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 k1-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 Fp and Fq for independent sources, revealing that the choice of field can heavily impact the performance of a linear network code.