2007 Fiscal Year Final Research Report Summary
Analysis of Large-scale Discrete Optimization Problems and Development of Efficient Algorithms Based on Submodularity Structures
| Project/Area Number |
16310111
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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 |
Social systems engineering/Safety system
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| Research Institution | Kyoto University |
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Principal Investigator |
FUJISHIGE Satoru Kyoto University, Research Institute for Mathematical Sciences, Professor (10092321)
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| Co-Investigator(Kenkyū-buntansha) |
TAMURA Akihisa Keio University, Department of Mathematics, Professor (50217189)
MAKINO Kazuhisa University of Tokyo, Graduate School of Information and Technology, Associate Professor (60294162)
HIRAI Hiroshi Kyoto University, Research Institute for Mathematical Sciences, Assistant Professor (20378962)
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| Project Period (FY) |
2004 – 2007
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| Keywords | Discrete Optimization / Algorithms / Submodular Functions / Combinatorial Optimization / Mathematical Programming |
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| Research Abstract |
Major results of our research are the following.
1. By utilizing discrete concave functions and considering possibly bounded side payments, we have established a common generalization of the marriage model due to Gale and Shapley and the assignment model due to Shapley and Shubik that are standard in the theory of two-sided matching markets, and have shown the existence of a pairwise stable outcome in our model.
2. We have presented a first polynomial-time algorithm for the monotone min-max connected partitioning problem and have shown that the min-max connected partitioning problem is NP-hard if the cost function is not monotone, and that the min-sum connected partitioning problem is NP-hard even if the cost function is monotone. We also considered an evacuation problem in dynamic networks as an application of the tree partitioning problem.
3. Bisubmodular functions are a natural "directed", or "signed", extension of submodular functions with several applications. We have investigated th
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e difficulty of extending the strongly polynomial version of the ordinary submodular function minimization algorithms to bisubmodular function minimization (BSFM), and we have showen a way around the difficulty. This new method gives the first combinatorial strongly polynomial algorithm for BSFM.
4. We have considered minimum-cost source-location problems and their generalizations with three connectivity requirements (arc-connectivity requirements and two kinds of vertex-connectivity requirements), and have shown that the source location problem with edge-connectivity requirements in undirected networks is strongly NP-hard. Moreover, we have shown that the source location problems with three connectivity requirements are inapproximable within a ratio of c In D for some constant c, unless every problem in NP has an O (N log^2 N) -time deterministic algorithm. Here D denotes the sum of given demands. We have also devised (1+ In D) -approximation algorithms for all the extended source location problems if we have the integral capacity and demand functions.
5. We have investigated support vector machine (SVM) with a discrete kernel, named electric network kernel, mined on the vertex set of an undirected graph. Emphasis is laid on mathematical analysis of its theoretical properties with the aid of electric network theory and the theory of discrete metrics. SVM with this kernel admits physical interpretations in terms of resistive eletric networks; in particular, the SVM decision function corresponds to an electric potential.
6. We nave considered a problem of finding a minimum transversal that can be regarded as a natural generalization of source location problems and external network problems in (undirected) graphs and hypergraphs. We have found an interesting structural characterization of minimal deficient sets and have shown a necessary and sufficient condition for such sets to form a tree hypergraph. By using this characterization, we have obtained a polynomial-time algorithm, which provides first polynomial-time algorithms for source location problem in hypergraphs and external network problems in graphs and hypergraphs. Less
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