| Project/Area Number |
17K00029
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Mathematical informatics
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| Research Institution | The University of Tokyo |
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Principal Investigator |
Hirai Hiroshi 東京大学, 大学院情報理工学系研究科, 准教授 (20378962)
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| Co-Investigator(Kenkyū-buntansha) |
岩政 勇仁 京都大学, 情報学研究科, 助教 (70854602)
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| Project Period (FY) |
2017-04-01 – 2021-03-31
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| Project Status |
Completed (Fiscal Year 2020)
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| Budget Amount *help |
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2019: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2018: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | 離散凸解析 / 劣モジュラ最適化 / 多項式時間アルゴリズム / 代数的アルゴリズム / CAT(0)空間 / 組合せ最適化 |
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| Outline of Final Research Achievements |
In this research project, we explored new types of discrete convexity, which will be useful for discrete optimization, and designed algorithms based on it. We introduced the problem of computing the Dieudonne determinant of a matrix having noncommutative variables, and showed that it generalizes fundamental combinatorial optimization problems and can be efficiently solved by discrete convex optimization on a Euclidean building. We introduced a new class of lattices, called uniform semimodular lattices, and showed that it is equivalent to valuated matroids, which is an important class of discrete convex functions. We studied systematically a class of graphs, called weakly modular graphs, which is expected as ground structures of discrete convex functions, and clarified its relationships to nonpositively curved spaces.
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| Academic Significance and Societal Importance of the Research Achievements |
これまでは整数格子のうえでの最適化のための離散凸性として研究されてきたことを，本研究課題では，より一般的なグラフや多面体を貼り合わせた空間上の最適化のための離散凸性へ拡張すること目標としている．そのような視点が有効となる問題が理論計算機科学の最先端においてじょじょに現れてきている．実際，本研究でも扱った非可換ランクの概念は，不変式論，量子情報，幾何学的計算量理論など広い分野にまたがる応用が見出されつつある．本課題の成果がそうした問題を扱う際の礎となることが期待される．
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