Abstract
The Flatness theorem states that the maximum lattice width \(\mathrm {Flt}(d)\) of a d-dimensional lattice-free convex set is finite. It is the key ingredient for Lenstra’s algorithm for integer programming in fixed dimension, and much work has been done to obtain bounds on \(\mathrm {Flt}(d)\). While most results have been concerned with upper bounds, only few techniques are known to obtain lower bounds. In fact, the previously best known lower bound \(\mathrm {Flt}(d) \ge 1.138d\) arises from direct sums of a 3-dimensional lattice-free simplex.
In this work, we establish the lower bound \(\mathrm {Flt}(d) \ge 2d - O(\sqrt{d})\), attained by a family of lattice-free simplices. Our construction is based on a differential equation that naturally appears in this context.
Additionally, we provide the first local maximizers of the lattice width of 4- and 5-dimensional lattice-free convex bodies.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Andersen, K., Louveaux, Q., Weismantel, R., Wolsey, L.A.: Inequalities from two rows of a simplex tableau. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 1–15. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-72792-7_1
Averkov, G., Basu, A., Paat, J.: Approximation of corner polyhedra with families of intersection cuts. SIAM J. Optim. 28(1), 904–929 (2018)
Averkov, G., Codenotti, G., Macchia, A., Santos, F.: A local maximizer for lattice width of 3-dimensional hollow bodies. Discrete Appl. Math. 298, 129–142 (2021)
Baes, M., Oertel, T., Weismantel, R.: Duality for mixed-integer convex minimization. Math. Prog. 158(1), 547–564 (2015). https://doi.org/10.1007/s10107-015-0917-y
Balas, E.: Intersection cuts-a new type of cutting planes for integer programming. Oper. Res. 19(1), 19–39 (1971)
Banaszczyk, W., Litvak, A.E., Pajor, A., Szarek, S.J.: The flatness theorem for nonsymmetric convex bodies via the local theory of banach spaces. Math. oper. res. 24(3), 728–750 (1999)
Basu, A., Conforti, M., Cornuéjols, G., Weismantel, R., Weltge, S.: Optimality certificates for convex minimization and helly numbers. Oper. Res. Lett. 45(6), 671–674 (2017)
Basu, A., Conforti, M., Di Summa, M.: A geometric approach to cut-generating functions. Math. Program. 151(1), 153–189 (2015). https://doi.org/10.1007/s10107-015-0890-5
Blair, C.E., Jeroslow, R.G.: Constructive characterizations of the value-function of a mixed-integer program i. Discrete Appl. Math. 9(3), 217–233 (1984)
Blair, C.E., Jeroslow, R.G.: Constructive characterizations of the value function of a mixed-integer program ii. Discrete Appl. Math. 10(3), 227–240 (1985)
Borozan, V., Cornuéjols, G.: Minimal valid inequalities for integer constraints. Math. Oper. Res. 34(3), 538–546 (2009)
Cevallos, A., Weltge, S., Zenklusen, R.: Lifting linear extension complexity bounds to the mixed-integer setting. In: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms. pp. 788–807. SIAM (2018)
Codenotti, G., Santos, F.: Hollow polytopes of large width. Proc. Am. Math. Soc. 148(2), 835–850 (2020)
Dash, S., Dobbs, N.B., Günlük, O., Nowicki, T.J., Świrszcz, G.M.: Lattice-free sets, multi-branch split disjunctions, and mixed-integer programming. Math. Program. 145(1), 483–508 (2013). https://doi.org/10.1007/s10107-013-0654-z
Del Pia, A., Weismantel, R.: Relaxations of mixed integer sets from lattice-free polyhedra. Ann. Oper. Res. 240(1), 95–117 (2015). https://doi.org/10.1007/s10479-015-2024-0
Doolittle, J., Katthän, L., Nill, B., Santos, F.: Empty simplices of large width. arXiv preprint. arXiv:2103.14925 (2021)
Fukasawa, R., Poirrier, L., Xavier, Á.S.: The (not so) trivial lifting in two dimensions. Math. Prog. Comp. 11(2), 211–235 (2018). https://doi.org/10.1007/s12532-018-0146-5
Herr, K., Rehn, T., Schürmann, A.: On lattice-free orbit polytopes. Discrete Comput. Geom. 53(1), 144–172 (2014). https://doi.org/10.1007/s00454-014-9638-x
Hurkens, C.: Blowing up convex sets in the plane. Linear Algebra Appl. 134, 121–128 (1990)
Khinchine, A.: A quantitative formulation of kronecker’s theory of approximation. Izv. Akad. Nauk SSR Ser. Matem. 12(2), 113–122 (1948)
Lenstra, H.W., Jr.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)
Morán, R., Diego, A., Dey, S.S., Vielma, J.P.: A strong dual for conic mixed-integer programs. SIAM J. Optim. 22(3), 1136–1150 (2012)
Rudelson, M.: Distances between non-symmetric convex bodies and the \( mm^* \)-estimate. Positivity 24, 161–178 (2000)
Sebő, A.: An introduction to empty lattice simplices. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds.) IPCO 1999. LNCS, vol. 1610, pp. 400–414. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48777-8_30
Wolsey, L.A.: The b-hull of an integer program. Discrete Appl. Math. 3(3), 193–201 (1981)
Wolsey, L.A.: Integer programming duality: price functions and sensitivity analysis. Math. Program. 20(1), 173–195 (1981). https://doi.org/10.1007/BF01589344
Acknowledgements
The authors would like to thank Gennadiy Averkov and Paco Santos for valuable feedback and discussions on earlier stages of this work, and Amitabh Basu for discussions about applications of the Flatness theorem within optimization. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project number 451026932.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Mayrhofer, L., Schade, J., Weltge, S. (2022). Lattice-Free Simplices with Lattice Width \(2d - o(d)\). In: Aardal, K., Sanità, L. (eds) Integer Programming and Combinatorial Optimization. IPCO 2022. Lecture Notes in Computer Science, vol 13265. Springer, Cham. https://doi.org/10.1007/978-3-031-06901-7_28
Download citation
DOI: https://doi.org/10.1007/978-3-031-06901-7_28
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-06900-0
Online ISBN: 978-3-031-06901-7
eBook Packages: Computer ScienceComputer Science (R0)