This paper presents a heuristic algorithm for minimizing the makespan required to route a set of agents inhabiting a shared guidepath network from their initial locations to their respective destinations while observing a set of regulations that seek to ensure the safety and the integrity of the generated traffic. From an application standpoint, the presented developments are motivated by the traffic coordination challenges that arise in the context of many automated unit-load material handling systems and also in the transport of the ionized atoms that takes place in the context of quantum computing. From a methodological standpoint, our developments constitute a customization of the general “local-search” framework of combinatorial optimization theory to the traffic management problem that is considered in this paper. Hence, the presented results include a rigorous characterization of the considered problem and its solution space, detailed algorithms for the construction of the necessary initial solutions and the improving step for the pursued search, a complexity analysis of these algorithms, and a set of computational experiments that reveal and assess the computational efficiency of the presented algorithms and the efficacy of the derived solutions. The paper concludes with some suggestions for potential extensions of the presented results.