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# Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes

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LIPIcs.SoCG.2022.18.pdf
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## Acknowledgements

This work was done in part while the authors were participating in the Probability, Geometry and Computation in High Dimensions semester at the Simons Institute for the Theory of Computing and in the Interplay between High-Dimensional Geometry and Probability trimester at the Hausdorff Institute for Mathematics.

## Cite As

Gilles Bonnet, Daniel Dadush, Uri Grupel, Sophie Huiberts, and Galyna Livshyts. Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.18

## Abstract

The combinatorial diameter diam(P) of a polytope P is the maximum shortest path distance between any pair of vertices. 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 compared to the dimension. More precisely, for an n-dimensional polytope P defined by the intersection of m i.i.d. half-spaces whose normals are chosen uniformly from the sphere, we show that diam(P) is Ω(n m^{1/(n-1)}) and O(n² m^{1/(n-1)} + n⁵ 4ⁿ) with high probability when m ≥ 2^{Ω(n)}. For the upper bound, we first prove that the number of vertices in any fixed two dimensional projection sharply concentrates around its expectation when m is large, where we rely on the Θ(n² m^{1/(n-1)}) bound on the expectation due to Borgwardt [Math. Oper. Res., 1999]. To obtain the diameter upper bound, we stitch these "shadows paths" together over a suitable net using worst-case diameter bounds to connect vertices to the nearest shadow. For the lower bound, we first reduce to lower bounding the diameter of the dual polytope P^∘, corresponding to a random convex hull, by showing the relation diam(P) ≥ (n-1)(diam(P^∘)-2). We then prove that the shortest path between any "nearly" antipodal pair vertices of P^∘ has length Ω(m^{1/(n-1)}).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
##### Keywords
• Random Polytopes
• Combinatorial Diameter
• Hirsch Conjecture

