Abstract

We derive a new upper bound on the diameter of the graph of a polyhedron P = {x ∈ Rn : Ax ≤ b}, where A ∈ Zm×n. The bound is polynomial in n and the largest absolute value of a sub-determinant of A, denoted by Δ. More precisely, we show that the diameter of P is bounded by O(Δ2 n4 log nΔ). If P is bounded, then we show that the diameter of P is at most O(Δ2 n3.5 log nΔ).For the special case in which A is a totally unimodular matrix, the bounds are O(n4 log n) and O(n3.5 log n) respectively. This improves over the previous best bound of O(m16n3(log mn)3) due to Dyer and Frieze [MR1274170].

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