Approximation of Pareto optima in multiple-objective, shortest-path problems
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We study the problem of finding all Pareto-optimal solutions for the multi-criteria single-source shortest-path problem with nonnegative edge lengths. The standard approaches are generalizations of labelsetting (Dijkstra) and label-correcting algorithms, in which the distance labels are multi-dimensional and more than one distance label is maintained for each node. The crucial parameter for the run time and space consumption is the total number of Pareto optima. In general, this value can be exponentially large in the input size. However, in various practical applications one can observe that the input data has certain characteristics, which may lead to a much smaller number -- small enough to make the problem efficiently tractable from a practical viewpoint. In this paper, we identify certain key characteristics, which occur in various applications. These key characteristics are evaluated on a concrete application scenario (computing the set of best train connections in view of travel time, fare, and number of train changes) and on a simplified randomized model, in which these characteristics occur in a very purist form. In the applied scenario, it will turn out that the number of Pareto optima on each visited node is restricted by a small constant. To counter-check the conjecture that these characteristics are the cause of these uniformly positive results, we will also report negative results from another application, in which these characteristics do not occur.