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A Spectral Approach to Polytope Diameter

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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 polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint matrix, which in some cases improves previously known results. 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.

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Notes

  1. Our proof yields a slightly stronger conclusion regarding continuity of the formal Hessian than [17].

  2. The reader may consult e.g. [16, Chap. 6] for an introduction to continuous time Markov processes.

References

  1. Alexandrov, A.D.: Zur theorie der gemischten volumina von konvexen körpern ii. Mat. Sbornik N.S. 2, 1205–1238 (1937)

    Google Scholar 

  2. Alexandrov, A.D.: Zur theorie der gemischten volumina von konvexen körpern iv. Mat. Sbornik N.S. 3, 227–251 (1938)

    Google Scholar 

  3. Anari, N., Liu, K., Gharan, S.O., Vinzant, C.: 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 (pp. 1–12). ACM, New York (2019)

  4. Barnette, D.: An upper bound for the diameter of a polytope. Discret. Math. 10, 9–13 (1974)

    Article  MathSciNet  Google Scholar 

  5. Bonifas, N., Di Summa, M., Eisenbrand, F., Hähnle, N., Niemeier, M.: On sub-determinants and the diameter of polyhedra. Discrete Comput. Geom. 52(1), 102–115 (2014)

    Article  MathSciNet  Google Scholar 

  6. Borgwardt, K., Huhn, P.: A lower bound on the average number of pivot-steps for solving linear programs valid for all variants of the simplex-algorithm. Math. Methods OR 49, 175–210 (1999)

    Article  MathSciNet  Google Scholar 

  7. Borgwardt, K.-H.: Untersuchungen zur Asymptotik der mittleren Schrittzahl von Simplexverfahren in der linearen Optimierung. In: 2nd Symposium on Operations Research (Rheinisch-Westfälische Tech. Hochsch. Aachen, Aachen, 1977), Teil 1, Operations Res. Verfahren, XXVIII, pp. 332–345. Hain, Königstein/Ts (1978)

  8. Borgwardt, K.-H.: The Simplex Method. Algorithms and Combinatorics: Study and Research Texts, vol. 1. Springer, Berlin (1987). A probabilistic analysis

  9. Brunsch, T., Röglin, H.: Finding short paths on polytopes by the shadow vertex algorithm. In: Automata, Languages, and Programming. Part I. Lecture Notes in Computer Sciences, vol. 7965, pp. 279–290. Springer, Heidelberg (2013)

  10. Chung, F.R.K.: Diameters and eigenvalues. J. Am. Math. Soc. 2(2), 187–196 (1989)

    Article  MathSciNet  Google Scholar 

  11. Dadush, D., Hähnle, N.: On the shadow simplex method for curved polyhedra. Discrete Comput. Geom. 56(4), 882–909 (2016)

    Article  MathSciNet  Google Scholar 

  12. Dadush, D., Huiberts, S.: A friendly smoothed analysis of the simplex method. SIAM J. Comput. (2020). https://doi.org/10.1137/18M119720

    Article  MathSciNet  Google Scholar 

  13. Deshpande, A., Spielman, D.A.: 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)

  14. Dyer, M., Frieze, A.: Random walks, totally unimodular matrices, and a randomised dual simplex algorithm. Math. Program. 64(1, Ser. A), 1–16 (1994)

    Article  MathSciNet  Google Scholar 

  15. Eisenbrand, F., Vempala, S.: Geometric random edge. Math. Program. 164(1–2, Ser. A), 325–339 (2017)

    Article  MathSciNet  Google Scholar 

  16. Grimmett, G.R., Stirzaker, D.R.: Probability and Random Processes. Oxford University Press, Oxford (2020). 4th edn [of 0667520]

  17. Izmestiev, I.: The Colin de Verdière number and graphs of polytopes. Isr. J. Math. 178, 427–444 (2010)

    Article  MathSciNet  Google Scholar 

  18. Kalai, G., Kleitman, D.J.: A quasi-polynomial bound for the diameter of graphs of polyhedra. Bull. Am. Math. Soc. (N.S.) 26(2), 315–316 (1992)

