The two-unicast problem

TL;DR

This paper investigates the capacity limits of two-unicast networks, introduces a new outer bound called GNS, and demonstrates the computational complexity and fundamental difficulty of achieving capacity in such networks.

Contribution

It proposes the GNS outer bound, proves its tightness in certain cases, and shows the computational hardness and complexity of the two-unicast problem.

Findings

GNS bound is the tightest edge-cut bound for two-unicast networks.

Computing the GNS bound is NP-complete.

Linear coding is insufficient for general two-unicast networks.

Abstract

We consider the communication capacity of wireline networks for a two-unicast traffic pattern. The network has two sources and two destinations with each source communicating a message to its own destination, subject to the capacity constraints on the directed edges of the network. We propose a simple outer bound for the problem that we call the Generalized Network Sharing (GNS) bound. We show this bound is the tightest edge-cut bound for two-unicast networks and is tight in several bottleneck cases, though it is not tight in general. We also show that the problem of computing the GNS bound is NP-complete. Finally, we show that despite its seeming simplicity, the two-unicast problem is as hard as the most general network coding problem. As a consequence, linear coding is insufficient to achieve capacity for general two-unicast networks, and non-Shannon inequalities are necessary for characterizing capacity of general two-unicast networks.