On sub-determinants and the diameter of polyhedra

Authors:
Nicolas Bonifas;Marco Di Summa;Friedrich Eisenbrand;Nicolai Hähnle;Martin Niemeier
Affiliations:
LIX & IBM, Palaiseau & Gentilly, France;Università degli Studi di Padova, Padova, Italy;Technische Universität Berlin, Berlin, Germany;Technische Universität Berlin, Berlin, Germany;Technische Universität Berlin, Berlin, Germany
Venue:
Proceedings of the twenty-eighth annual symposium on Computational geometry
Year:
2012

Citing 7
Cited 1

The Hirsch conjecture is true for (0,1)-polytopes

Mathematical Programming: Series A and B
Random walks, totally unimodular matrices, and a randomised dual simplex algorithm

Mathematical Programming: Series A and B
A polynomial time primal network simplex algorithm for minimum cost flows

Mathematical Programming: Series A and B
A Linear Bound On The Diameter Of The Transportation Polytope*

Combinatorica
Graphs of transportation polytopes

Journal of Combinatorial Theory Series A
Beyond Hirsch Conjecture: Walks on Random Polytopes and Smoothed Complexity of the Simplex Method

SIAM Journal on Computing
Diameter of Polyhedra: Limits of Abstraction

Mathematics of Operations Research

Finding short paths on polytopes by the shadow vertex algorithm

ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I

Quantified Score

Hi-index	0.00

Visualization

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].