Algorithms for optimal min hop and foremost paths in interval temporal graphs

Path problems are fundamental to the study of graphs. Temporal graphs are graphs in which the edges connecting the vertices change with time. Min hop paths problem in a temporal graph is the problem of finding time respecting paths from source vertex to every destination vertex such that the path goes through minimum number of edges. Foremost paths problem in a temporal graph requires to find time respecting paths that arrive at the destination vertices at earliest possible time. In this paper we present algorithms to find min hop paths and foremost paths in interval temporal graphs. These algorithms are benchmarked against the fastest algorithms known for foremost and min-hop paths by Wu et al. (IEEE Trans Knowl Data Eng 28(11):2927-2942, 2016a. https://doi.org/10.1109/TKDE.2016.2594065) that work on contact sequence temporal graph model. On the available test data, our foremost path algorithm provides a speedup of up to 1800 over the fastest algorithm for contact sequence graphs; the speedup for our min-hop algorithm is up to 6700. We also demonstrate that interval temporal graph model on which our algorithms work represents a superset of contact sequence temporal graphs. We show that path problems exist that are NP-hard in interval temporal graph model but polynomial in the contact sequence temporal graph model in terms of the number of vertices and edges in the input graph. This is due to the fact that the contact sequence graph model may require much larger number of edges than the corresponding interval temporal graph to represent a given temporal graph.