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arXiv · 2606.01684

Submodular flows and extreme flows on measurable spaces

Jing Yu·Junchi Zhang·Mingyang Zhou
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Abstract

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 Lov\'asz 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.

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Jing Yu, Junchi Zhang, Mingyang Zhou. 2026-06-01. Submodular flows and extreme flows on measurable spaces. https://arxiv.org/abs/2606.01684

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