On Sub-determinants and the Diameter of Polyhedra

Abstract

We derive a new upper bound on the diameter 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(Δ2n4 log nΔ). If P is bounded, then we show that the diameter of P is at most O(Δ2n3.5 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 (Math Program 64:1-16, 1994). © 2014 Springer Science+Business Media New York.