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Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes
[article]

2021
*
arXiv
*
pre-print

In this paper, we provide upper and lower

arXiv:2112.13027v1
fatcat:6c7ucu6xvnh2npl4tu3wgcmw4y
*bounds**on**the**combinatorial**diameter**of*a*random*"spherical"*polytope*, which is tight to within*one*factor*of*dimension when*the*number*of*inequalities is large ...*The**combinatorial**diameter*diam(P)*of*a*polytope*P is*the*maximum shortest path distance between any pair*of*vertices. ...*Asymptotic**Bounds**on**the**Combinatorial**Diameter**of**Random**Polytopes*...##
###
Comments on: Recent progress on the combinatorial diameter of polytopes and simplicial complexes

2013
*
TOP - An Official Journal of the Spanish Society of Statistics and Operations Research
*

*On*

*the*other hand, it features new results (exponential lower

*bounds*)

*on*

*the*

*diameter*

*of*simplicial complexes and re-interprets

*the*recent activities

*of*

*the*polymath 3 project that has been coordinated ...

*On*

*the*

*one*hand, it gives an impressive survey

*on*

*the*progress

*on*

*the*

*diameter*problem, including

*the*breakthrough

*of*

*the*author with his disproof

*of*

*the*Hirsch conjecture among many other recent results ... It turns out that all so far best-known

*asymptotic*upper

*bounds*for

*the*

*diameter*

*of*polyhedra such as

*the*n log d+1 -

*bound*

*of*Kalai and Kleitman (1992) and

*the*linear

*bounds*in fixed dimension

*of*Barnette ...

##
###
Questions About Polytopes
[chapter]

2001
*
Mathematics Unlimited — 2001 and Beyond
*

Thanks to Michael Joswig and Marc Pfetsch also for

doi:10.1007/978-3-642-56478-9_63
fatcat:wirx62rwivh6ff6symxv7rioiu
*the*polymake/javaview pictures*of**polytopes*. ... Thanks to*the*Computational Geometry community, whose support gives me*the*courage to start sentences by "We should . . . ". ...*On**the*other hand, Bárány & Pór (2000) produced a first superexponential lower*bound*for*the*number*of*suitable*random*0/1-*polytopes*with a rather large (exponential) number*of*vertices. ...##
###
Automorphic Forms on O s +2,2(ℝ+ and Generalized Kac-Moody Algebras
[chapter]

1995
*
Proceedings of the International Congress of Mathematicians
*

*The*rst part

*of*this paper deals with convexity in general and

*the*second part deals with

*the*combinatorics

*of*convex

*polytopes*. ... There are many excellent surveys 20, 9] and collections

*of*open problems 13, 29]. I try to discuss several speci c topics and to zoom in

*on*issues which I am more familiar with. ... In another independent development Slater, Tarjan and Thurston 43] proved a sharp lower

*bound*

*on*

*the*(

*combinatorial*)

*diameter*

*of*

*the*associahedron using volume estimates

*of*hyperbolic

*polytopes*. ...

##
###
Combinatorics and Convexity
[chapter]

1995
*
Proceedings of the International Congress of Mathematicians
*

*The*rst part

*of*this paper deals with convexity in general and

*the*second part deals with

*the*combinatorics

*of*convex

*polytopes*. ... There are many excellent surveys 20, 9] and collections

*of*open problems 13, 29]. I try to discuss several speci c topics and to zoom in

*on*issues which I am more familiar with. ... In another independent development Slater, Tarjan and Thurston 43] proved a sharp lower

*bound*

*on*

*the*(

*combinatorial*)

*diameter*

*of*

*the*associahedron using volume estimates

*of*hyperbolic

*polytopes*. ...

##
###
Page 4705 of Mathematical Reviews Vol. , Issue 2004f
[page]

2004
*
Mathematical Reviews
*

This paper shows that this also holds for

*randomized*algorithms, and hence*the**asymptotic*relative accuracy*of**randomized*algorithms is not superior to that*of*deterministic algorithms for*diameter*or ... Summary: “We study*combinatorial**bounds*for geometric permu- tations*of*balls with*bounded*size disparity in d-space. ...##
###
The Graph of the Pedigree Polytope is Asymptotically Almost Complete (Extended Abstract)
[article]

2016
*
arXiv
*
pre-print

We show that in

arXiv:1611.08419v2
fatcat:sfwymizqynaszo4er6fkoob2fe
*the*graph*of**the*pedigree*polytope*,*the*quotient minimum degree over number*of*vertices tends to 1 as*the*number*of*cities tends to infinity. ... Pedigree*polytopes*are extensions*of**the*classical Symmetric Traveling Salesman Problem*polytopes*(Arthanari 2000) whose graphs contain*the*TSP*polytope*graphs as spanning subgraphs (Arthanari 2013). ...*The*famous Hirsch conjecture in*the**combinatorial*study*of**polytopes*, settled by Santos [18] , concerned*the**diameter**of*graphs*of**polytopes*. ...##
###
Doubly random polytopes
[article]

