Barvinok's algorithm uses signed decomposition of cones in the dual space, to avoid dealing with all the lower-dimensional subcones arising in an inclusion-exclusion formula. By making use of lattice basis reduction, cones are recursively decomposed until only unimodular cones appear.
Beck and Sottile (2005) introduced the concept of irrational (positive) decompositions of cones to give simplified proofs for some theorems of Stanley. Irrational decompositions are decompositions into cones whose proper faces do not contain any lattice points. Counting formulas for lattice points based on irrational decompositions therefore do not need to take any inclusion-exclusion principle into account.
Our new variant of Barvinok's algorithm uses irrational signed decompositions to avoid having to deal with lower-dimensional subcones. This enables us to remove the polarization step, and to perform the signed decomposition in primal space. Due to the computation in primal space, it is possible to stop the recursive decomposition algorithm at some point before unimodular cones are reached.
We illustrate the advantages of the new method using computational results with a preliminary implementation within the software package LattE (De Loera et al., 2004).
e-print: arXiv math.CO/0603308