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.
References
N. Alon and R. B. Boppana, The monotone circuit complexity of boolean functions, Combinatorica 7 (1987), 1–22.
E. Balas, Disjunctive programming: properties of the convex hull of feasible points, Discrete Applied Mathematics 89 (1998), 3–44.
M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf, Computational Geometry: Algorithms and Applications, Springer, Berlin, 2000.
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.
A. Borodin and S. Cook, On the number of additions needed to cumpute specific polynomial, SIAM Journal on Computing 5 (1976), 146–157.
R. P. Brent, The parallel evaluation of general arithmetic expressions, Journal of the Association for Computing Machinery 21 (1974), 201–206.
P. Bürgisser, Completeness and Reduction in Algebraic Complexity Theory, Algorithms and Computation in Mathematics, Vol. 7, Springer, Berlin, 2000.
P. Büirgisser, M. Clausen and M. A. Shokrollahi, Algebraic Complexity Theory, Grundlehren der mathematischen Wissenschaften, Vol. 315, Springer, Berlin, 1997.
P. Carstensen, Complexity of some parametric integer and network programming problems, Mathematical Programming 26 (1983), 64–75.
P. Carstensen, The complexity of some problems in parametric linear and combinatorial programming, PhD thesis, University of Michigan, 1983.
B. Chazelle, H. Edelsbrunner and L. J. Guibas, The complexity of cutting complexes, Discrete & Computational Geometry 4 (1989), 139–181.
M. Confronti, M. D. Summa and Y. Faenza, Balas formulation for the union of polytopes is optimal, Mathematical Programming 180 (2020), 311–326.
J. Edmonds, Matroids and the greedy algorithm, Mathematical Programming 1 (1971), 127–136.
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.
S. Fomin, D. Grigoriev, and G. Koshevoy. Subtraction-free complexity, cluster transformations, and spanning trees, Foundations of Computational Mathematics 16 (2016), 1–31.
D. Gale, Optimal assignments in an ordered set: an application of matroid theory, Journal of Combinatorial Theory 4 (1968), 1073–1082.
S. Gao, Absolute irreducibility of polynomials via newton polytopes, Journal of Algebra 237 (2001), 501–520.
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.
J. A. Grochow, Monotone projection lower bounds from extended formulation lower bounds, Theory of Computing 13 (2017), 1–15.
P. Hrubeš, On the distribution of runners on a circle, European Journal of Combinatorics 89 (2020). Article no. 103137.
L. Hyafil, On the parallel evaluation of multivariate polynomials, SIAM Journal on Computing 8 (1979), 120–123.
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.
S. Jukna, Lower bounds for tropical circuits and dynamic programs, Theory of Computing Systems 57 (2015), 160–194.
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.
V. Klee, On a conjecture of Lindenstrauss, Israel Journal of Mathematics 1 (1963), 1–4.
P. Koiran, Shallow circuits with high-powered inputs, in Symposium on Innovations in Computer Science, https://arxiv.org/abs/1004.4960.
P. Koiran, Arithmetic circuits: the chasm at depth four gets wider, Theoretical Computer Science 448 (2012), 56–65.
P. Koiran, N. Portier, S. Tavenas and S. Thomassé, A τ-conjecture for Newton polygons, Foundations of computational mathematics 15 (2015), 187–197.
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.
J. G. Lagarias, Y. Luo and A. Padrol, Moser’s shadow problem, L’Enseignement Mathématique 64 (2018), 477–496.
E. H. Moore, A two-fold generalization of fermat’s theorem, Bulletin of the American Mathematical Society 2 (1896), 189–199.
L. Moser, Poorly formulated unsolved problems in combinatorial geometry, Mimeographed Notes, East Lansing conference, 1966.
K. Mulmuley, Lower bounds in a parallel model without bit operations, SIAM Journal on Computing 28 (1999), 1460–1509.
K. Mulmuley and P. Shah, A lower bound for the shortest path problem, Journal of Computer and System Sciences 62 (2001), 253–267.
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.
A. Rao and A. Yehudayoff, Communication Complexity, Cambridge University Press, Cambridge, 2020.
R. Raz and A. Yehudayoff, Multilinear formulas, maximal-partition discrepancy and mixed-sources extractors, Journal of Computer and System Sciences 77 (2011), 167–190.
T. Rothvoß, Some 0/1 polytopes need exponential size extended formulations, Mathematical Programming 142 (2013), 255–268.
T. Rothvoß, The matching polytope has exponential extension complexity, Journal of the Association for Computing Machinery 64 (2017), Article no. 41.
E. Shamir and M. Snir, On the depth complexity of formulas, Mathematical Systems Theory 13 (1979), 301–322.
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.
H. R. Tiwary, On computing the shadows and slices of polytopes, https://arxiv.org/abs/0804.4150.
L. G. Valiant, Negation can be exponentially powerful, Theoretical Computer Science 12 (1980), 303–314.
A. Vince, A framework for the greedy algorithm, Discrete Applied Mathematics 121 (2002), 247–260.
M. Yannakakis, Expressing combinatorial optimization problems by linear programs, Journal of Computer and System Sciences 43 (1991), 441–466.
Acknowledgement
We thank Michael Forbes for pointing out the connection between shadow complexity and Conjecture 1.
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
Hrubeš, P., Yehudayoff, A. Shadows of Newton polytopes. Isr. J. Math. 256, 311–343 (2023). https://doi.org/10.1007/s11856-023-2510-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-023-2510-z