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A Spectral Approach to Polytope Diameter

Authors Hariharan Narayanan, Rikhav Shah, Nikhil Srivastava

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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.

Cite AsGet 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

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.

Subject Classification

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

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References

  1. Paul Alexandroff. Zur homologie. theorie der kompakten. Compositio Mathematica, 4:256-270, 1937. URL: http://eudml.org/doc/88654.
  2. 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 STOC'19 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 1-12. ACM, New York, 2019. URL: https://doi.org/10.1145/3313276.3316385.
  3. David Barnette. An upper bound for the diameter of a polytope. Discrete Math., 10:9-13, 1974. URL: https://doi.org/10.1016/0012-365X(74)90016-8.
  4. Nicolas Bonifas, Marco Di Summa, Friedrich Eisenbrand, Nicolai Hähnle, and Martin Niemeier. On sub-determinants and the diameter of polyhedra. Discrete Comput. Geom., 52(1):102-115, 2014. URL: https://doi.org/10.1007/s00454-014-9601-x.
  5. Karl-Heinz Borgwardt. Untersuchungen zur Asymptotik der mittleren Schrittzahl von Simplexverfahren in der linearen Optimierung. In Second Symposium on Operations Research (Rheinisch-Westfälische Tech. Hochsch. Aachen, Aachen, 1977), Teil 1, Operations Res. Verfahren, XXVIII, pages 332-345. Hain, Königstein/Ts., 1978. Google Scholar
  6. Karl-Heinz Borgwardt. The simplex method, volume 1 of Algorithms and Combinatorics: Study and Research Texts. Springer-Verlag, Berlin, 1987. A probabilistic analysis. URL: https://doi.org/10.1007/978-3-642-61578-8.
  7. Tobias Brunsch and Heiko Röglin. Finding short paths on polytopes by the shadow vertex algorithm. In Automata, languages, and programming. Part I, volume 7965 of Lecture Notes in Comput. Sci., pages 279-290. Springer, Heidelberg, 2013. URL: https://doi.org/10.1007/978-3-642-39206-1_24.
  8. Daniel Dadush and Nicolai Hähnle. On the shadow simplex method for curved polyhedra. Discrete Comput. Geom., 56(4):882-909, 2016. URL: https://doi.org/10.1007/s00454-016-9793-3.
  9. Daniel Dadush and Sophie Huiberts. A friendly smoothed analysis of the simplex method. SIAM J. Comput., 49(5):STOC18-449-STOC18-499, 2020. URL: https://doi.org/10.1137/18M1197205.
  10. A. Deshpande and D.A. Spielman. Improved smoothed analysis of the shadow vertex simplex method. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), pages 349-356, 2005. URL: https://doi.org/10.1109/SFCS.2005.44.
  11. Martin Dyer and Alan Frieze. Random walks, totally unimodular matrices, and a randomised dual simplex algorithm. Math. Programming, 64(1, Ser. A):1-16, 1994. URL: https://doi.org/10.1007/BF01582563.
  12. Friedrich Eisenbrand and Santosh Vempala. Geometric random edge. Math. Program., 164(1-2, Ser. A):325-339, 2017. URL: https://doi.org/10.1007/s10107-016-1089-0.
  13. Geoffrey R. Grimmett and David R. Stirzaker. Probability and random processes. Oxford University Press, Oxford, 2020. Fourth edition [of 0667520]. Google Scholar
  14. Ivan Izmestiev. The Colin de Verdière number and graphs of polytopes. Israel J. Math., 178:427-444, 2010. URL: https://doi.org/10.1007/s11856-010-0070-5.
  15. Gil Kalai and Daniel J. Kleitman. A quasi-polynomial bound for the diameter of graphs of polyhedra. Bull. Amer. Math. Soc. (N.S.), 26(2):315-316, 1992. URL: https://doi.org/10.1090/S0273-0979-1992-00285-9.
  16. D. G. Larman. Paths of polytopes. Proc. London Math. Soc. (3), 20:161-178, 1970. URL: https://doi.org/10.1112/plms/s3-20.1.161.
  17. Milena Mihail. On the expansion of combinatorial polytopes. In Mathematical foundations of computer science 1992 (Prague, 1992), volume 629 of Lecture Notes in Comput. Sci., pages 37-49. Springer, Berlin, 1992. URL: https://doi.org/10.1007/3-540-55808-X_4.
  18. Denis Naddef. The Hirsch conjecture is true for (0,1)-polytopes. Math. Programming, 45(1, (Ser. B)):109-110, 1989. URL: https://doi.org/10.1007/BF01589099.
  19. Francisco Santos. A counterexample to the Hirsch conjecture. Ann. of Math. (2), 176(1):383-412, 2012. URL: https://doi.org/10.4007/annals.2012.176.1.7.
  20. Rolf Schneider. Polytopes and Brunn-Minkowski theory. In Polytopes: abstract, convex and computational (Scarborough, ON, 1993), volume 440 of NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., pages 273-299. Kluwer Acad. Publ., Dordrecht, 1994. Google Scholar
  21. Rolf Schneider. Convex bodies: the Brunn-Minkowski theory, volume 151 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, expanded edition, 2014. Google Scholar
  22. Daniel A. Spielman and Shang-Hua Teng. Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J. ACM, 51(3):385-463, 2004. URL: https://doi.org/10.1145/990308.990310.
  23. Noriyoshi Sukegawa. An asymptotically improved upper bound on the diameter of polyhedra. Discrete Comput. Geom., 62(3):690-699, 2019. URL: https://doi.org/10.1007/s00454-018-0016-y.
  24. V. A. Timorin. An analogue of the Hodge-Riemann relations for simple convex polyhedra. Uspekhi Mat. Nauk, 54(2(326)):113-162, 1999. URL: https://doi.org/10.1070/rm1999v054n02ABEH000134.
  25. Michael J. Todd. An improved Kalai-Kleitman bound for the diameter of a polyhedron. SIAM J. Discrete Math., 28(4):1944-1947, 2014. URL: https://doi.org/10.1137/140962310.
  26. Edwin R. Van Dam and Willem H. Haemers. Eigenvalues and the diameter of graphs. Linear and Multilinear Algebra, 39(1-2):33-44, 1995. URL: https://doi.org/10.1080/03081089508818378.
  27. Roman Vershynin. Beyond Hirsch conjecture: walks on random polytopes and smoothed complexity of the simplex method. SIAM J. Comput., 39(2):646-678, 2009. URL: https://doi.org/10.1137/070683386.
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