On approximating the sum-rate for multiple-unicasts

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(log² k) factor from the GNS cut. It further yields a vector-linear network code that achieves joint source entropy rate within an O(log² k) factor from the GNS cut, but 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 F there exist networks for which the optimum sum-rate supported by vector-linear codes over F for independent sources can be multiplicatively separated by a factor of k^1-δ, for any constant δ > 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 F_p and F_q for independent sources, revealing that the choice of field can heavily impact the performance of a linear network code.