Tighter Cut-based Bounds for k -pairs Communication Problems Nicholas J. A. Harvey ∗ Robert Kleinberg † Abstract We study the extent to which combinatorial cut conditions determine the maxi- mum network coding rate of k-pairs communication problems. We seek a combina- torial 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 consider- ing 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 combina- torial 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. 1 Introduction A k-pairs communication problem is a special type of network coding problem in which each message has a single source node and a single sink node. Such problems warrant study for both practical and theoretical reasons. On the practical side, k-pairs communi- cation problems model the vast majority of contemporary communication sessions, which typically involve a single sender and single receiver. On the theoretical side, character- izing the capacity of k-pairs communication problems has implications for long-standing open problems in computational complexity [1]. The term “multiple unicast sessions” has also been used to refer to k-pairs communication problems [7, 11]. A key goal of the existing work on k-pairs communication problems is to compare the network coding rate to the corresponding multicommodity flow rate, i.e., the maximum transmittion rate achievable when data is treated as a fluid that cannot be copied or coded. On the lower-bound side, there are examples [5, 6, 11] showing that the use of coding can increase the rate by an unbounded factor in directed graphs (as the size of the graph tends to infinity). In contrast, no undirected graph is known where the network coding rate exceeds the multicommodity flow rate. The undirected k-pairs conjecture claims that actually these two rates are always equal [1, 5, 6, 11]. On the upper-bound side, various approaches have been proposed to bound the net- work coding rate. A general outer bound on the capacity of multi-commodity information networks can be found in standard textbooks [4]. This bound was used by Borade [3] * MIT Computer Science and Artificial Intelligence Laboratory. nickh@mit.edu. Supported by a Natural Sciences and Engineering Research Council of Canada Fellowship. † MIT Department of Mathematics. rdk@math.mit.edu. Supported by a Fannie and John Hertz Foundation Fellowship.