A primal Barvinok algorithm based on irrational decompositions

Computing parametric rational generating functions with a primal Barvinok algorithm

Computations with Barvinok's short rational generating functions are traditionally being performed in the dual space, to avoid the combinatorial complexity of inclusion--exclusion formulas for the intersecting proper faces of cones. We prove that, on the level of indicator functions of polyhedra, there is no need for using inclusion--exclusion formulas to account for boundary effects: All linear identities in the space of indicator functions can be purely expressed using half-open variants of the full-dimensional polyhedra in the identity. This gives rise to a practically efficient, parametric…

单纯锥的闭式分解与用于格点计数的PDBarv算法

Closed-Form Decomposition for Simplicial Cones and PDBarv Algorithm for Lattice Point Counting

本文提出闭式分解表达式和混合算法PDBarv，结合原始与对偶Barvinok算法并加速，实现维度5以上格点计数性能提升超20%。