  We examine the effectiveness of packet routing in this model for the broad
class next-hop preferences with filtering. Here each node v has a filtering
list D(v) consisting of nodes it does not want its packets to route through.
Acceptable paths (those that avoid nodes in the filtering list) are ranked
according to the next-hop, that is, the neighbour of v that the path begins
with. On the negative side, we present a strong inapproximability result. For
filtering lists of cardinality at most one, given a network in which an
equilibrium is guaranteed to exist, it is NP-hard to approximate the maximum
number of packets that can be routed to within a factor of O(n^{1-\epsilon}),
for any constant \epsilon >0. On the positive side, we give algorithms to show
that in two fundamental cases every packet will eventually route with
probability one. The first case is when each node's filtering list contains
only itself, that is, D(v)={v}. Moreover, with positive probability every
packet will be routed before the control plane reaches an equilibrium. The
second case is when all the filtering lists are empty, that is,
$\mathcal{D}(v)=\emptyset$. Thus, with probability one packets will route even
when the nodes don't care if their packets cycle! Furthermore, with probability
one every packet will route even when the control plane has em no equilibrium
at all.
