  In this paper, we consider a cognitive multi-hop relay secondary user (SU)
system sharing the spectrum with some primary users (PU). The transmit power as
well as the hop selection of the cognitive relays can be dynamically adapted
according to the local (and causal) knowledge of the instantaneous channel
state information (CSI) in the multi-hop SU system. We shall determine a low
complexity, decentralized algorithm to maximize the average end-to-end
throughput of the SU system with dynamic spatial reuse. The problem is
challenging due to the decentralized requirement as well as the causality
constraint on the knowledge of CSI. Furthermore, the problem belongs to the
class of stochastic Network Utility Maximization (NUM) problems which is quite
challenging. We exploit the time-scale difference between the PU activity and
the CSI fluctuations and decompose the problem into a master problem and
subproblems. We derive an asymptotically optimal low complexity solution using
divide-and-conquer and illustrate that significant performance gain can be
obtained through dynamic hop selection and power control. The worst case
complexity and memory requirement of the proposed algorithm is O(M^2) and
O(M^3) respectively, where $M$ is the number of SUs.
