On the enumeration of bicriteria temporal paths

Abstract

We discuss the complexity of path enumeration in weighted temporal graphs. In a weighted temporal graph, each edge has an availability time, a traversal time and some cost. We introduce two bicriteria temporal min-cost path problems in which we are interested in the set of all efficient paths with low costs and short duration or early arrival times, respectively. Unfortunately, the number of efficient paths can be exponential in the size of the input. For the case of strictly positive edge costs, however, we are able to provide algorithms that enumerate the set of efficient paths with polynomial time delay and polynomial space. If we are only interested in the set of Pareto-optimal solutions (not in the paths themselves), then we show that in the case of nonnegative edge costs these sets can be found in polynomial time. In addition, for each Pareto-optimal solution, we are able to find an efficient path in polynomial time.