On the Hardness of Approximating the Network Coding Capacity

Abstract—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 NPhard. 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 αC for any universal (i.e., independent of the size of the instance) constant α ≤ 1 is “hard”. 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 s...

• Type: Journal
• Year: 2011
• Where: TIT
• Authors: Michael Langberg, Alexander Sprintson
• Keywords: General Network, Linear Capacity, Network Coding, TIT 2011