The Edge-Removal Problem’s Connections to the Zero-Error and $\delta$ -Dependence Problems in Network Coding

The edge-removal problem addresses the loss in capacity obtained by the removal of an edge of capacity <inline-formula> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> from a given network. In the context of network coding, the edge-removal problem has been solved for certain families of networks (such as instances with co-located sources). However, the problem is open for general instances. This work ties the edge-removal problem to two additional problems in the context of network communication: (a) the “ zero-error” network coding problem, which asks whether the zero-error network coding capacity differs from the capacity when error probability is allowed to tend to zero as the blocklength grows; and (b) the “<inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>-dependence” problem, which measures the advantage of message dependence in network coding capacity.

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