On lattice point counting in $\varDelta$-modular polyhedra

Abstract

Let a polyhedron P be defined by one of the following ways:

• (i) $P = \{x \in {{\,\mathrm{{\mathbb {R}}}\,}}^n :A x \le b\}$, where $A \in {{\,\mathrm{{\mathbb {Z}}}\,}}^{(n+k) \times n}$, $b \in {{\,\mathrm{{\mathbb {Z}}}\,}}^{(n+k)}$ and ${{\,\mathrm{rank}\,}}A = n$,

• (ii) $P = \{x \in {{\,\mathrm{{\mathbb {R}}}\,}}_+^n :A x = b\}$, where $A \in {{\,\mathrm{{\mathbb {Z}}}\,}}^{k \times n}$, $b \in {{\,\mathrm{{\mathbb {Z}}}\,}}^{k}$ and ${{\,\mathrm{rank}\,}}A = k$,

and let all rank order minors of A be bounded by $\varDelta$ in absolute values. We show that the short rational generating function for the power series

$$\begin{aligned} \sum \limits _{m \in P \cap {{\,\mathrm{{\mathbb {Z}}}\,}}^n} {{\,\mathrm{{\mathbf {x}}}\,}}^m \end{aligned}$$

can be computed with the arithmetical complexity $O\left( T_{{\mathrm{SNF}}}(d) \cdot d^{k} \cdot d^{\log _2 \varDelta }\right) ,$ where k and $\varDelta$ are fixed, $d = \dim P$, and $T_{{\mathrm{SNF}}}(m)$ is the complexity of computing the Smith Normal Form for $m \times m$ integer matrices. In particular, $d = n$, for the case (i), and $d = n-k$, for the case (ii). The simplest examples of polyhedra that meet the conditions (i) or (ii) are the simplices, the subset sum polytope and the knapsack or multidimensional knapsack polytopes. Previously, the existence of a polynomial time algorithm in varying dimension for the considered class of problems was unknown already for simplicies ($k = 1$). We apply these results to parametric polytopes and show that the step polynomial representation of the function $c_P({{\,\mathrm{{\mathbf {y}}}\,}}) = |P_{{{\,\mathrm{{\mathbf {y}}}\,}}} \cap {{\,\mathrm{{\mathbb {Z}}}\,}}^n|$, where $P_{{{\,\mathrm{{\mathbf {y}}}\,}}}$ is a parametric polytope, whose structure is close to the cases (i) or (ii), can be computed in polynomial time even if the dimension of $P_{{{\,\mathrm{{\mathbf {y}}}\,}}}$ is not fixed. As another consequence, we show that the coefficients $e_i(P,m)$ of the Ehrhart quasi-polynomial

$$\begin{aligned} \left| mP \cap {{\,\mathrm{{\mathbb {Z}}}\,}}^n\right| = \sum \limits _{j = 0}^n e_j(P,m)m^j \end{aligned}$$

can be computed with a polynomial-time algorithm, for fixed k and $\varDelta$.