A new and faster representation for counting integer points in parametric polyhedra

Dmitry V. Gribanov, Dmitry S. Malyshev, Panos M. Pardalos, Nikolai Yu. Zolotykh

Abstract: In this paper, we consider the counting function \({{\,\mathrm{{{\,\mathrm{\mathcal {E}}\,}}_{{{\,\mathrm{\mathcal {P}}\,}}}}\,}}(y) = |{{\,\mathrm{\mathcal {P}}\,}}_{y} \cap {{\,\mathrm{\mathbb {Z}}\,}}^{n_x}|\) for a parametric polyhedron \({{\,\mathrm{\mathcal {P}}\,}}_{y} = \{ x \in {{\,\mathrm{\mathbb {R}}\,}}^{n_x} :A x \le b + B y\}\), where \(y \in {{\,\mathrm{\mathbb {R}}\,}}^{n_y}\). We give a new representation of \({{\,\mathrm{{{\,\mathrm{\mathcal {E}}\,}}_{{{\,\mathrm{\mathcal {P}}\,}}}}\,}}(y)\), called a piece-wise step-polynomial with periodic coefficients, which is a generalization of piece-wise step-polynomials and integer/rational Ehrhart’s quasi-polynomials. It gives the fastest way to calculate \({{\,\mathrm{{{\,\mathrm{\mathcal {E}}\,}}_{{{\,\mathrm{\mathcal {P}}\,}}}}\,}}(y)\) in certain scenarios.

External IDs:dblp:journals/coap/GribanovMPZ25