Tighter Cut-based Bounds for k-pairs Communication Problems

We study the extent to which combinatorial cut conditions determine the maximum network coding rate of k-pairs communication problems. We seek a combinatorial parameter of directed networks which constitutes a valid upper bound on the network coding rate but exceeds this rate by only a small factor in the worst case. (This worst-case ratio is called the gap of the parameter.) We begin by considering vertex-sparsity and meagerness, showing that both of these parameters have a gap which is linear in the network size. Due to the weakness of these bounds, we propose a new bound called informational meagerness. This bound generalizes both vertex-sparsity and meagerness and is potentially the first known combinatorial cut condition with a sublinear gap. However, we prove that informational meagerness does not tightly characterize the network coding rate: its gap can be super-constant.