2026-07-21T14:13:07Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:15704 2024-03-06T10:56:00Z ddc:004 open_access

A Spectral Approach to Polytope Diameter Narayanan, Hariharan Shah, Rikhav Srivastava, Nikhil Polytope diameter Markov Chain 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Hariharan Narayanan and Rikhav Shah and Nikhil Srivastava 2022 Is Part Of LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2022.108 urn:nbn:de:0030-drops-157044 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.108 eng https://creativecommons.org/licenses/by/4.0/legalcode