2021
*
arXiv
*
pre-print

We establish results

arXiv:2006.07000v2
fatcat:yqwubva2ezcxhibacfwl7tns6a
*on*how well Q approximates*the*unit sphere in terms*of*m and p as well as*asymptotics**on**the**combinatorial*complexity*of*Q for certain regimes*of*p. ... A two-step model for generating*random**polytopes*is considered. ... Given d, m, ε and π,*one*can apply Lemma 14 to find p so that every vertex*of*Q ∼ P 2 (d, m, p) is within distance 1 + ε*of**the*origin. This p correspond to N 1 = pF (d)m. ...##
###
Page 3575 of Mathematical Reviews Vol. , Issue 2004e
[page]

2004
*
Mathematical Reviews
*

*The*authors prove, however, that this lower

*bound*can be matched by an upper

*bound*

*of*

*the*same

*asymptotic*value. ... ; Berlin)

*On*

*the*complexity

*of*

*polytope*isomorphism problems. ...

##
###
On the graph-density of random 0/1-polytopes
[article]

2003
*
arXiv
*
pre-print

Let D_d,n be

arXiv:math/0306246v1
fatcat:tlqtdrkkkvgshdysrfcpmwgkf4
*the*density*of**the*graph*of*P_d,n (i.e.,*the*number*of**one*-dimensional faces*of*P_d,n divided by n(n-1)/2). ... Let X_d,n be an n-element subset*of*0,1^d chosen uniformly at*random*, and denote by P_d,n := conv X_d,n its convex hull. ... Acknowledgements We thank*one**of**the*referees for several suggestions that helped to improve*the*presentation. ...##
###
Page 6906 of Mathematical Reviews Vol. , Issue 2003i
[page]

2003
*
Mathematical Reviews
*

We obtain tight

*bounds**on**the*same quantity when*the**random*rectangles are*of*a specified volume.” 2003i:52008 52A35 52A10 Swanepoel, Konrad J. ... Further, for*the*case where {a;}i<i<n is*the*canonical basis {e;})<i<,*of*R” and q = 1, they obtain*asymptotically*sharp estimates*of**the*most important parameters, such as type and co-type constants, ...##
###
Linear programming, the simplex algorithm and simple polytopes

1997
*
Mathematical programming
*

In

doi:10.1007/bf02614318
fatcat:egfjuwbfsvfabomu2fq6ggteua
*the*second part we discuss some recent developments concerning*the*simplex algorithm. We describe subexponential*randomized*pivot rules and upper*bounds**on**the**diameter**of*graphs*of**polytopes*. ... In*the*rst part*of**the*paper we survey some far-reaching applications*of**the*basic facts*of*linear programming to*the**combinatorial*theory*of*simple*polytopes*. ... We are short*of*polynomial*bounds*for*the**diameter*, and despite*the*simplicity*of**the*proofs for*the*known*bounds*we cannot push them any further. ...##
###
On the Expansion of Graphs of 0/1-Polytopes
[article]

2001
*
arXiv
*
pre-print

*Bounding*

*the*edge expansion from below is important for

*bounding*

*the*"mixing time"

*of*a

*random*walk

*on*

*the*graph from above. ... We present different techniques for

*bounding*

*the*edge expansion

*of*a 0/1-

*polytope*from below. ... This was even outperformed by

*the*cut

*polytope*

*of*

*the*complete graph

*on*n nodes that has

*diameter*

*one*for all n ≥ 2 (Barahona and Mahjoub [6] ). ...

##
###
On the Graph-Density of Random 0/1-Polytopes
[chapter]

2003
*
Lecture Notes in Computer Science
*

Let ∆ d,n be

doi:10.1007/978-3-540-45198-3_27
fatcat:dg4mxyvv5fhfzkzdftebd72pr4
*the*density*of**the*graph*of*P d,n (i.e.,*the*number*of**one*-dimensional faces*of*P d,n divided by'n 2´) . ... Let X d,n be an n-element subset*of*{0, 1} d chosen uniformly at*random*, and denote by P d,n := conv X d,n its convex hull. ... Acknowledgements We thank*one**of**the*referees for several suggestions that helped to improve*the*presentation. ...##
###
Some Algorithmic Problems in Polytope Theory
[chapter]

2003
*
Algebra, Geometry and Software Systems
*

Acknowledgment: We thank

doi:10.1007/978-3-662-05148-1_2
fatcat:i6zq6zfvrrgm5bgve2d6rxwxoi
*the*referee for many valuable comments and Günter M. Ziegler for carefully reading*the*manuscript. ...*Diameter*Input:*Polytope*P in H-description Output:*The**diameter**of*P Status (general): N P-hard Status (fixed dim.): Polynomial time Frieze and Teng [21] gave*the*proof*of*N P-hardness. ...*One**of**the*basic properties*of*a*polytope*is its dimension. ...
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