On lattice point counting in Delta-modular polyhedra

Let a polyhedron P be defined by one of the following ways: (i) P={x is an element of R-n:Ax <= b}, where A is an element of Z((n+k)xn), b is an element of Z((n+k)) and rankA=n, (ii) P={x is an element of R-+(n): Ax=b}, where A is an element of Z(kxn), b is an element of Z(k) and rankA=k, and let all rank order minors of A be bounded by Delta in absolute values. We show that the short rational generating function for the power series Sigma(xm)(m is an element of P boolean AND Zn) can be computed with the arithmetical complexity O(T-SNF(d) center dot d(k) center dot d(log2 Delta)), where k and Delta are fixed, d = dim P, and T-SNF(m) is the complexity of computing the Smith Normal Form for m xm 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 boolean AND 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 |mP boolean AND Z(n)| = Sigma(n)(j=0) e(j) (P, m)m(j) can be computed with a polynomial-time algorithm, for fixed k and Delta.