Unified Label-Setting Algorithm for variants of the Shortest Path Problem with Resource Constraints

The Shortest Path Problem with Resource Constraints is a common subproblem in many Branch-Cut-and-Price algorithms (BCP), e.g., routing and scheduling problems. This paper presents a general label-setting algorithm for solving various shortest path problems. An integration of the ‘Shortest Path Problem with Resource Constraints and k-cycle elimination’ with the ‘Partial Elementarity’ is introduced, and so is a bi-directional label-setting algorithm for solving general Shortest Path Problem with Resource Constraints. Moreover it is shown how to handle the socalled weighted subset penalties; which say that given a subset of nodes T each weighted with wt ∈ Q : ∀t ∈ T a penalty σ has to be paid for every whole visit to T , i.e., penalty for path P is ⌊∑i∈P∩T wi⌋σ. These additional constraints makes it possible to handle Chvatal Rank 1 cuts applied to a Set-Packing like master problem of a BCP algorithm.