Taking into account the multi-modality of urban transportation networks for computing the itinerary of an individual passenger introduces a number of additional constraints such as mode restrictions and various objective functions. In this paper, constraints on modes are gathered under the concept of viable path, modeled by a non deterministic finite state automaton (NFA). The goal is to find the non-dominated viable shortest paths considering the minimization of the travel time and of the number of modal transfers. We show that the problem, initially considered by Lozano and Storchi [15], is a polynomially-solvable bi-objective variant of the mono-objective regular language-constrained shortest path problem [2,8].We propose several label setting algorithms for solving the problem: a topological label-setting algorithm improving on algorithms proposed by Pallottino and Scutellà [23] and Lozano and Storchi [15], a multi-label algorithm using buckets and its bidirectional variant, as well as dedicated goal oriented techniques. Furthermore, we propose a new NFA state-based dominance rule. The computational experiments, carried-out on a realistic urban network, show that the state-based dominance rule associated with bidirectional search yields significant average speed-ups. On an expanded graph comprising 1 859 350 nodes, we obtain on average 3.5 non-dominated shortest paths in less than 180 ms.keywords: bi-objective regular language-constrained shortest path problem, multimodal transportation, finite state automaton, label-setting algorithms, state-based dominance rule, bidirectional search, state-based estimated travel times.
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