An Asymptotically Improved Upper Bound on the Diameter of Polyhedra

Sukegawa, Noriyoshi

doi:10.1007/s00454-018-0016-y

Cited by 19 publications

(18 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another quantity that has attracted attention, due to its connection with the complexity of the simplex algorithm [15,23,26,27], is the largest diameter δ(d, k) a lattice polytope contained in [0, k] d can have [11,12,13,17,20]. Here, by the diameter of a polytope, we mean the diameter of the graph made of its vertices and edges.…”

Section: Introductionmentioning

confidence: 99%

The diameter of lattice zonotopes

Deza

Pournin

Sukegawa

2020

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

We establish sharp asymptotic estimates for the diameter of primitive zonotopes when their dimension is fixed. We also prove that, for infinitely many integers k, the largest possible diameter of a lattice zonotope contained in the hypercube [0, k] d is uniquely achieved by a primitive zonotope. As a consequence, we obtain that this largest diameter grows like k d/(d+1) up to an explicit multiplicative constant, when d is fixed and k goes to infinity, providing a new lower bound on the largest possible diameter of a lattice polytope contained in [0, k] d .

show abstract

Section: Introductionmentioning

confidence: 99%

The diameter of lattice zonotopes

Deza

Pournin

Sukegawa

2020

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…This bound contrasts the best-known upper bounds on polytope diameters, which are linear in fixed dimension, but grow exponentially in the dimension (e.g., [18] and [10]). For a survey of the best bounds and more updates about diameters of polytopes see [7,8,10,26,28] and the references therein.…”

Section: Our Resultsmentioning

confidence: 99%

Diameters of Cocircuit Graphs of Oriented Matroids: An Update

Adler¹,

Loera²,

Klee³

et al. 2021

Electron. J. Combin.

View full text Add to dashboard Cite

Oriented matroids are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid. The motivation for our investigations is the complexity of the simplex method and the criss-cross method. We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the latest exact bounds for oriented matroids of low rank and low corank, and for all oriented matroids with up to nine elements (this part required a large computer-based proof). For arbitrary oriented matroids, we present an improvement to a quadratic bound of Finschi. Our discussion highlights an old conjecture that states a linear bound for the diameter is possible. On the positive side, we show the conjecture is true for oriented matroids of low rank and low corank, and, verified with computers, for all oriented matroids with up to nine elements. On the negative side, our computer search showed two natural strengthenings of the main conjecture are false.

show abstract

“…This bound contrasts the best-known upper bounds on polytope diameters, which are linear in fixed dimension, but grow exponentially in the dimension (e.g., [16] and [8]). For a survey of the best bounds and more updates about diameters of polytopes see [5,6,8,22,24] and the references therein.…”

Section: Our Resultsmentioning

confidence: 99%

Diameters of Cocircuit Graphs of Oriented Matroids: An Update

Adler¹,

Loera²,

Klee³

et al. 2020

Preprint

View full text Add to dashboard Cite

Oriented matroids (often called order types) are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid.We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the latest exact bounds for oriented matroids of low rank and low corank, and for all oriented matroids with up to nine elements (this part required a large computer-based proof). The motivation for our investigations is the complexity of the simplex method and the criss-cross method. For arbitrary oriented matroids, we present an improvement to a quadratic bound of Finschi. Our discussion highlights two very important conjectures related to the polynomial Hirsch conjecture for polytope diameters.

show abstract

An Asymptotically Improved Upper Bound on the Diameter of Polyhedra

Cited by 19 publications

References 27 publications

The diameter of lattice zonotopes

The diameter of lattice zonotopes

Diameters of Cocircuit Graphs of Oriented Matroids: An Update

Diameters of Cocircuit Graphs of Oriented Matroids: An Update

Contact Info

Product

Resources

About