, 教授 (20247810)
TERAI Naoki Saga University, School of Education, Associate Professor, 文化教育学部, 助教授 (90259862)
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| Project Period (FY) |
2003 – 2005
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| Project Status |
Completed (Fiscal Year 2005)
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| Budget Amount *help |
¥8,600,000 (Direct Cost: ¥8,600,000)
Fiscal Year 2005: ¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2004: ¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2003: ¥3,000,000 (Direct Cost: ¥3,000,000)
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| Keywords | Grobner Basis / Universal Grobner Basis / Lattice Ideal / Integer Programming / Corner Polytope / Finite Graph / Toric Ideal / Initial Ideal / 隣接行列 / 符号理論 / 組合せ最適化問題 / Gomory relaxation / corner polyhedron |
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| Research Abstract |
The lattice ideal of dimension zero appears in the research of both pure mathematics and applied mathematics. The original purpose of the present research project was to establish the algebraic theory of universal Groebner bases of lattice ideal of dimension zero and to study of its theoretical effectivity to commutative algebra and algebraic geometry as well as its practical effectivity to integer programming, coding theory together with algebraic statistics. First, we investigated the universal Groebner basis of the lattice ideal of dimension zero arising from the toric ideal of a finite graph and succeeded in describing its structure in terms of the finite graph. Second, in the study of a problem on integer programming arising from a finite graph for which Gomory's relaxation can be applied, we developed the technique to decide the estimation of the computational complexity of finding an optimal solution by using combinatorics on finite graphs. Third, we achieved the study of finding an explicit expression of the corner polyhedron of a lattice ideal of dimension zero in terms of the Minkowski sum of a bounded convex polytope and a convex cone, and obtained some results on the combinatorics of the polyhedral structure of the corner polyhedron. Finally, we developed the algebraic study on the Markov basis of the contingency table in algebraic statistics and presented a statistic model arising from a complete multipartite graph. These research results will contribute to the development of the algebraic study of integer programming. In addition, we organized two international meetings related with computational commutative algebra and Groebner bases.
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