Extension complexity of low-dimensional polytopes

Kwan MA, Sauermann L, Zhao Y. 2022. Extension complexity of low-dimensional polytopes. Transactions of the American Mathematical Society. 375(6), 4209–4250.

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Author
Kwan, Matthew AlanISTA; Sauermann, Lisa; Zhao, Yufei
Department
Abstract
Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope P is defined to be the minimum number of facets of a (possibly higher-dimensional) polytope from which P can be obtained as a (linear) projection. This notion is motivated by its relevance to combinatorial optimisation, and has been studied intensively for various specific polytopes associated with important optimisation problems. In this paper we study extension complexity as a parameter of general polytopes, more specifically considering various families of low-dimensional polytopes. First, we prove that for a fixed dimension d, the extension complexity of a random d-dimensional polytope (obtained as the convex hull of random points in a ball or on a sphere) is typically on the order of the square root of its number of vertices. Second, we prove that any cyclic n-vertex polygon (whose vertices lie on a circle) has extension complexity at most 24√n. This bound is tight up to the constant factor 24. Finally, we show that there exists an no(1)-dimensional polytope with at most n vertices and extension complexity n1−o(1). Our theorems are proved with a range of different techniques, which we hope will be of further interest.
Publishing Year
Date Published
2022-06-01
Journal Title
Transactions of the American Mathematical Society
Acknowledgement
The research of the first author was supported by SNSF Project 178493 and NSF Award DMS-1953990. The research of the second author supported by NSF Award DMS-1953772. The research of the third author was supported by NSF Award DMS-1764176, NSF CAREER Award DMS-2044606, a Sloan Research Fellowship, and the MIT Solomon Buchsbaum Fund.
Volume
375
Issue
6
Page
4209-4250
ISSN
eISSN
IST-REx-ID

Cite this

Kwan MA, Sauermann L, Zhao Y. Extension complexity of low-dimensional polytopes. Transactions of the American Mathematical Society. 2022;375(6):4209-4250. doi:10.1090/tran/8614
Kwan, M. A., Sauermann, L., & Zhao, Y. (2022). Extension complexity of low-dimensional polytopes. Transactions of the American Mathematical Society. American Mathematical Society. https://doi.org/10.1090/tran/8614
Kwan, Matthew Alan, Lisa Sauermann, and Yufei Zhao. “Extension Complexity of Low-Dimensional Polytopes.” Transactions of the American Mathematical Society. American Mathematical Society, 2022. https://doi.org/10.1090/tran/8614.
M. A. Kwan, L. Sauermann, and Y. Zhao, “Extension complexity of low-dimensional polytopes,” Transactions of the American Mathematical Society, vol. 375, no. 6. American Mathematical Society, pp. 4209–4250, 2022.
Kwan MA, Sauermann L, Zhao Y. 2022. Extension complexity of low-dimensional polytopes. Transactions of the American Mathematical Society. 375(6), 4209–4250.
Kwan, Matthew Alan, et al. “Extension Complexity of Low-Dimensional Polytopes.” Transactions of the American Mathematical Society, vol. 375, no. 6, American Mathematical Society, 2022, pp. 4209–50, doi:10.1090/tran/8614.

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