1997 Fiscal Year Final Research Report Summary
Fundamental studies on methods for analyzing large-scale combinatorial systems with submodular structures
| Project/Area Number |
08680450
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
社会システム工学
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| Research Institution | Osaka University (1997)
University of Tsukuba (1996) |
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Principal Investigator |
FUJISIGE Satoru Osaka Univ., DEPT.of CHEM.SCI.ENG., professor, 大学院・基礎工学研究科, 教授 (10092321)
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| Project Period (FY) |
1996 – 1997
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| Keywords | Submodular functions / Combinatorial optimization |
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| Research Abstract |
Submodular structures often play fundamental and important roles in a lot of problems related to discrete, combinatorial and algebraic systems such as graphs and networks through matroidal structures and flow boundary polyhedra. In the present research project we have investigated the fundamental properties of submodular structures that are useful in analyzing large-scale discrete systems with submodular structures, and developed algorithms for solving problems for such discrete systems.
The submodular flow problem is one of the most powerful mathematical models for optimization problems with submodular structures and its solution algorithms have been intensively investigated through the world. In the present research project we extended the push/relabel technique for network flows of A.V.Goldberg to the submodular flow problem and showed the effectiveness of the algorithm.
We also investigated a combinatorial system, called bisubmodular system, that generalizes the combinatorial system
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with submodular structures. We obtained several useful results. First, we showed each domain of a bisubmodular function can be expressed as a collection of ideals of a signed poset. We proposed an efficient algorithm for finding a minimum-weight ideal of a poset. We also developed an efficient algorithm for decomposing a bidirected graph into strongly connected components and showed the poset structure on the set of strongly connected components. Secondly, we revealed the fundamental properties of bisubmodular functions and associated polyhedra called bisubmodular polyhedra. In particular, we showed a min-max theorem with integrality for characterizing the mimimum-norm point, relative to l_1 norm, on a bisubmodular, polyhedron. This is an extremely powerful theorem with integrality that includes as special cases a lot of combinatorial min-max theorems such as intersection theorem for matroids, polymatroids and submodular systems. We expect that it will play an important role in dealing with various combinatorial optimization problems. We also revealed the relationship between bidirected network flows and bisubmodular systems. Less
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