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From Trees to Polynomials and Back Again: New Capacity Bounds with Applications to TSP

Authors Leonid Gurvits, Nathan Klein , Jonathan Leake

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Author Details

Leonid Gurvits
  • City College New York, NY, USA
Nathan Klein
  • Institute for Advanced Study, Princeton, NJ, USA
Jonathan Leake
  • University of Waterloo, Canada

Funding

  • Klein, Nathan: This author is supported by the National Science Foundation under Grant No. DMS-1926686.
  • Leake, Jonathan: This author acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number RGPIN-2023-03726]. Cette recherche a été partiellement financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), [numéro de référence RGPIN-2023-03726].

Acknowledgements

The third author would like to thank Vijay Bhattiprolu for many helpful and interesting discussions.

Cite AsGet BibTex

Leonid Gurvits, Nathan Klein, and Jonathan Leake. From Trees to Polynomials and Back Again: New Capacity Bounds with Applications to TSP. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 79:1-79:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.79

Abstract

We give simply exponential lower bounds on the probabilities of a given strongly Rayleigh distribution, depending only on its expectation. This resolves a weak version of a problem left open by Karlin-Klein-Oveis Gharan in their recent breakthrough work on metric TSP, and this resolution leads to a minor improvement of their approximation factor for metric TSP. Our results also allow for a more streamlined analysis of the algorithm. To achieve these new bounds, we build upon the work of Gurvits-Leake on the use of the productization technique for bounding the capacity of a real stable polynomial. This technique allows one to reduce certain inequalities for real stable polynomials to products of affine linear forms, which have an underlying matrix structure. In this paper, we push this technique further by characterizing the worst-case polynomials via bipartitioned forests. This rigid combinatorial structure yields a clean induction argument, which implies our stronger bounds. In general, we believe the results of this paper will lead to further improvement and simplification of the analysis of various combinatorial and probabilistic bounds and algorithms.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
  • Mathematics of computing → Probability and statistics
  • Theory of computation → Routing and network design problems
Keywords
  • traveling salesman problem
  • strongly Rayleigh distributions
  • polynomial capacity
  • probability lower bounds
  • combinatorial lower bounds

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