Faster algorithm for counting of the integer points number in Δ-modular polyhedra
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Let a polytope 𝒫 be defined by one of the following ways: (i) 𝒫 = {x ∈ℝ^n A x ≤ b}, where A ∈ℤ^(n+m) × n, b ∈ℤ^(n+m), and rank(A) = n, (ii) 𝒫 = {x ∈ℝ_+^n A x = b}, where A ∈ℤ^m × n, b ∈ℤ^m, and rank(A) = m, and let all the rank minors of A be bounded by Δ in the absolute values. We show that |𝒫∩ℤ^n| can be computed with an algorithm, having the arithmetic complexity bound O( ν(d,m,Δ) · d^3 ·Δ^4 ·log(Δ) ), where d = (𝒫) and ν(d,m,Δ) is the maximal possible number of vertices in a d-dimensional polytope P, defined by one of the systems above. Using the obtained result, we have the following arithmetical complexity bounds to compute |P ∩ℤ^n|: 1) The bound O(d/m+1)^m · d^3 ·Δ^4 ·log(Δ) that is polynomial on d and Δ, for any fixed m; 2) The bound O(m/d+1)^d/2· d^3 ·Δ^4 ·log(Δ) that is polynomial on m and Δ, for any fixed d; 3) The bound O(d)^3 + d/2·Δ^4+d·log(Δ) that is polynomial on Δ, for any fixed d. Given bounds can be used to obtain faster algorithms for the ILP feasibility problem, and for the problem to count integer points in a simplex or in an unbounded Subset-Sum polytope. Unbounded and parametric versions of the above problem are also considered.
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