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Shadows of Newton polytopes

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Abstract

We define the shadow complexity of a polytope P as the maximum number of vertices in a linear projection of P to the plane. We describe connections to algebraic complexity and to parametrized optimization. We also provide several basic examples and constructions, and develop tools for bounding shadow complexity.

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References

  1. N. Alon and R. B. Boppana, The monotone circuit complexity of boolean functions, Combinatorica 7 (1987), 1–22.

    Article  MathSciNet  MATH  Google Scholar 

  2. E. Balas, Disjunctive programming: properties of the convex hull of feasible points, Discrete Applied Mathematics 89 (1998), 3–44.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf, Computational Geometry: Algorithms and Applications, Springer, Berlin, 2000.

    Book  MATH  Google Scholar 

  4. S. Berkowitz, L. Valiant, S. Skyum and C. Rackoff, Fast parallel computation of polynomials using few processors, SIAM Journal on Computing 12 (1983), 641–644.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Borodin and S. Cook, On the number of additions needed to cumpute specific polynomial, SIAM Journal on Computing 5 (1976), 146–157.

    Article  MathSciNet  MATH  Google Scholar 

  6. R. P. Brent, The parallel evaluation of general arithmetic expressions, Journal of the Association for Computing Machinery 21 (1974), 201–206.

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Bürgisser, Completeness and Reduction in Algebraic Complexity Theory, Algorithms and Computation in Mathematics, Vol. 7, Springer, Berlin, 2000.

    MATH  Google Scholar 

  8. P. Büirgisser, M. Clausen and M. A. Shokrollahi, Algebraic Complexity Theory, Grundlehren der mathematischen Wissenschaften, Vol. 315, Springer, Berlin, 1997.

    Book  Google Scholar 

  9. P. Carstensen, Complexity of some parametric integer and network programming problems, Mathematical Programming 26 (1983), 64–75.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Carstensen, The complexity of some problems in parametric linear and combinatorial programming, PhD thesis, University of Michigan, 1983.

  11. B. Chazelle, H. Edelsbrunner and L. J. Guibas, The complexity of cutting complexes, Discrete & Computational Geometry 4 (1989), 139–181.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Confronti, M. D. Summa and Y. Faenza, Balas formulation for the union of polytopes is optimal, Mathematical Programming 180 (2020), 311–326.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Edmonds, Matroids and the greedy algorithm, Mathematical Programming 1 (1971), 127–136.

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Fiorini, S. Massar, S. Pokutta, H. R. Tiwary and R. de Wolf, Linear vs. semidefinite extended formulations: Exponential separation and strong lower bounds, in STOC’12-Proceedings of the 2012 ACM Symposium on Theory of Computing, ACM, New York, 2012, 95–106.

    Google Scholar 

  15. S. Fomin, D. Grigoriev, and G. Koshevoy. Subtraction-free complexity, cluster transformations, and spanning trees, Foundations of Computational Mathematics 16 (2016), 1–31.

    Article  MathSciNet  MATH  Google Scholar 

  16. D. Gale, Optimal assignments in an ordered set: an application of matroid theory, Journal of Combinatorial Theory 4 (1968), 1073–1082.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Gao, Absolute irreducibility of polynomials via newton polytopes, Journal of Algebra 237 (2001), 501–520.

    Article  MathSciNet  MATH  Google Scholar 

  18. P. Gritzmann and B. Sturmfels, Minkowski addition of polytopes: Computational complexity and applications to Gröbner bases, SIAM Journal on Discrete Mathematics 6 (1993), 246–269.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. A. Grochow, Monotone projection lower bounds from extended formulation lower bounds, Theory of Computing 13 (2017), 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  20. P. Hrubeš, On the distribution of runners on a circle, European Journal of Combinatorics 89 (2020). Article no. 103137.

  21. L. Hyafil, On the parallel evaluation of multivariate polynomials, SIAM Journal on Computing 8 (1979), 120–123.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Jerrum and M. Snir, Some exact complexity results for straight-line computations over semirings, Journal of the Association for Computing Machinery 29 (1982), 874–897.

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Jukna, Lower bounds for tropical circuits and dynamic programs, Theory of Computing Systems 57 (2015), 160–194.

    Article  MathSciNet  MATH  Google Scholar 

  24. E. Kaltofen, Uniform closure properties of p-computable functions, in STOC’ 86: Proceedings of the eighteenth annual ACM symposium on Theory of computing, ACM, New York, 1987, pp. 330–337.

    Google Scholar 

  25. V. Klee, On a conjecture of Lindenstrauss, Israel Journal of Mathematics 1 (1963), 1–4.

    Article  MathSciNet  MATH  Google Scholar 

  26. P. Koiran, Shallow circuits with high-powered inputs, in Symposium on Innovations in Computer Science, https://arxiv.org/abs/1004.4960.

