A note on the multiple unicast capacity of directed acyclic networks

TL;DR

This paper investigates the conditions under which multiple unicast sessions can be simultaneously achieved in directed acyclic networks with unit capacity edges, providing characterizations for specific cases and insights for general scenarios.

Contribution

It offers new capacity characterizations and feasibility conditions for multiple unicast network coding in directed acyclic networks, especially for three sessions and beyond.

Findings

Characterized achievable and unachievable connectivity 3-tuples for three unicast sessions.

Identified network topologies and capacity conditions relevant to general n sessions.

Abstract

We consider the multiple unicast problem under network coding over directed acyclic networks with unit capacity edges. There is a set of n source-terminal (s_i - t_i) pairs that wish to communicate at unit rate over this network. The connectivity between the s_i - t_i pairs is quantified by means of a connectivity level vector, [k_1 k_2 ... k_n] such that there exist k_i edge-disjoint paths between s_i and t_i. Our main aim is to characterize the feasibility of achieving this for different values of n and [k_1 ... k_n]. For 3 unicast connections (n = 3), we characterize several achievable and unachievable values of the connectivity 3-tuple. In addition, in this work, we have found certain network topologies, and capacity characterizations that are useful in understanding the case of general n.