### G-2003-55

# The Shortest Path Problem with Resource Constraints and *k*-Cycle Elimination for *k* ≥ 3

## and BibTeX reference

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 implicitly generate the set of all feasible routes or schedules, as e.g. in the column generation formulation of the vehicle routing problem with time windows (VRPTW).
The ESPPRC problem being NP-hard in the strong sense, 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. 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 the 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 proved to be a key ingredient for getting exact integer solutions for well-known hard problems from the literature.

Published **September 2003**
,
31 pages

### Document

**G-2003-55.pdf**
(400 KB)