Combinatiorial Algorithms for Wireless Information Flow

TL;DR

This paper presents a polynomial time algorithm to determine the unicast capacity of linear deterministic wireless relay networks, enabling simple encoding strategies to achieve optimal information flow.

Contribution

It introduces a polynomial time algorithm for capacity discovery and simple encoding strategies in linear deterministic wireless relay networks.

Findings

The algorithm computes capacity in polynomial time.

Simple one-symbol processing at relays suffices to achieve capacity.

The approach generalizes min-cut max-flow to wireless networks.

Abstract

A long-standing open question in information theory is to characterize the unicast capacity of a wireless relay network. The difficulty arises due to the complex signal interactions induced in the network, since the wireless channel inherently broadcasts the signals and there is interference among transmissions. Recently, Avestimehr, Diggavi and Tse proposed a linear deterministic model that takes into account the shared nature of wireless channels, focusing on the signal interactions rather than the background noise. They generalized the min-cut max-flow theorem for graphs to networks of deterministic channels and proved that the capacity can be achieved using information theoretical tools. They showed that the value of the minimum cut is in this case the minimum rank of all the adjacency matrices describing source-destination cuts. In this paper, we develop a polynomial time algorithm that discovers the relay encoding strategy to achieve the min-cut value in linear deterministic (wireless) networks, for the case of a unicast connection. Our algorithm crucially uses a notion of linear independence between channels to calculate the capacity in polynomial time. Moreover, we can achieve the capacity by using very simple one-symbol processing at the intermediate nodes, thereby constructively yielding finite length strategies that achieve the unicast capacity of the linear deterministic (wireless) relay network.