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

Authors Leonid Gurvits, Nathan Klein , Jonathan Leake



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.79.pdf
  • Filesize: 0.81 MB
  • 20 pages

Document Identifiers

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

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Kareem Adiprasito, June Huh, and Eric Katz. Hodge theory for combinatorial geometries. Annals of Mathematics, 188(2):381-452, 2018. Google Scholar
  2. Yeganeh Alimohammadi, Nima Anari, Kirankumar Shiragur, and Thuy-Duong Vuong. Fractionally log-concave and sector-stable polynomials: counting planar matchings and more. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 433-446, 2021. Google Scholar
  3. Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials iii: Mason’s ultra-log-concavity conjecture for independent sets of matroids. arXiv preprint, 2018. URL: https://arxiv.org/abs/1811.01600.
  4. Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials ii: High-dimensional walks and an fpras for counting bases of a matroid. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 1-12, 2019. Google Scholar
  5. Nima Anari, Tung Mai, Shayan Oveis Gharan, and Vijay V Vazirani. Nash social welfare for indivisible items under separable, piecewise-linear concave utilities. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2274-2290. SIAM, 2018. Google Scholar
  6. Nima Anari and Shayan Oveis Gharan. The kadison-singer problem for strongly rayleigh measures and applications to asymmetric tsp. arXiv preprint, 2014. URL: https://arxiv.org/abs/1412.1143.
  7. Nima Anari and Shayan Oveis Gharan. A generalization of permanent inequalities and applications in counting and optimization. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 384-396, 2017. Google Scholar
  8. Nima Anari, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials, entropy, and a deterministic approximation algorithm for counting bases of matroids. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 35-46. IEEE, 2018. Google Scholar
  9. Arash Asadpour, Michel X Goemans, Aleksander Madry, Shayan Oveis Gharan, and Amin Saberi. An o (log n/log log n)-approximation algorithm for the asymmetric traveling salesman problem. In Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete algorithms, pages 379-389, 2010. Google Scholar
  10. Alexander Barvinok. Enumerating contingency tables via random permanents. Combinatorics, Probability and Computing, 17(1):1-19, 2008. Google Scholar
  11. Alexander Barvinok. Computing the permanent of (some) complex matrices. Foundations of Computational Mathematics, 16:329-342, 2016. Google Scholar
  12. Julius Borcea, Petter Brändén, and Thomas Liggett. Negative dependence and the geometry of polynomials. Journal of the American Mathematical Society, 22(2):521-567, 2009. Google Scholar
  13. Petter Brändén. Hyperbolic polynomials and the kadison-singer problem. arXiv preprint, 2018. URL: https://arxiv.org/abs/1809.03255.
  14. Petter Brändén and June Huh. Lorentzian polynomials. Annals of Mathematics, 192(3):821-891, 2020. Google Scholar
  15. Petter Brändén, Jonathan Leake, and Igor Pak. Lower bounds for contingency tables via lorentzian polynomials. Israel Journal of Mathematics, 253(1):43-90, 2023. Google Scholar
  16. Gregory P Egorychev. The solution of van der waerden’s problem for permanents. Advances in Mathematics, 42(3):299-305, 1981. Google Scholar
  17. Dmitry I Falikman. Proof of the van der waerden conjecture regarding the permanent of a doubly stochastic matrix. Mathematical notes of the Academy of Sciences of the USSR, 29:475-479, 1981. Google Scholar
  18. Leonid Gurvits. Hyperbolic polynomials approach to van der waerden/schrijver-valiant like conjectures: sharper bounds, simpler proofs and algorithmic applications. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 417-426, 2006. Google Scholar
  19. Leonid Gurvits. The van der waerden conjecture for mixed discriminants. Advances in Mathematics, 200(2):435-454, 2006. Google Scholar
  20. Leonid Gurvits. On multivariate newton-like inequalities. In Advances in Combinatorial Mathematics: Proceedings of the Waterloo Workshop in Computer Algebra 2008, pages 61-78. Springer, 2009. Google Scholar
  21. Leonid Gurvits. A polynomial-time algorithm to approximate the mixed volume within a simply exponential factor. Discrete & Computational Geometry, 41:533-555, 2009. Google Scholar
  22. Leonid Gurvits and Jonathan Leake. Capacity lower bounds via productization. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 847-858, 2021. Google Scholar
  23. Anna Karlin, Nathan Klein, and Shayan Oveis Gharan. A (slightly) improved bound on the integrality gap of the subtour lp for tsp. In FOCS, pages 844-855. IEEE Computer Society, 2022. Google Scholar
  24. Anna R. Karlin, Nathan Klein, and Shayan Oveis Gharan. A (slightly) improved approximation algorithm for metric tsp. In STOC. ACM, 2021. Google Scholar
  25. Anna R. Karlin, Nathan Klein, and Shayan Oveis Gharan. A deterministic better-than-3/2 approximation algorithm for metric tsp. In Alberto Del Pia and Volker Kaibel, editors, Integer Programming and Combinatorial Optimization, pages 261-274, Cham, 2023. Springer International Publishing. Google Scholar
  26. Adam W Marcus, Daniel A Spielman, and Nikhil Srivastava. Interlacing families ii: Mixed characteristic polynomials and the kadison—singer problem. Annals of Mathematics, pages 327-350, 2015. Google Scholar
  27. Shayan Oveis Gharan, Amin Saberi, and Mohit Singh. A randomized rounding approach to the traveling salesman problem. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 550-559. IEEE, 2011. Google Scholar
  28. Damian Straszak and Nisheeth K Vishnoi. Real stable polynomials and matroids: Optimization and counting. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 370-383, 2017. Google Scholar