On Approximating the Sum-Rate for Multiple-Unicasts

TL;DR

This paper develops a polynomial-time approximation algorithm for the sum-rate of multiple-unicasts network coding, demonstrating bounds within an $O( ext{log}^2 k)$ factor and revealing significant separation between independent and correlated source coding.

Contribution

It introduces an efficient approximation method for the GNS cut and shows a fundamental separation in vector-linear coding capabilities for independent versus correlated sources.

Findings

Approximation algorithm within $O( ext{log}^2 k)$ factor of GNS cut

Existence of networks with large separation between independent and correlated source sum-rates

Field choice significantly impacts vector-linear network code performance

Abstract

We study upper bounds on the sum-rate of multiple-unicasts. We approximate the Generalized Network Sharing Bound (GNS cut) of the multiple-unicasts network coding problem with $k$ independent sources. Our approximation algorithm runs in polynomial time and yields an upper bound on the joint source entropy rate, which is within an $O(\log^2 k)$ factor from the GNS cut. It further yields a vector-linear network code that achieves joint source entropy rate within an $O(\log^2 k)$ factor from the GNS cut, but \emph{not} with independent sources: the code induces a correlation pattern among the sources. Our second contribution is establishing a separation result for vector-linear network codes: for any given field $\mathbb{F}$ there exist networks for which the optimum sum-rate supported by vector-linear codes over $\mathbb{F}$ for independent sources can be multiplicatively separated by a factor of $k^{1-\delta}$, for any constant ${\delta>0}$, from the optimum joint entropy rate supported by a code that allows correlation between sources. Finally, we establish a similar separation result for the asymmetric optimum vector-linear sum-rates achieved over two distinct fields $\mathbb{F}_{p}$ and $\mathbb{F}_{q}$ for independent sources, revealing that the choice of field can heavily impact the performance of a linear network code.