On the hardness of approximating the network coding capacity

This work addresses the computational complexity of achieving the capacity of a general network coding instance. We focus on the linear capacity, namely the capacity of the given instance when restricted to linear encoding functions. It has been shown [Lehman and Lehman, SODA 2005] that determining the (scalar) linear capacity of a general network coding instance is NP-hard. In this work we initiate the study of approximation in this context. Namely, we show that given an instance to the general network coding problem of linear capacity C, constructing a linear code of rate alphaC for any universal (i.e., independent of the size of the instance) constant alphales1 is ldquohardrdquo. Specifically, finding such network codes would solve a long standing open problem in the field of graph coloring. In addition, we consider the problem of determining the (scalar) linear capacity of a planar network coding instance (i.e., a general instance in which the underlying graph is planar). We show that even for planar networks this problem remains NP-hard.