A note on the parametric integer programming in the average case: sparsity, proximity, and FPT-algorithms
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We consider the Integer Linear Programming (ILP) problem max{c^ x : A x ≤ b, x ∈ Z^n }, parameterized by a right-hand side vector b ∈ Z^m, where A ∈ Z^m × n is a matrix of the rank n. Let v be an optimal vertex of the Linear Programming (LP) relaxation max{c^ x : A x ≤ b} and B be a corresponding optimal base. We show that, for almost all b ∈ Z^m, an optimal point of the square ILP problem max{c^ x : A_B x ≤ b_B, x ∈ Z^n } satisfies the constraints A x ≤ b of the original problem. From works of R. Gomory it directly follows that the square ILP problem max{c^ x : A_B x ≤ b_B, x ∈ Z^n } can be solved by an algorithm of the arithmetic complexity O(n ·δ·logδ), where δ = | A_B|. Consequently, it can be shown that, for almost all b ∈ Z^m, the original problem max{c^ x : A x ≤ b, x ∈ Z^n } can be solved by an algorithm of the arithmetic complexity O(n ·Δ·logΔ), where Δ is the maximum absolute value of n × n minors of A. By the same technique, we give new inequalities on the integrality gap and sparsity of a solution and slack variables. Another ingredient is a known lemma that states the equality of the maximum absolute values of rank minors of matrices with orthogonal columns. This lemma gives us a way to transform ILP problems of the type max{c^ x : A x = b, x ∈ Z^n_+} to problems of the previous type, here we assume that rank(A) = m and all the m × m minors of A are coprime. Consequently, it follows that, for almost all b ∈ Z^m, there exists an algorithm with the arithmetic complexity O((n-m) ·Δ·logΔ) to solve the problem in the equality form. Sparsity and integrality gap bounds are also presented.
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