State-based accelerations and bidirectional search for bi-objective multi-modal shortest paths

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 nondeterministic 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 (2001), is a polynomially-solvable hi-objective variant of the mono-objective regular language-constrained shortest path problem (Barett et al., 2000; Delling et al., 2009). We propose several label setting algorithms for solving the problem: a topological label-setting algorithm improving on algorithms proposed by Pallottino and Scutella (1998) and Lozano and Storchi (2001), 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. (C) 2012 Elsevier Ltd. All rights reserved.