  Routing is a widespread approach to transfer information from a source node
to a destination node in many deployed wireless ad-hoc networks. Today's
implemented routing algorithms seek to efficiently find the path/route with the
largest Full-Duplex (FD) capacity, which is given by the minimum among the
point-to-point link capacities in the path. Such an approach may be suboptimal
if then the nodes in the selected path are operated in Half-Duplex (HD) mode.
Recently, the capacity (up to a constant gap that only depends on the number of
nodes in the path) of an HD line network i.e., a path) has been shown to be
equal to half of the minimum of the harmonic means of the capacities of two
consecutive links in the path. This paper asks the questions of whether it is
possible to design a polynomial-time algorithm that efficiently finds the path
with the largest HD capacity in a relay network. This problem of finding that
path is shown to be NP-hard in general. However, if the number of cycles in the
network is polynomial in the number of nodes, then a polynomial-time algorithm
can indeed be designed.
