The Shortest Path Problem with k-Cycle Elimination (k ≥ 3): Improving Branch and Price Algorithms for Vehicle Routing and Scheduling

The elementary shortest path problem with resource constraints (ESPPRC) is a widely used modeling tool in formulating vehicle routing and crew scheduling applications, see e.g. [3]. The ESPPRC consists of finding shortest paths from a source to all other nodes of a network that do not contain any cycles, i.e. duplicate nodes. The ESPPRC occurs as a subproblem of an enclosing problem and is used to implicitly generate the set of all feasible routes or schedules, as in the column generation formulation of the vehicle routing problem with time windows (VRPTW), see [2]. The ESPPRC problem being NP-hard in the strong sense [6], classical solution approaches are based on the corresponding non-elementary shortest path problem with resource constraints (SPPRC), which can be solved using a pseudo-polynomial labeling algorithm [5]. While solving the enclosing master problem by branch-and-price (see [1] for an introduction to the methodology), this subproblem relaxation leads to weak lower bounds and sometimes impractically large branch-and-bound trees. A compromise between solving ESPPRC and SPPRC is to forbid cycles of small lengths. In the SPPRC with k-cycle elimination (SPPRC-k-cyc) only paths with cycles of length at least k + 1 are allowed. The case k = 2 which forbids sequences of the form i − j − i is well known [7], and has been used successfully to reduce integrality gaps for the VRPTW [10, 4]. We propose a new definition of the dominance rule among labels for dealing with arbitrary values