    Article  MathSciNet  Google Scholar 

  19. Larman, D.G.: Paths of polytopes. Proc. Lond. Math. Soc. 3(20), 161–178 (1970)

    Article  MathSciNet  Google Scholar 

  20. McMullen, P.: On simple polytopes. Invent. Math. 113, 419–444 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  21. McMullen, P.: Weights on polytopes. Discrete Comput. Geom. 15, 363–388 (1996)

    Article  MathSciNet  Google Scholar 

  22. Mihail, M.: On the expansion of combinatorial polytopes. In: Mathematical Foundations of Computer Science 1992 (Prague, 1992). Lecture Notes in Computer Science, vol. 629, pp. 37–49. Springer, Berlin (1992)

  23. Naddef, D.: The Hirsch conjecture is true for \((0,1)\)-polytopes. Math. Program. 45(1, (Ser. B)), 109–110 (1989)

    Article  MathSciNet  Google Scholar 

  24. Narayanan, H., Shah, R., Srivastava, N.: A spectral approach to polytope diameter. arXiv preprint (2021). arXiv:2101.12198

  25. Santos, F.: A counterexample to the Hirsch conjecture. Ann. Math. 176(1), 383–412 (2012)

    Article  MathSciNet  Google Scholar 

  26. Schneider, R.: Polytopes and Brunn–Minkowski theory. In: Polytopes: Abstract, Convex and Computational (Scarborough, ON, 1993). NATO Advanced Study Institute, Nonstandard Analysis and Its Applications, vol. 440, pp. 273–299. Kluwer, Dordrecht (1994)

  27. Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 151, expanded edn. Cambridge University Press, Cambridge (2014)

  28. Sokal, A.D., Thomas, L.E.: Absence of mass gap for a class of stochastic contour models. J. Stat. Phys. 51(5), 907–947 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  29. Spielman, D.A., Teng, S.-H.: Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J. ACM 51(3), 385–463 (2004)

    Article  MathSciNet  Google Scholar 

  30. Stanley, R.P.: The numbers of faces of a simplicial convex polytope. Adv. Math. 35, 236–238 (1980)

    Article  MathSciNet  Google Scholar 

  31. Sukegawa, N.: An asymptotically improved upper bound on the diameter of polyhedra. Discrete Comput. Geom. 62(3), 690–699 (2019)

    Article  MathSciNet  Google Scholar 

  32. Timorin, V.A.: An analogue of the Hodge–Riemann relations for simple convex polyhedra. Uspekhi Mat. Nauk, 54(2(326)), 113–162 (1999)

  33. Todd, M.J.: An improved Kalai–Kleitman bound for the diameter of a polyhedron. SIAM J. Discret. Math. 28(4), 1944–1947 (2014)

    Article  MathSciNet  Google Scholar 

  34. Van Dam, E.R., Haemers, W.H.: Eigenvalues and the diameter of graphs. Linear Multilinear Algebra 39(1–2), 33–44 (1995)

    MathSciNet  Google Scholar 

  35. Vershynin, R.: Beyond Hirsch conjecture: walks on random polytopes and smoothed complexity of the simplex method. SIAM J. Comput. 39(2), 646–678 (2009)

    Article  MathSciNet  Google Scholar 

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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 [17]. We thank the IUSSTF virtual center on “Polynomials as an Algorithmic Paradigm” for supporting this collaboration.

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Authors and Affiliations

  1. TIFR Mumbai, Mumbai, India

    Hariharan Narayanan

  2. UC Berkeley, Berkeley, USA

    Rikhav Shah & Nikhil Srivastava

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  1. Hariharan Narayanan
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  2. Rikhav Shah
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Correspondence to Hariharan Narayanan.

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Narayanan, H., Shah, R. & Srivastava, N. A Spectral Approach to Polytope Diameter. Discrete Comput Geom (2024). https://doi.org/10.1007/s00454-024-00636-y

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