A Simple Necessary and Sufficient Condition for the Double Unicast Problem

TL;DR

This paper presents a simple necessary and sufficient condition for achieving rate (1,1) in a double unicast network using vector network coding, simplifying previous complex conditions and characterizing the capacity region.

Contribution

It introduces a straightforward condition for double unicast capacity, applicable to XOR network coding, simplifying prior complex criteria and characterizing the capacity region.

Findings

The condition is both necessary and sufficient for rate (1,1) with vector network coding.

The condition is sufficient for XOR network coding.

It provides a simple characterization of the capacity region.

Abstract

We consider a directed acyclic network where there are two source-terminal pairs and the terminals need to receive the symbols generated at the respective sources. Each source independently generates an i.i.d. random process over the same alphabet. Each edge in the network is error-free, delay-free, and can carry one symbol from the alphabet per use. We give a simple necessary and sufficient condition for being able to simultaneously satisfy the unicast requirements of the two source-terminal pairs at rate-pair $(1,1)$ using vector network coding. The condition is also sufficient for doing this using only "XOR" network coding and is much simpler compared to the necessary and sufficient conditions known from previous work. Our condition also yields a simple characterization of the capacity region of a double-unicast network which does not support the rate-pair $(1,1)$.