2018 Fiscal Year Final Research Report

Developments of discrete optimization theory and efficient algorithms based on submodular structures

Research Project

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Project/Area Number	25280004
Research Category	Grant-in-Aid for Scientific Research (B)
Allocation Type	Partial Multi-year Fund
Section	一般
Research Field	Mathematical informatics
Research Institution	Kyoto University
Principal Investigator	Fujishige Satoru 京都大学, 数理解析研究所, 特任教授 (10092321)
Co-Investigator(Kenkyū-buntansha)	牧野和久京都大学, 数理解析研究所, 教授 (60294162) 平井広志東京大学, 大学院情報理工学系研究科, 准教授 (20378962) 高澤兼二郎法政大学, 理工学部, 准教授 (10583859) 谷川眞一東京大学, 大学院情報理工学系研究科, 准教授 (30623540)
Project Period (FY)	2013-04-01 – 2019-03-31
Keywords	離散最適化 / 組合せ最適化 / 劣モジュラ関数 / 離散アルゴリズム
Outline of Final Research Achievements	From the view points of the discrete structures associated with submodularity, we have investigated the theory and algorithms for discrete and combinatorial optimization problems which has been drawing researchers' attention in the world. We have developed a new theory of discrete convex functions, based on submodular structures, and effectively applied the theory to combinatorial and discrete optimization problems. We have also examined a general class of submodular-like discrete structures such as transversal-submodular functions and skew-bisubmodular functions. We then have shown fundamental min-max theorems for such discrete systems and investigated discrete algorithmic structures. We have also shown that the submodular structures play crucial roles in economy with indivisible commodities and a class of allocation problems in gaming situations.
Free Research Field	組合せ最適化
Academic Significance and Societal Importance of the Research Achievements	大規模な離散最適化・組合せ最適化問題を厳密に、あるいは、近似的に解くための効率的アルゴリズムの構築に向けて、劣モジュラ的な離散構造の有効性を明らかにした。学術的な意義としては、離散最適化・組合せ最適化の理論と効率的アルゴリズムの基礎となる、離散凸関数の新たな理論構築や、新たな劣モジュラ的関数に関連する最大・最小定理の導出等であり、社会的意義としては、非分割財の経済やゲーム論的配分の問題における劣モジュラ構造の有用性の解明等である。