Network coding is highly non-approximable

We address the network coding problem, in which messages available to a set of sources must be passed through a network to a set of sinks with specified demands. It is known that the problem of deciding whether the demands can be met using a linear code is NP-hard [Lehman and Lehman, SODA '04]. Such a result is not known if we allow general (=nonlinear) codes to be used. Despite this, network coding is believed to be a very hard problem both when restricting to linear codes, and when considering general codes. In the current paper we give some evidence for this hardness. Call a sink happy if it receives all of the data it demands. We show that the problem of maximizing the number of happy sinks by a general network code is NP-hard to approximate to within a multiplicative factor of n1−∊, for any ∊ > 0. Here, n is the number of sinks. To our knowledge, this is the first hardness result known for general network coding. The same holds for maximizing the number of happy sources. Let ns be the number of sinks. We also prove a stronger result about linear codes: that given a network that can be satisfied by a linear code, it is NP-hard to find a linear code that makes at least 22 sinks happy. In particular, this means that the problem of maximizing the number of happy sinks in a linear code cannot be approximated to any factor better than ns/22, even for arbitrarily large ns — this is the harshest kind of inapproximability possible.