Fairness in Communication for Omniscience

TL;DR

This paper introduces a method to fairly distribute communication rates among users in communication for omniscience by optimizing a quadratic fairness measure using submodular function minimization, ensuring efficient and equitable strategies.

Contribution

It formulates a new quadratic fairness optimization problem over the rate region and provides a polynomial-time algorithm to find the lexicographically optimal fair rate vector.

Findings

The proposed algorithm finds the fair rate vector efficiently.

The method guarantees a lex-optimal and fair distribution of rates.

Complexity is reduced by focusing on fundamental partitions.

Abstract

We consider the problem of how to fairly distribute the minimum sum-rate among the users in communication for omniscience (CO). We formulate a problem of minimizing a weighted quadratic function over a submodular base polyhedron which contains all achievable rate vectors, or transmission strategies, for CO that have the same sum-rate. By solving it, we can determine the rate vector that optimizes the Jain's fairness measure, a more commonly used fairness index than the Shapley value in communications engineering. We show that the optimizer is a lexicographically optimal (lex-optimal) base and can be determined by a decomposition algorithm (DA) that is based on submodular function minimization (SFM) algorithm and completes in strongly polynomial time. We prove that the lex-optimal minimum sum-rate strategy for CO can be determined by finding the lex-optimal base in each user subset in the fundamental partition and the complexity can be reduced accordingly.