Nonbipartite Dulmage-Mendelsohn Decomposition for Berge Duality (Foundations and Applications of Algorithms and Computation)
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- Kita, Nanao
- National Institute of Informatics
Bibliographic Information
- Other Title
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- ベルジュ双対のための非二部的Dulmage-Mendelsohn分解 (アルゴリズムと計算理論の基礎と応用)
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Abstract
The Dulmage-Mendelsohn decomposition is a classical canonical decomposition in matching theory applicable for bipartite graphs and is famous not only for its application in the field of matrix computation, but also for providing a prototypal structure in matroidal optimization theory. The Dulmage-Mendelsohn decomposition is stated and proved using the two color classes of a bipartite graph, and therefore generalizing this decomposition for nonbipartite graphs has been a difficult task. In our study, we obtain a new canonical decomposition that is a generalization of the Dulmage-Mendelsohn decomposition for arbitrary graphs using a recently introduced tool in matching theory, the basilica decomposition. Our result enables us to understand all known canonical decompositions in a unified way. Furthermore, we apply our result to derive a new theorem regarding barriers. The duality theorem for the maximum matching problem is the celebrated Berge formula, in which dual optimizers are known as barriers. Several results regarding maximal barriers have been derived by known canonical decompositions; however, no characterization has been known for general graphs. In our study, we provide a characterization of the family of maximal barriers in general graphs, in which the known results are developed and unified.
Journal
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- RIMS Kokyuroku
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RIMS Kokyuroku 2088 36-43, 2018-08
京都大学数理解析研究所
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Details 詳細情報について
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- CRID
- 1050285299787405952
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- NII Article ID
- 120006861217
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- NII Book ID
- AN00061013
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- ISSN
- 18802818
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- HANDLE
- 2433/251596
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- Text Lang
- en
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- Article Type
- departmental bulletin paper
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- Data Source
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- IRDB
- CiNii Articles