The shortest-path problem with resource constraints and k-cycle elimination for k ≥ 3

The elementary shortest-path problem with resource constraints (ESPPRC) is a widely used modeling tool in formulating vehicle-routing and crew-scheduling applications. The ESPPRC often occurs as a subproblem of an enclosing problem, where it is used to generate implicitly the set of all feasible-routes or schedules, as in the column-generation formulation of the vehicle-routing problem with time windows (VRPTW). As the ESPPRC problem is NP-hard in the strong sense, classical solution approaches are based on the corresponding nonelementary shortest-path problem with resource constraints (SPPRC), which can be solved using a pseudo-polynomial labeling algorithm. While solving the enclosing problem by branch and price, 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 length. In the SPPRC with k-cycle elimination (SPPRC-k-cyc), paths with cycles are allowed only if cycles have length at least k + 1. The case k = 2 forbids sequences of the form i - j - i and has been successfully used to reduce integrality gaps. We propose a new definition of the dominance rule among labels for dealing with arbitrary values of k >= 2. The numerical experiments on the linear relaxation of some hard VRPTW instances from Solomon's benchmark show that k-cycle elimination with k >= 3 can substantially improve the lower bounds of vehicle-routing problems with side constraints. The new algorithm has proven to be a key ingredient for getting exact integer solutions for well-known hard problems from the literature.