On a Simple Connection Between Δ-modular ILP and LP, and a New Bound on the Number of Integer Vertices
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Let A ∈ Z^m × n, rank(A) = n, b ∈ Z^m, and P be an n-dimensional polyhedron, induced by the system A x ≤ b. It is a known fact that if F is a k-face of P, then there exist at least n-k linearly independent inequalities of the system A x ≤ b that become equalities on F. In other words, there exists a set of indices J, such that |J| = n-k, rank(A_JC) = n-k, and A_J x - b_J = 0, for any x ∈ F. We show that a similar fact holds for the integer polyhedron P_I = conv(P ∩ Z^n) if we additionally suppose that P is Δ-modular, for some Δ∈{1,2,…}. More precisely, if F is a k-face of P_I, then there exists a set of indices J, such that |J| = n-k, rank(A_J) = n-k, and A_J x - b_J_∞ < Δ, for any x ∈ F ∩ Z^n. In other words, there exist at least n-k linearly independent inequalities of the system A x ≤ b that almost become equalities on F ∩ Z^n. When we say almost, we mean that the slacks are at most Δ-1 in the absolute value. Using this fact, we prove the inequality |vert(P_I)| ≤ 2 ·mn·Δ^n-1, for the number of vertices of P_I, which is better, than the state of the art bound for Δ = O(n^2).
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