A New and Faster Representation for Counting Integer Points in Parametric Polyhedra
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| Format: | Preprint |
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2023
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| author | Gribanov, D. Malyshev, D. Pardalos, P. Zolotykh, N. |
| author_facet | Gribanov, D. Malyshev, D. Pardalos, P. Zolotykh, N. |
| contents | In this paper, we consider the counting function $E_P(y) = |P_{y} \cap Z^{n_x}|$ for a parametric polyhedron $P_{y} = \{x \in R^{n_x} \colon A x \leq b + B y\}$, where $y \in R^{n_y}$. We give a new representation of $E_P(y)$, called a \emph{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 $E_P(y)$ in certain scenarios. The most important cases are the following:
1) We show that, for the parametric polyhedron $P_y$ defined by a standard-form system $A x = y,\, x \geq 0$ with a fixed number of equalities, the function $E_P(y)$ can be represented by a polynomial-time computable function. In turn, such a representation of $E_P(y)$ can be constructed by an $poly\bigl(n, \|A\|_{\infty}\bigr)$-time algorithm;
2) Assuming again that the number of equalities is fixed, we show that integer/rational Ehrhart's quasi-polynomials of a polytope can be computed by FPT-algorithms, parameterized by sub-determinants of $A$ or its elements;
3) Our representation of $E_P$ is more efficient than other known approaches, if $A$ has bounded elements, especially if it is sparse in addition.
Additionally, we provide a discussion about possible applications in the area of compiler optimization. In some "natural" assumptions on a program code, our approach has the fastest complexity bounds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_13788 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A New and Faster Representation for Counting Integer Points in Parametric Polyhedra Gribanov, D. Malyshev, D. Pardalos, P. Zolotykh, N. Data Structures and Algorithms Computational Geometry Discrete Mathematics Combinatorics In this paper, we consider the counting function $E_P(y) = |P_{y} \cap Z^{n_x}|$ for a parametric polyhedron $P_{y} = \{x \in R^{n_x} \colon A x \leq b + B y\}$, where $y \in R^{n_y}$. We give a new representation of $E_P(y)$, called a \emph{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 $E_P(y)$ in certain scenarios. The most important cases are the following: 1) We show that, for the parametric polyhedron $P_y$ defined by a standard-form system $A x = y,\, x \geq 0$ with a fixed number of equalities, the function $E_P(y)$ can be represented by a polynomial-time computable function. In turn, such a representation of $E_P(y)$ can be constructed by an $poly\bigl(n, \|A\|_{\infty}\bigr)$-time algorithm; 2) Assuming again that the number of equalities is fixed, we show that integer/rational Ehrhart's quasi-polynomials of a polytope can be computed by FPT-algorithms, parameterized by sub-determinants of $A$ or its elements; 3) Our representation of $E_P$ is more efficient than other known approaches, if $A$ has bounded elements, especially if it is sparse in addition. Additionally, we provide a discussion about possible applications in the area of compiler optimization. In some "natural" assumptions on a program code, our approach has the fastest complexity bounds. |
| title | A New and Faster Representation for Counting Integer Points in Parametric Polyhedra |
| topic | Data Structures and Algorithms Computational Geometry Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2310.13788 |