A general label setting algorithm and tractability analysis for the multiobjective temporal shortest path problem

Abstract Given a directed temporal graph, a start node , and objectives, the task in the single‐source multiobjective temporal shortest path problem (SSMTSPP) consists of computing the set of nondominated images of temporal ‐‐paths for each node as well as one corresponding efficient path for each of these images. This problem generalizes both the multiobjective shortest path problem in static graphs and the single‐objective temporal shortest path problem. In this article, we provide a general label setting algorithm for the SSMTSPP that can handle a large variety of different objectives. The only condition imposed on the objectives is a monotonicity property that generalizes the nonnegativity of the arc costs required for the well‐known label setting algorithm for solving the static single‐source shortest path problem in both the single objective and the multiobjective case. Our analysis of the presented algorithm shows that its worst‐case running time is polynomial in the sum of the input size of the problem instance and the number of nondominated images, which implies that it runs in polynomial time as long as the number of nondominated images is polynomial in the instance size (i.e., for all tractable versions of the problem). To complement this result, we provide a complete classification into tractable and intractable problems for all SSMTSPPs involving a large variety of objectives. In particular, using our general analysis, this provides a large range of specific SSMTSPPs for which our general label setting algorithm runs in polynomial time.