On Sub-determinants and the Diameter of Polyhedra

Abstract

We derive a new upper bound on the diameter of a polyhedron $$P = \{x {\in } {\mathbb {R}}^n :Ax\le b\}$$ P = { x ? R n : A x ≤ b } , where $$A \in {\mathbb {Z}}^{m\times n}$$ A ? Z m × n . The bound is polynomial in $$n$$ n and the largest absolute value of a sub-determinant of $$A$$ A , denoted by $$\Delta $$ Δ . More precisely, we show that the diameter of $$P$$ P is bounded by $$O(\Delta ^2 n^4\log n\Delta )$$ O ( Δ 2 n 4 log n Δ ) . If $$P$$ P is bounded, then we show that the diameter of $$P$$ P is at most $$O(\Delta ^2 n^{3.5}\log n\Delta )$$ O ( Δ 2 n 3.5 log n Δ ) . For the special case in which $$A$$ A is a totally unimodular matrix, the bounds are $$O(n^4\log n)$$ O ( n 4 log n ) and $$O(n^{3.5}\log n)$$ O ( n 3.5 log n ) respectively. This improves over the previous best bound of $$O(m^{16}n^3(\log mn)^3)$$ O ( m 16 n 3 ( log m n ) 3 ) due to Dyer and Frieze (Math Program 64:1---16, 1994).