Abstract
Under consideration is the Successive Minima Problem for the 2-dimensional lattice with respect to the order given by some conic function f. We propose an algorithm with complexity of 3.32 log2R + O(1) calls to the comparison oracle of f, where R is the radius of the circular searching area, while the best known lower oracle complexity bound is 3 log2R + O(1). Wegivean efficient criterion for checking that given vectors of a 2-dimensional lattice are successive minima and form a basis for the lattice. Moreover, we show that the similar Successive Minima Problem for dimension n can be solved by an algorithm with at most O(n)2n log R calls to the comparison oracle. The results of the article can be applied to searching successive minima with respect to arbitrary convex functions defined by the comparison oracle.
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References
A. Yu. Chirkov, “Minimization of a Quasiconvex Function on 2-Dimensional Lattice,” Vestnik of Lobachevsky State University of Nizhny Novgorod, Ser. Modeling and Optimal Control 1, 227–238 (2003).
A. Ahmadi, A. Olshevsky, P. Parrilo, and J. Tsitsiklis, “NP-Hardness of Deciding Convexity of Quadratic Polynomials and Related Problems,” Math. Program. 137 (1–2), 453–476 (2013).
D. Dadush, Integer Programming, Lattice Algorithms, and Deterministic Volume Estimation, Ph. D. Thesis (ProQuest LLC, Ann Arbor, MI; Georgia Institute of Technology, 2012).
D. Dadush, C. Peikert, and S. Vempala, “Enumerative Lattice Algorithms in Any Norm via M-Ellipsoid Coverings,” in Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science (Palm Springs, California, October 23–25, 2011), pp. 580–589.
L. Khachiyan and L. Porkolab, “Integer Optimization on Convex Semialgebraic Sets,” Discrete and Comput. Geom. 23 (2), 207–224 (2000).
H. Lenstra, “Integer Programming with a Fixed Number of Variables,” Math. Oper. Res. 8 (4), 538–548 (1983).
J. A. de Loera, R. Hemmecke, M. Koppe, and R. Weismantel, “Integer Polynomial Optimization in Fixed Dimension,” Math. Oper. Res. 31 (1), 147–153 (2006).
S. Heinz, “Complexity of Integer Quasiconvex Polynomial Optimization,” J. Complexity 21 (4), 543–556 (2005).
S. Heinz, “Quasiconvex Functions Can Be Approximated by Quasiconvex Polynomials,” ESAIM Control Optim. Calc. Var. 14 (4), 795–801 (2008).
R. Hemmecke, S. Onn, and R. Weismantel, “A Polynomial Oracle-Time Algorithm for Convex Integer Minimization,” Math. Program. 126 (1), 97–117 (2011).
R. Hildebrand and M. Köppe, “A New Lenstra-Type Algorithm for Quasiconvex Polynomial Integer Minimization with Complexity 2O(n log n),” Discrete Optim. 10 (1), 69–84 (2013).
T. Oertel, Integer Convex Minimization in Low Dimensions, Thes. Doct. Phylosophy (Eidgenössische Technische Hochschule, Zürich, 2014).
T. Oertel, C. Wagner, and R. Weismantel, “Integer Convex Minimization by Mixed Integer Linear Optimization,” Oper. Res. Lett. 42 (6), 424–428 (2014).
A. Basu and T. Oertel, “Centerpoints: A Link Between Optimization and Convex Geometry,” SIAM J. Optim. 27 (2), 866–889 (2017).
A. Yu. Chirkov, D. V. Gribanov, D. S. Malyshev, P. M. Pardalos, S. I. Veselov, and A. Yu. Zolotykh, “On the Complexity of Quasiconvex Integer Minimization Problem,” J. Global Optim. 73 (4), 761–788 (2018).
S. I. Veselov, D. V. Gribanov, N. Yu. Zolotykh, and A. Yu. Chirkov, “Minimizing a Symmetric Quasiconvex Function on a Two-Dimensional Lattice,” Discret. Anal. Issled. Oper. 25 (3), 23–35 (2018)
S. I. Veselov, D. V. Gribanov, N. Yu. Zolotykh, and A. Yu. Chirkov, J. Appl. Indust. Math. 12 (3), 587–594 (2018)].
D. Micciancio, “Efficient Reductions Among Lattice Problems,” in Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (San Francisco, California, January 20–22, 2008), pp. 84–93.
D. Micciancio and P. Voulgaris, “A Deterministic Single Exponential Time Algorithm for Most Lattice Problems Based on Voronoi Cell Computations,” SIAM J. Comput. 42 (3), 1364–1391 (2010).
D. Aggarwal, D. Dadush, O. Regev, and N. Stephens-Davidowitz, “Solving the Shortest Vector Problem in 2n Time via Discrete Gaussian Sampling,” in STOC’15. Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing (Portland, Oregon, USA, June 14–17, 2015), pp. 733–742.
D. Aggarwal, D. Dadush, and N. Stephens-Davidowitz, “Solving the Closest Vector Problem in 2n Time—The Discrete Gaussian Strikes Again!” in IEEE 56th Annual Symposium on Foundations of Computer Science (Berkeley, California, October 18–20, 2015), pp. 563–582.
R. Graham, D. Knuth, and O. Patashnik, Concrete Mathematics—A Foundation for Computer Science, 2nd ed. (Addison-Wesley Prof., Reading, MA, USA, 1994).
J. Cassels, An Introduction to the Geometry of Numbers (Springer, Berlin, 1997).
J. Edmonds, “Systems of Distinct Representatives and Linear Algebra,” J. Res. National Bureau of Stand. B: Math. Math. Phys. 71 B (4), 241–245 (1967).
M. Grötschel, L. Lovász, and A. Schrijver, “Geometric Algorithms and Combinatorial Optimization,” in Algorithms and Combinatorics, Vol. 2, 2nd corr. ed. (Springer, Berlin, 1993).
Acknowledgments
The authors are especially grateful to S. I. Veselov, N. Yu. Zolotykh, and A. Yu. Chirkov for their invaluable help in the preparation of the manuscript.
Funding
The authors were supported by the Russian Foundation for Basic Research (project no. 18-31-2001-mol-a-ved).
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Russian Text © The Author(s), 2020, published in Diskretnyi Analiz i Issledovanie Operatsii, 2020, Vol. 27, No. 1, pp. 17–42.
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Gribanov, D.V., Malyshev, D.S. Minimization of Even Conic Functions on the Two-Dimensional Integral Lattice. J. Appl. Ind. Math. 14, 56–72 (2020). https://doi.org/10.1134/S199047892001007X
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DOI: https://doi.org/10.1134/S199047892001007X