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

Gilles Bonnet, Daniel Dadush, Uri Grupel, Sophie Huiberts, Galyna Livshyts
2021 arXiv   pre-print
In this paper, we provide upper and lower 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  ... 
arXiv:2112.13027v1 fatcat:6c7ucu6xvnh2npl4tu3wgcmw4y

Comments on: Recent progress on the combinatorial diameter of polytopes and simplicial complexes

Friedrich Eisenbrand
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  ... 
doi:10.1007/s11750-013-0292-x fatcat:rug4bg632nffdf4ds3i4lmpt2y

Questions About Polytopes [chapter]

Günter M. Ziegler
2001 Mathematics Unlimited — 2001 and Beyond  
Thanks to Michael Joswig and Marc Pfetsch also for 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.  ... 
doi:10.1007/978-3-642-56478-9_63 fatcat:wirx62rwivh6ff6symxv7rioiu

Automorphic Forms on O s +2,2(ℝ+ and Generalized Kac-Moody Algebras [chapter]

Richard E. Borcherds
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.  ... 
doi:10.1007/978-3-0348-9078-6_67 fatcat:cxsazvmamfehdenbvhxkk74yyy

Combinatorics and Convexity [chapter]

Gil Kalai
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.  ... 
doi:10.1007/978-3-0348-9078-6_131 fatcat:k4ja37y3lbcqzakf7rqtraqpwe

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]

Abdullah Makkeh and Mozhgan Pourmoradnasseri and Dirk Oliver Theis
2016 arXiv   pre-print
We show that in 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.  ... 
arXiv:1611.08419v2 fatcat:sfwymizqynaszo4er6fkoob2fe

Doubly random polytopes [article]

Andrew Newman
2021 arXiv   pre-print
We establish results 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.  ... 
arXiv:2006.07000v2 fatcat:yqwubva2ezcxhibacfwl7tns6a

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]

Volker Kaibel, Anja Remshagen
2003 arXiv   pre-print
Let D_d,n be 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.  ... 
arXiv:math/0306246v1 fatcat:tlqtdrkkkvgshdysrfcpmwgkf4

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

Gil Kalai
1997 Mathematical programming  
In 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.  ... 
doi:10.1007/bf02614318 fatcat:egfjuwbfsvfabomu2fq6ggteua

On the Expansion of Graphs of 0/1-Polytopes [article]

Volker Kaibel
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] ).  ... 
arXiv:math/0112146v1 fatcat:im4m42numnbllhfkwncm7jqqiu

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

Volker Kaibel, Anja Remshagen
2003 Lecture Notes in Computer Science  
Let ∆ d,n be 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.  ... 
doi:10.1007/978-3-540-45198-3_27 fatcat:dg4mxyvv5fhfzkzdftebd72pr4

Some Algorithmic Problems in Polytope Theory [chapter]

Volker Kaibel, Marc E. Pfetsch
2003 Algebra, Geometry and Software Systems  
Acknowledgment: We thank 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.  ... 
doi:10.1007/978-3-662-05148-1_2 fatcat:i6zq6zfvrrgm5bgve2d6rxwxoi
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