Symmetric Splendor: Unraveling Universally Closest Refinements and Fisher Market Equilibrium through Density-Friendly Decomposition

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.