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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2022.108
URN: urn:nbn:de:0030-drops-157044
URL: https://drops.dagstuhl.de/opus/volltexte/2022/15704/
Narayanan, Hariharan ;
Shah, Rikhav ;
Srivastava, Nikhil
A Spectral Approach to Polytope Diameter
Abstract
We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for integer polytopes in terms of the length of the description of the polytope (in bits) and the minimum angle between facets of its polar. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high probability there exists a "giant component" of vertices, with measure 1-o(1) and polynomial diameter. Both bounds rely on spectral gaps - of a certain Schrödinger operator in the first case, and a certain continuous time Markov chain in the second - which arise from the log-concavity of the volume of a simple polytope in terms of its slack variables.
BibTeX - Entry
@InProceedings{narayanan_et_al:LIPIcs.ITCS.2022.108,
author = {Narayanan, Hariharan and Shah, Rikhav and Srivastava, Nikhil},
title = {{A Spectral Approach to Polytope Diameter}},
booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
pages = {108:1--108:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-217-4},
ISSN = {1868-8969},
year = {2022},
volume = {215},
editor = {Braverman, Mark},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/15704},
URN = {urn:nbn:de:0030-drops-157044},
doi = {10.4230/LIPIcs.ITCS.2022.108},
annote = {Keywords: Polytope diameter, Markov Chain}
}
Keywords: |
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Polytope diameter, Markov Chain |
Collection: |
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13th Innovations in Theoretical Computer Science Conference (ITCS 2022) |
Issue Date: |
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2022 |
Date of publication: |
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25.01.2022 |