On the multiple unicast capacity of 3-source, 3-terminal directed acyclic networks

TL;DR

This paper classifies 3-source, 3-terminal directed acyclic networks based on connectivity levels, providing network coding schemes or counterexamples for supporting unit-rate communication, and demonstrates improved throughput over routing in overlapping path scenarios.

Contribution

It offers a comprehensive classification of network connectivity levels and introduces new linear network coding schemes for cases with less than full connectivity.

Findings

Unit-rate transmission supported when all connectivity levels are at least 3.

Constructive network coding schemes for most connectivity vectors with less than full connectivity.

Enhanced throughput achieved by combining network coding with routing in overlapping path networks.

Abstract

We consider the multiple unicast problem with three source-terminal pairs over directed acyclic networks with unit-capacity edges. The three $s_i-t_i$ pairs wish to communicate at unit-rate via network coding. The connectivity between the $s_i - t_i$ pairs is quantified by means of a connectivity level vector, $[k_1 k_2 k_3]$ such that there exist $k_i$ edge-disjoint paths between $s_i$ and $t_i$. In this work we attempt to classify networks based on the connectivity level. It can be observed that unit-rate transmission can be supported by routing if $k_i \geq 3$, for all $i = 1, \dots, 3$. In this work, we consider, connectivity level vectors such that $\min_{i = 1, \dots, 3} k_i < 3$. We present either a constructive linear network coding scheme or an instance of a network that cannot support the desired unit-rate requirement, for all such connectivity level vectors except the vector $[1~2~4]$ (and its permutations). The benefits of our schemes extend to networks with higher and potentially different edge capacities. Specifically, our experimental results indicate that for networks where the different source-terminal paths have a significant overlap, our constructive unit-rate schemes can be packed along with routing to provide higher throughput as compared to a pure routing approach.