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A Spectral Approach to Polytope Diameter

Authors Hariharan Narayanan, Rikhav Shah, Nikhil Srivastava

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  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Mathematical optimization
Keywords
  • Polytope diameter
  • Markov Chain

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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.

Cite As Get BibTex

Hariharan Narayanan, Rikhav Shah, and Nikhil Srivastava. A Spectral Approach to Polytope Diameter. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 108:1-108:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.108

Author Details

Hariharan Narayanan
  • Tata Institute of Fundamental Research, Mumbai, India
Rikhav Shah
  • University of California Berkeley, CA, USA
Nikhil Srivastava
  • University of California Berkeley, CA, USA

Funding

  • Narayanan, Hariharan: Supported by a Swarna Jayanti Fellowship.
  • Srivastava, Nikhil: Supported by NSF Grants CCF-1553751 and CCF-2009011.

Acknowledgements

We thank Daniel Dadush, Bo'az Klartag, and Ramon van Handel for helpful comments and suggestions on an earlier version of this manuscript. We thank Ramon van Handel for pointing out the important reference [Izmestiev, 2010]. We thank the IUSSTF virtual center on "Polynomials as an Algorithmic Paradigm" for supporting this collaboration.

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