On lattice point counting in Δ-modular polyhedra
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Let a polyhedron P be defined by one of the following ways: (i) P = {x ∈ R^n A x ≤ b}, where A ∈ Z^(n+k) × n, b ∈ Z^(n+k) and rank A = n; (ii) 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 arithmetic complexity O(T_SNF(d) · d^k· d^log_2 Δ), where k and Δ are fixed, d = P, and T_SNF(m) is the complexity to compute the Smith Normal Form for m × m integer matrix. In particular, d = n for the case (i) and d = n-k for the case (ii). The simplest examples of polyhedra that meet conditions (i) or (ii) are the simplicies, the subset sum polytope and the knapsack or multidimensional knapsack polytopes. 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 parametric polytope, can be computed by a polynomial time even in varying dimension if P_y has a close structure to the cases (i) or (ii). As another consequence, we show that the coefficients e_i(P,m) of the Ehrhart quasi-polynomial | mP ∩ Z^n| = ∑_j = 0^n e_i(P,m)m^j can be computed by a polynomial time algorithm for fixed k and Δ.
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