The Exact Subgraph Recoverable Robust Shortest Path Problem

Abstract

Passengers of a public transportation system are often forced to change their planned route due to deviation in travel times. Rerouting is mostly done by simple means such as announcements. We introduce a model, in which the passenger computes his optimal route on his mobile device in a given subnetwork according to the actual travel times. Those travel times are sent to him as soon as a delay occurs.

The main focus of this paper is on the calculation of a small subnetwork. This subnetwork shall contain for every realization of travel times a shortest path of the original network and minimize the number of arcs. For this so called $\textsc{Exact Subgraph Recoverable Robust Shortest Path}$ problem we introduce an approximation algorithm with an approximation factor of $\frac{m}{\ell}$, for any fixed constant ℓ ∈ ℕ. This is the best possible approximation factor for the interval- and the Γ-scenario case, in which all realizations of travel times are given indirectly by lower and upper bounds on the arc cost. Unless P = NP, for those two scenario sets the problems is not approximable with a factor better than m ^{(1 − ε)}, where m is the number of arcs in the given graph and ε> 0.