Submodular flows and extreme flows on measurable spaces
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866916072252243968 |
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| author | Yu, Jing Zhang, Junchi Zhou, Mingyang |
| author_facet | Yu, Jing Zhang, Junchi Zhou, Mingyang |
| contents | The theory of submodular flows, introduced by Edmonds and Giles, is a cornerstone of combinatorial optimization, unifying network flows, matroid intersections and directed cut coverings. In this paper, we establish a measurable-space version of this framework, addressing the structural existence and duality questions raised as part of Problem~10.6 by Lovász in \textit{Submodular setfunctions on sigma-algebras, version 2}.
We develop a theory of submodular flows on standard Borel spaces and establish the measurable analogues of the existence and optimality theorems. Furthermore, we introduce a measurable notion of the residual graph and characterize extreme flows by combining a base-polytope intersection condition with an acyclicity condition for the measurable residual graph, generalizing the discrete geometric intuition to the infinite-dimensional setting. Finally, we apply the theory to constrained supply-demand problems on measurable bipartite graphs and to fractional measurable orientations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01684 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Submodular flows and extreme flows on measurable spaces Yu, Jing Zhang, Junchi Zhou, Mingyang Combinatorics The theory of submodular flows, introduced by Edmonds and Giles, is a cornerstone of combinatorial optimization, unifying network flows, matroid intersections and directed cut coverings. In this paper, we establish a measurable-space version of this framework, addressing the structural existence and duality questions raised as part of Problem~10.6 by Lovász in \textit{Submodular setfunctions on sigma-algebras, version 2}. We develop a theory of submodular flows on standard Borel spaces and establish the measurable analogues of the existence and optimality theorems. Furthermore, we introduce a measurable notion of the residual graph and characterize extreme flows by combining a base-polytope intersection condition with an acyclicity condition for the measurable residual graph, generalizing the discrete geometric intuition to the infinite-dimensional setting. Finally, we apply the theory to constrained supply-demand problems on measurable bipartite graphs and to fractional measurable orientations. |
| title | Submodular flows and extreme flows on measurable spaces |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2606.01684 |