We present a comprehensive framework that unifies several research areas within the context of vertex-weighted bipartite graphs, providing deeper insights and improved solutions. The fundamental solution concept for each problem involves refinement, where vertex weights on one side are distributed among incident edges. The primary objective is to identify a refinement pair with specific optimality conditions that can be verified locally. This framework connects existing and new problems that are traditionally studied in different contexts. We explore three main problems: (1) density-friendly hypergraph decomposition, (2) universally closest distribution refinements problem, and (3) symmetric Fisher Market equilibrium. Our framework presents a symmetric view of density-friendly hypergraph decomposition, wherein hyperedges and nodes play symmetric roles. This symmetric decomposition serves as a tool for deriving precise characterizations of optimal solutions for other problems and enables the application of algorithms from one problem to another.

翻译：我们提出了一个在顶点赋权二分图背景下统一多个研究领域的综合性框架，该框架提供了更深入的见解和更优的解决方案。每个问题的基本解概念都涉及细化过程，即一侧的顶点权重被分配到其关联边上。主要目标是找到满足特定最优性条件的细化对，这些条件可以在局部进行验证。该框架将传统上在不同背景下研究的现有问题及新问题联系起来。我们探讨了三个主要问题：(1) 密度友好超图分解，(2) 通用最邻近分布细化问题，以及 (3) 对称Fisher市场均衡。我们的框架为密度友好超图分解提供了一个对称视角，其中超边与节点扮演对称角色。这种对称分解可作为工具，用于推导其他问题最优解的精确刻画，并使得一个问题的算法能够应用于另一个问题。