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## References

1. Michel L Balinski. The Hirsch conjecture for dual transportation polyhedra. Mathematics of Operations Research, 9(4):629-633, 1984.
2. Imre Bárány and Zoltán Füredi. On the shape of the convex hull of random points. Probability Theory and Related Fields, 77(2):231-240, February 1988. URL: https://doi.org/10.1007/bf00334039.
3. David Barnette. Wv paths on 3-polytopes. Journal of Combinatorial Theory, 7(1):62-70, July 1969. URL: https://doi.org/10.1016/s0021-9800(69)80007-4.
4. David Barnette. An upper bound for the diameter of a polytope. Discrete Mathematics, 10(1):9-13, 1974. URL: https://doi.org/10.1016/0012-365x(74)90016-8.
5. Nicolas Bonifas, Marco Di Summa, Friedrich Eisenbrand, Nicolai Hähnle, and Martin Niemeier. On sub-determinants and the diameter of polyhedra. Discrete & Computational Geometry, 52(1):102-115, 2014.
6. Karl Heinz Borgwardt. The simplex method: a probabilistic analysis, volume 1 of Algorithms and Combinatorics: Study and Research Texts. Springer-Verlag, Berlin, 1987. URL: https://doi.org/10.1007/978-3-642-61578-8.
7. Karl Heinz Borgwardt. Erratum: A sharp upper bound for the expected number of shadow vertices in lp-polyhedra under orthogonal projection on two-dimensional planes. Mathematics of Operations Research, 24(4):925-984, 1999. URL: http://www.jstor.org/stable/3690611.
8. Karl Heinz Borgwardt and Petra Huhn. A lower bound on the average number of pivot-steps for solving linear programs valid for all variants of the simplex-algorithm. Mathematical Methods of Operations Research, 49(2):175-210, April 1999. URL: https://doi.org/10.1007/s186-1999-8373-5.
9. Steffen Borgwardt, Jesús A De Loera, and Elisabeth Finhold. The diameters of network-flow polytopes satisfy the Hirsch conjecture. Mathematical Programming, 171(1):283-309, 2018.
10. Graham Brightwell, Jan Van den Heuvel, and Leen Stougie. A linear bound on the diameter of the transportation polytope. Combinatorica, 26(2):133-139, 2006.
11. Daniel Dadush and Nicolai Hähnle. On the shadow simplex method for curved polyhedra. Discrete Computational Geometry, 56(4):882-909, June 2016. URL: https://doi.org/10.1007/s00454-016-9793-3.
12. Daniel Dadush and Sophie Huiberts. A friendly smoothed analysis of the simplex method. SIAM Journal on Computing, 49(5):STOC18-449, 2019.
13. Alberto Del Pia and Carla Michini. On the diameter of lattice polytopes. Discrete & Computational Geometry, 55(3):681-687, 2016.
14. Antoine Deza and Lionel Pournin. Improved bounds on the diameter of lattice polytopes. Acta Mathematica Hungarica, 154(2):457-469, 2018.
15. Martin Dyer and Alan Frieze. Random walks, totally unimodular matrices, and a randomised dual simplex algorithm. Mathematical Programming, 64(1):1-16, 1994.
16. Friedrich Eisenbrand, Nicolai Hähnle, Alexander Razborov, and Thomas Rothvoss. Diameter of polyhedra: Limits of abstraction. Mathematics of Operations Research, 35(4):786-794, 2010.
17. Marc Glisse, Sylvain Lazard, Julien Michel, and Marc Pouget. Silhouette of a random polytope. Journal of Computational Geometry, 7(1):14, 2016.
18. Richard C Grinold. The Hirsch conjecture in Leontief substitution systems. SIAM Journal on Applied Mathematics, 21(3):483-485, 1971.
19. Gil Kalai and Daniel J. Kleitman. A quasi-polynomial bound for the diameter of graphs of polyhedra. Bull. Amer. Math. Soc., 26(2):315-317, July 1992. URL: https://doi.org/10.1090/s0273-0979-1992-00285-9.
20. Victor Klee, David W Walkup, et al. The d-step conjecture for polyhedra of dimension d < 6. Acta Mathematica, 117:53-78, 1967.
21. Peter Kleinschmidt and Shmuel Onn. On the diameter of convex polytopes. Discrete mathematics, 102(1):75-77, 1992.
22. Jean-Philippe Labbé, Thibault Manneville, and Francisco Santos. Hirsch polytopes with exponentially long combinatorial segments. Mathematical Programming, 165(2):663-688, 2017.
23. D.G. Larman. Paths on polytopes. Proc. London Math. Soc. (3), s3-20(1):161-178, January 1970. URL: https://doi.org/10.1112/plms/s3-20.1.161.
24. Carla Michini and Antonio Sassano. The Hirsch Conjecture for the fractional stable set polytope. Mathematical Programming, 147(1):309-330, 2014.
25. Denis Naddef. The Hirsch conjecture is true for (0, 1)-polytopes. Mathematical Programming: Series A and B, 45(1-3):109-110, 1989.
26. Hariharan Narayanan, Rikhav Shah, and Nikhil Srivastava. A spectral approach to polytope diameter, 2021. URL: http://arxiv.org/abs/2101.12198.
27. A Reznikov and EB Saff. The covering radius of randomly distributed points on a manifold. International Mathematics Research Notices, 2016(19):6065-6094, 2016.
28. Laura Sanità. The diameter of the fractional matching polytope and its hardness implications. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 910-921. IEEE, 2018.
29. Francisco Santos. A counterexample to the Hirsch Conjecture. Annals of Mathematics, 176(1):383-412, July 2012. URL: https://doi.org/10.4007/annals.2012.176.1.7.
30. Rolf Schneider and Wolfgang Weil. Stochastic and integral geometry. Probability and its Applications (New York). Springer-Verlag, Berlin, 2008. URL: https://doi.org/10.1007/978-3-540-78859-1.
31. Daniel A Spielman and Shang-Hua Teng. Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. Journal of the ACM (JACM), 51(3):385-463, 2004.
32. Noriyoshi Sukegawa. An asymptotically improved upper bound on the diameter of polyhedra. Discrete & Computational Geometry, 62(3):690-699, 2019.
33. Michael J Todd. An improved Kalai-Kleitman bound for the diameter of a polyhedron. SIAM Journal on Discrete Mathematics, 28(4):1944-1947, 2014.
34. Roman Vershynin. Beyond Hirsch conjecture: walks on random polytopes and smoothed complexity of the simplex method. SIAM J. Comput., 39(2):646-678, 2009. Preliminary version in FOCS `06. URL: https://doi.org/10.1137/070683386.
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