Mission impossible: Computing the network coding capacity region

TL;DR

This paper explores the complexity of computing network coding capacity regions, showing that even single-source networks with hierarchical or secrecy constraints involve the complex set of entropy functions, surpassing linear code capabilities.

Contribution

It demonstrates that the difficulty in computing capacity regions extends to single-source networks with specific constraints, highlighting the limitations of linear codes.

Findings

Capacity regions involve the set of all entropy functions.

Linear codes are insufficient for certain single-source networks.

Complexity similar to multi-source cases even in simpler networks.

Abstract

One of the main theoretical motivations for the emerging area of network coding is the achievability of the max-flow/min-cut rate for single source multicast. This can exceed the rate achievable with routing alone, and is achievable with linear network codes. The multi-source problem is more complicated. Computation of its capacity region is equivalent to determination of the set of all entropy functions $\Gamma^*$, which is non-polyhedral. The aim of this paper is to demonstrate that this difficulty can arise even in single source problems. In particular, for single source networks with hierarchical sink requirements, and for single source networks with secrecy constraints. In both cases, we exhibit networks whose capacity regions involve $\Gamma^*$. As in the multi-source case, linear codes are insufficient.