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 $O(\log^2 k)$ 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.Comment: 10 pages; Shorter version appeared at ISIT (International Symposium on Information Theory) 2015; some typos correcte

Compute-and-Forward: Finding the Best Equation

Compute-and-Forward is an emerging technique to deal with interference. It allows the receiver to decode a suitably chosen integer linear combination of the transmitted messages. The integer coefficients should be adapted to the channel fading state. Optimizing these coefficients is a Shortest Lattice Vector (SLV) problem. In general, the SLV problem is known to be prohibitively complex. In this paper, we show that the particular SLV instance resulting from the Compute-and-Forward problem can be solved in low polynomial complexity and give an explicit deterministic algorithm that is guaranteed to find the optimal solution.Comment: Paper presented at 52nd Allerton Conference, October 201