We apply a distributed, message-passing scheme for sparsely-coupled linear programming problems to the stabilization of positive switched linear systems. We first develop exact conditions for the existence of a stabilizing path-dependent controller for positive switched linear systems in terms of an increasing family of linear programming (LP) problems. These results are of independent interest as a special case of the stabilization of arbitrary switched linear systems. The size of each member of this family of LPs grows combinatorially in both the number of switching modes and the path-length of the controller. We then make use of a problem decomposition for sparsely-coupled linear programming problems by decomposing the induced switching graph for the system. The resulting problem decomposition can be solved in parallel using distributed computing resources. An example problem and its decomposition are presented to facilitate discussion of this approach and to highlight future areas of interest for our particular problem formation.