A new graph perspective on max-min fairness in Gaussian parallel channels

TL;DR

This paper introduces a graph-based framework to analyze and characterize max-min fairness in Gaussian parallel channels, linking it to graph theory concepts like Lovasz functions and odd cycles.

Contribution

It provides a novel characterization of max-min fair performance using graph theory, including Lovasz functions and 2-norm distances, for the first time in this context.

Findings

Max-min fair performance relates to Lovasz function of the sharing graph.

Odd cycles in the sharing graph influence fairness outcomes.

A 2-norm distance based on power allocations characterizes performance bounds.

Abstract

In this work we are concerned with the problem of achieving max-min fairness in Gaussian parallel channels with respect to a general performance function, including channel capacity or decoding reliability as special cases. As our central results, we characterize the laws which determine the value of the achievable max-min fair performance as a function of channel sharing policy and power allocation (to channels and users). In particular, we show that the max-min fair performance behaves as a specialized version of the Lovasz function, or Delsarte bound, of a certain graph induced by channel sharing combinatorics. We also prove that, in addition to such graph, merely a certain 2-norm distance dependent on the allowable power allocations and used performance functions, is sufficient for the characterization of max-min fair performance up to some candidate interval. Our results show also a specific role played by odd cycles in the graph induced by the channel sharing policy and we present an interesting relation between max-min fairness in parallel channels and optimal throughput in an associated interference channel.