  27. P. Koiran, Arithmetic circuits: the chasm at depth four gets wider, Theoretical Computer Science 448 (2012), 56–65.

    Article  MathSciNet  MATH  Google Scholar 

  28. P. Koiran, N. Portier, S. Tavenas and S. Thomassé, A τ-conjecture for Newton polygons, Foundations of computational mathematics 15 (2015), 187–197.

    Article  MathSciNet  MATH  Google Scholar 

  29. U. H. Kortenkamp, J. Richter-Gebert, A. Sarangajan and G. M. Ziegler, Extremal properties of 0/1-polytopes, Discrete & Computational Geometry 17 (1997), 439–448.

    Article  MathSciNet  MATH  Google Scholar 

  30. J. G. Lagarias, Y. Luo and A. Padrol, Moser’s shadow problem, L’Enseignement Mathématique 64 (2018), 477–496.

    Article  MathSciNet  MATH  Google Scholar 

  31. E. H. Moore, A two-fold generalization of fermat’s theorem, Bulletin of the American Mathematical Society 2 (1896), 189–199.

    Article  MathSciNet  MATH  Google Scholar 

  32. L. Moser, Poorly formulated unsolved problems in combinatorial geometry, Mimeographed Notes, East Lansing conference, 1966.

  33. K. Mulmuley, Lower bounds in a parallel model without bit operations, SIAM Journal on Computing 28 (1999), 1460–1509.

    Article  MathSciNet  MATH  Google Scholar 

  34. K. Mulmuley and P. Shah, A lower bound for the shortest path problem, Journal of Computer and System Sciences 62 (2001), 253–267.

    Article  MathSciNet  MATH  Google Scholar 

  35. N. Nisan, Lower bounds for non-commutative computation, in STOC’ 91: Proceedings of the Twenty-Third Annual ACM Symposium on Theory of Computing, ACM, New York, 1991, pp. 410–418.

    Chapter  Google Scholar 

  36. A. Rao and A. Yehudayoff, Communication Complexity, Cambridge University Press, Cambridge, 2020.

    Book  MATH  Google Scholar 

  37. R. Raz and A. Yehudayoff, Multilinear formulas, maximal-partition discrepancy and mixed-sources extractors, Journal of Computer and System Sciences 77 (2011), 167–190.

    Article  MathSciNet  MATH  Google Scholar 

  38. T. Rothvoß, Some 0/1 polytopes need exponential size extended formulations, Mathematical Programming 142 (2013), 255–268.

    Article  MathSciNet  MATH  Google Scholar 

  39. T. Rothvoß, The matching polytope has exponential extension complexity, Journal of the Association for Computing Machinery 64 (2017), Article no. 41.

  40. E. Shamir and M. Snir, On the depth complexity of formulas, Mathematical Systems Theory 13 (1979), 301–322.

    Article  MathSciNet  MATH  Google Scholar 

  41. A. Shpilka and A. Yehudayoff, Arithmetic circuits: A survey of recent results and open questions, Foundations and Trends in Theoretical Computer Science 5 (2010), 207–388.

    Article  MathSciNet  MATH  Google Scholar 

  42. H. R. Tiwary, On computing the shadows and slices of polytopes, https://arxiv.org/abs/0804.4150.

  43. L. G. Valiant, Negation can be exponentially powerful, Theoretical Computer Science 12 (1980), 303–314.

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Vince, A framework for the greedy algorithm, Discrete Applied Mathematics 121 (2002), 247–260.

    Article  MathSciNet  MATH  Google Scholar 

  45. M. Yannakakis, Expressing combinatorial optimization problems by linear programs, Journal of Computer and System Sciences 43 (1991), 441–466.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgement

We thank Michael Forbes for pointing out the connection between shadow complexity and Conjecture 1.

Author information

Authors and Affiliations

  1. Mathematical Institute of the Czech Academy of Science, 115 67, Praha 1, Czech Republic

    Pavel Hrubeš

  2. Department of Mathematics, Technion—Israel Institute of Technology, Haifa, 32000, Israel

    Amir Yehudayoff

Authors
  1. Pavel Hrubeš
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  2. Amir Yehudayoff
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Corresponding author

Correspondence to Amir Yehudayoff.

Additional information

This work is dedicated to celebrating Nati’s birthday. May the shadows we see be just of polytopes

Supported by the GACR grant 19-27871X.

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Hrubeš, P., Yehudayoff, A. Shadows of Newton polytopes. Isr. J. Math. 256, 311–343 (2023). https://doi.org/10.1007/s11856-023-2510-z

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  • DOI: https://doi.org/10.1007/s11856-023-2510-z

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