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Unraveling Universally Closest Refinements via Symmetric Density Decomposition and Fisher Market Equilibrium

Authors T-H. Hubert Chan , Quan Xue

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Author Details

T-H. Hubert Chan
  • Department of Computer Science, University of Hong Kong, Hong Kong
Quan Xue
  • Department of Computer Science, University of Hong Kong, Hong Kong

Funding

  • Chan, T-H. Hubert: This research was partially supported by the Hong Kong RGC grants 17201823, 17203122 and 17202121.

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T-H. Hubert Chan and Quan Xue. Unraveling Universally Closest Refinements via Symmetric Density Decomposition and Fisher Market Equilibrium. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 35:1-35:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.35

Abstract

We investigate the closest distribution refinements problem, which involves a vertex-weighted bipartite graph as input, where the vertex weights on each side sum to 1 and represent a probability distribution. A refinement of one side’s distribution is an edge distribution that corresponds to distributing the weight of each vertex from that side to its incident edges. The objective is to identify a pair of distribution refinements for both sides of the bipartite graph such that the two edge distributions are as close as possible with respect to a specific divergence notion. This problem is a generalization of transportation, in which the special case occurs when the two closest distributions are identical. The problem has recently emerged in the context of composing differentially oblivious mechanisms.
Our main result demonstrates that a universal refinement pair exists, which is simultaneously closest under all divergence notions that satisfy the data processing inequality. Since differential obliviousness can be examined using various divergence notions, such a universally closest refinement pair offers a powerful tool in relation to such applications.
We discover that this pair can be achieved via locally verifiable optimality conditions. Specifically, we observe that it is equivalent to the following problems, which have been traditionally studied in distinct research communities: (1) hypergraph density decomposition, and (2) symmetric Fisher Market equilibrium.
We adopt a symmetric perspective of hypergraph density decomposition, in which hyperedges and nodes play equivalent 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. This connection allows existing algorithms for computing or approximating the Fisher market equilibrium to be adapted for all the aforementioned problems. For example, this approach allows the well-known iterative proportional response process to provide approximations for the corresponding problems with multiplicative error in distributed settings, whereas previously, only absolute error had been achieved in these contexts. Our study contributes to the understanding of various problems within a unified framework, which may serve as a foundation for connecting other problems in the future.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Distribution functions
  • Theory of computation → Mathematical optimization
Keywords
  • closest distribution refinements
  • density decomposition
  • Fisher market equilibrium

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