On lattice point counting in Δ-modular polyhedra

Let a polyhedron P be defined by one of the following ways: P = { x ∈ R n : A x ≤ b } , where A ∈ Z ( n + k ) × n , b ∈ Z ( n + k ) and rank A = n , P = { x ∈ R + n : A x = b } , where A ∈ Z k × n , b ∈ Z k and rank A = k , and let all rank order minors of A be bounded by Δ in absolute values. We show that the short rational generating function for the power series ∑ m ∈ P ∩ Z n x m can be computed with the arithmetical complexity O T SNF ( d ) · d k · d log 2 Δ , where k and Δ are fixed, d = dim P , and T SNF ( m ) is the complexity of computing the Smith Normal Form for m × 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 ( y ) = | P y ∩ Z n | , where P 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 y is not fixed. As another consequence, we show that the coefficients e i ( P , m ) of the Ehrhart quasi-polynomial m P ∩ Z n = ∑ j = 0 n e j ( P , m ) m j can be computed with a polynomial-time algorithm, for fixed k and Δ .