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 worstcase 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 1o(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 logconcavity 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 = {108:1108:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772174},
ISSN = {18688969},
year = {2022},
volume = {215},
editor = {Braverman, Mark},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/15704},
URN = {urn:nbn:de:0030drops157044},
doi = {10.4230/LIPIcs.ITCS.2022.108},
annote = {Keywords: Polytope diameter, Markov Chain}
}
25.01.2022
Keywords: 

Polytope diameter, Markov Chain 
Seminar: 

13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Issue date: 

2022 
Date of publication: 

25.01.2022 