2002 Fiscal Year Final Research Report Summary
On Algorithms and Applications of Semidefinite Programming to Combinatorial Optimization
| Project/Area Number |
13640114
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
TAMURA Akihisa Research Institute for Mathematical Sciences, Associate Professor, 数理解析研究所, 助教授 (50217189)
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| Co-Investigator(Kenkyū-buntansha) |
FURIHATA Daisuke Osaka University, Cybermedia Center (2001), Lecturer, サイバーメディアセンター(13年度のみ), 講師 (80242014)
FUJIE Tetsuya Kobe University of Commerce, Institute of Economic Research, Research Associate, 商経学部, 助手 (40305678)
MUROTA Kazuo Research Institute for Mathematical Sciences, (2001), The University of Tokyo, Graduate School of Information Science and Technology (2002), Full Professor, 数理解析研究所, 教授 (50134466)
OOURA Takuya Research Institute for Mathematical Sciences, (2002), Research Associate, 数理解析研究所(14年度のみ), 助手 (50324710)
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| Project Period (FY) |
2001 – 2002
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| Keywords | Semidefinite Programming / Discrete Convex Analysis / Algorithms / Bidirected Graphs / Perfect Graphs |
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| Research Abstract |
We obtained several results on semidefinite programming and discrete convex analysis and presented these results at international conferences, e.g., International Symposium on Algorithms and Computation, Conference on Integer Programming and Combinatorial Optimization and so on. Here we explain seven results among these.
1) Fujie and Tamura generalized the theory of a convex set relaxation for the maximum weight stable set problem to the generalized stable set problem. They also gave simple proofs for several results for the maximum weight stable set problem.
2) Murota et al. proved that the numerically obtained solution of group symmetric semidefinite program is group symmetric.
3) Murota et al. gave a framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs.
4) Murota and Tamura gave an efficient algorithm to decide whether a competitive equilibrium exists or not in some economic model by utilizing M-convex submodular flow problem.
5) Tamura devised a new scaling technique and an efficient algorithm for M-convex function minimization problem.
6) Murota and Tamura proved proximity theorems for several discrete convex functions, for example, M2-convex functions, L2-convex functions and so on.
7) Tamura showed a technical result that any L2-convex function can be represented by the convolution of two L-convex functions attaining the infimum in the definition of the convolution. This result gives simple proofs for several known results on L2-convex functions.
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