We introduce a _p_-step formulation for the capacitated vehicle routing problem (CVRP). The parameter
_p_ indicates the length of partial paths corresponding to the used variables. This provides a family
of formulations including both the traditional arc-based and path-based formulations. Hence, it is a generalization
which unifies arc-based and path-based formulations, while also providing new formulations.
We show that the LP bound of the _p_-step formulation is increasing in _p_, although not monotonically.
Furthermore, we prove that computing the set partitioning bound is NP-hard. This is a meaningful result
in itself, but combined with the _p_-step formulation this also allows us to show that there does not exist
a strongest compact formulation for the CVRP, if _P ≠ NP_. While ending the search for a strongest
compact formulation, we propose th
