| Project/Area Number |
15360043
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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 |
Engineering fundamentals
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| Research Institution | The University of Tokyo |
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Principal Investigator |
MUROTA Kazuo The University of Tokyo, Graduate School of Information Sciences and Technology, Professor, 大学院・情報理工学系研究科, 教授 (50134466)
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| Co-Investigator(Kenkyū-buntansha) |
TAMURA Akihisa Keio University, Science and Technology, Assistant Professor, 理工学部, 教授 (50217189)
IWATA Satoru The University of Tokyo, Graduate School of Information Sciences and Technology, Assistant Professor, 大学院・情報理工学系研究科, 助教授 (00263161)
TSUCHIMURA Nobuyuki The University of Tokyo, Graduate School of Information Sciences and Technology, Research Associate, 大学院・情報理工学系研究科, 助手 (20345119)
塩浦 昭義 東北大学, 大学院・情報科学研究科, 助教授 (10296882)
松浦 史郎 東京大学, 大学院・情報理工学系研究科, 学術研究支援員 (00332619)
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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 |
¥10,200,000 (Direct Cost: ¥10,200,000)
Fiscal Year 2005: ¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2004: ¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2003: ¥3,800,000 (Direct Cost: ¥3,800,000)
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| Keywords | convex analysis / convex function / convex set / matroid / discrete optimization / mathematical programming / nonlinear programming / network flow / 双対性 / 劣モジュラ関数 / アルゴリズム / 離散凸関数 / 組合せ最適化 |
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| Research Abstract |
The main aim of this research project is to establish a novel paradigm of discrete convexity by exploiting applications of the theory of discrete convex analysis to fundamental problems in economics, system engineering, operations research, optimization, algorithm science, and others. In this research project, we obtained a number of results described below. All of the results are presented at refereed international conference or accepted for publications in international scientific journals.
(1)For optimization problems related to discrete convexity (in particular, for the submodular flow problem with M-convex cost function), we developed a number of efficient algorithms using the capacity scaling technique. The obtained algorithm has the best known complexity bound.
(2)We pointed out that the concept of multimodular functions, which had been conceived independently of discrete convex analysis, is essentially equivalent to the concept of L-convex functions. On the basis of this observation we derived some properties including discrete separation.
(3)We revealed a close relationship between M-convex functions and tree metrics. Specifically, we proved that the Hessian matrix of a quadratic M-convex function is essentially the same as the negative of the tree metric matrix.
(4)We established a discrete fixed point theorem for functions defined on integrally convex sets. The theorem finds applications in mathematical economics and game theory.
(5)We provided new characterizations of M-convex functions in terms of gross substitution and single improvement property known in mathematical economics.
(6)As applications of discrete convex analysis to mathematical economics, we developed a theory for economic equilibrium models with indivisible goods. We also developed an algorithm for computing a competitive equilibrium using the framework of M-convex submodular flow problem.
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