Reduction approaches for robust shortest path problems

We investigate the uncertain versions of two classical combinatorial optimization problems, namely the Single-Pair Shortest Path Problem (SP-SPP) and the Single-Source Shortest Path Problem (SS-SPP). The former consists of finding a path of minimum length connecting two specific nodes in a finite directed graph G; the latter consists of finding the shortest paths from a fixed node to the remaining nodes of G. When considering the uncertain versions of both problems we assume that cycles may occur in G and that arc lengths are (possibly degenerating) nonnegative intervals. We provide sufficient conditions for a node and an arc to be always or never in an optimal solution of the Minimax regret Single-Pair Shortest Path Problem (MSP-SPP). Similarly, we provide sufficient conditions for an arc to be always or never in an optimal solution of the Minimax regret Single-Source Shortest Path Problem (MSS-SPP). We exploit such results to develop pegging tests useful to reduce the overall running time necessary to exactly solve both problems. A classical problem in combinatorial optimization consists of finding a path of minimum length connecting two specific nodes of a given finite directed weighted graph G. This problem is known in the literature as the Shortest Path Problem (SPP) and arises in a wide variety of practical problem settings, both as stand-alone models and as subproblems in more complex problem settings. For example, the SPP arises in the telecommunications and transportation industries whenever one wants to send a message or a vehicle between two geographical locations as quickly or as cheaply as possible. Similarly, the SPP arises in urban traffic planning when drivers are assumed to move along shortest paths from their origins to their destinations [1]. Various versions of the SPP are described in the literature, mainly differing from each other in the type of origin-destination considered. In particular, we can distinguish between: the Single-Pair Shortest Path Problem (SP-SPP) in which one aims to find the path of minimum length connecting two specific nodes of G; the Single-Source Shortest Path Problem (SS-SPP) in which one aims to find the path of minimum length connecting a fixed node to the remaining nodes of G; and the All-Pairs Shortest Path Problem (AP-SPP) in which one aims to find the paths of minimum length connecting every pair of nodes of G. When arc lengths are deterministically known such versions can be solved efficiently, e.g., by using Dijkstra's algorithm if …