  We develop distributed algorithms to allocate resources in multi-hop wireless
networks with the aim of minimizing total cost. In order to observe the
fundamental duplexing constraint that co-located transmitters and receivers
cannot operate simultaneously on the same frequency band, we first devise a
spectrum allocation scheme that divides the whole spectrum into multiple
sub-bands and activates conflict-free links on each sub-band. We show that the
minimum number of required sub-bands grows asymptotically at a logarithmic rate
with the chromatic number of network connectivity graph. A simple distributed
and asynchronous algorithm is developed to feasibly activate links on the
available sub-bands. Given a feasible spectrum allocation, we then design
node-based distributed algorithms for optimally controlling the transmission
powers on active links for each sub-band, jointly with traffic routes and user
input rates in response to channel states and traffic demands. We show that
under specified conditions, the algorithms asymptotically converge to the
optimal operating point.
