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

TL;DR

This paper introduces a unified framework for vertex-weighted bipartite graph problems, connecting hypergraph decomposition, distribution refinements, and Fisher Market equilibrium to improve understanding and solutions.

Contribution

It unifies diverse problems through a symmetric density-friendly decomposition framework, enabling new insights and cross-application of algorithms.

Findings

Unified framework for multiple graph and market problems

Symmetric hypergraph decomposition characterizes optimal solutions

Framework facilitates algorithm transfer across problems

Abstract

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.