Restricted Primitive Sets and Simplicial Subdivisions with Arbitrary Refinement Factors

We discuss the implementation of a refinement procedure for fixed point computations using Scarf's primitive sets. The procedure, presented in Van der Heyden (Van der Heyden, L. 1982. A Refinement procedure for computing fixed points using Scarf's primitive sets. Math. Oper. Res.7 295–313.), moves between combinatorial objects called restricted primitive sets. We characterize restricted primitive sets when the vectors in these sets belong to a sequence of regular grids with increasing grid sizes. The refinement factor between successive grids is an arbitrary integer. These restricted primitive sets are associated with a set of easily implementable rules which guide the movement of the refinement procedure. We also present a geometrical interpretation of these restricted sets in terms of simplicial subdivisions.