Advertisement

Algorithmica

, Volume 81, Issue 9, pp 3494–3518 | Cite as

The Complexity of Optimization on Grids

  • Luis Barba
  • Malte MilatzEmail author
  • Jerri Nummenpalo
  • Xiaoming Sun
  • Antonis Thomas
  • Jialin Zhang
  • Zhijie Zhang
Original Research
  • 67 Downloads

Abstract

Unique-sink orientations of grids are models for linear optimization problems. Vertices of the grid represent possible solutions to the optimization problem; edge orientations indicate improving directions. The computational goal is to find the unique sink, representing the optimal solution. We study the query complexity of this model, where we consider two natural types of queries, vertex queries and edge queries. We describe a deterministic algorithm showing that the vertex query complexity of d-dimensional grids is \(O(n^{ \lceil d/2 \rceil })\). For the edge query complexity we obtain nearly the same bound, incurring only an \(n^{o(1)}\) overhead. Our algorithms rely on structural results with potential further applications in optimization theory.

Keywords

Unique-sink orientation Optimization Linear programming 

Notes

Acknowledgements

We would like to thank Bernd Gärtner for introducing the problem at the 2015 Gremo Workshop on Open Problems (GWOP), June 1–5, 2015, Feldis (GR), Switzerland. This work was supported in part by the National Natural Science Foundation of China Grants No. 61433014, 61832003, 61761136014, 61872334, 61502449, the 973 Program of China Grant No. 2016YFB1000201.

Supplementary material

References

  1. 1.
    Aggarwal, A., Klawe, M.M., Moran, S., Shor, P., Wilber, R.: Geometric applications of a matrix-searching algorithm. Algorithmica 2(1–4), 195–208 (1987).  https://doi.org/10.1007/BF01840359 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barba, L., Milatz, M., Nummenpalo, J., Thomas, A.: Deterministic algorithms for unique sink orientations of grids. In: Dinh, T.N., Thai, M.T. (eds.) Computing and Combinatorics, pp. 357–369. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-42634-1_29 CrossRefGoogle Scholar
  3. 3.
    Björner, A., Vergnas, M.L., Sturmfels, B., White, N., Ziegler, G.M.: Oriented matroids. Encyclopedia of Mathematics and its Applications, vol. 46, 2nd edn. Cambridge University Press, Cambridge (1999)Google Scholar
  4. 4.
    Chan, T.M.: Improved deterministic algorithms for linear programming in low dimensions. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’16, pp. 1213–1219. Society for Industrial and Applied Mathematics, Philadelphia (2016).  https://doi.org/10.1137/1.9781611974331.ch84
  5. 5.
    Chazelle, B., Matoušek, J.: On linear-time deterministic algorithms for optimization problems in fixed dimension. J. Algorithms 21(3), 579–597 (1996).  https://doi.org/10.1006/jagm.1996.0060 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Demaine, E.D., Langerman, S.: Optimizing a 2D function satisfying unimodality properties. In: Brodal, G.S., Leonardi, S. (eds.) Proceedings of ESA 2005, LNCS, vol. 3669, pp. 887–898. Springer, New York (2005).  https://doi.org/10.1007/11561071_78 Google Scholar
  7. 7.
    Felsner, S., Gärtner, B., Tschirschnitz, F.: Grid orientations, \((d, d + 2)\)-polytopes, and arrangements of pseudolines. Discrete Comput. Geom. 34(3), 411–437 (2005).  https://doi.org/10.1007/s00454-005-1187-x MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Galil, Z., Park, K.: Dynamic programming with convexity, concavity and sparsity. Theor. Comput. Sci. 92(1), 49–76 (1992).  https://doi.org/10.1016/0304-3975(92)90135-3 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gärtner, B., Matoušek, J., Rüst, L., Skovron, P.: Violator spaces: structure and algorithms. Discrete Appl. Math. (2008).  https://doi.org/10.1016/j.dam.2007.08.048 MathSciNetzbMATHGoogle Scholar
  10. 10.
    Gärtner, B., Morris Jr., W.D., Rüst, L.: Unique sink orientations of grids. Algorithmica 51(2), 200–235 (2008).  https://doi.org/10.1007/s00453-007-9090-x MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gärtner, B., Tschirschnitz, F., Welzl, E., Solymosi, J., Valtr, P.: One line and \(n\) points. Random Struct. Algorithm 23(4), 453–471 (2003).  https://doi.org/10.1002/rsa.10099 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Goodman, J.E., Pollack, R.: Upper bounds for configurations and polytopes in \(R^d\). Discrete Comput. Geom. 1(3), 219–227 (1986).  https://doi.org/10.1007/BF02187696 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grünbaum, B.: Convex Polytopes. Springer, New York (1967/2003).  https://doi.org/10.1007/978-1-4613-0019-9
  14. 14.
    Hansen, T.D., Paterson, M., Zwick, U.: Improved Upper Bounds for Random-Edge and Random-Jump on Abstract Cubes, pp. 874–881. Society for Industrial and Applied Mathematics, Philadelphia (2014).  https://doi.org/10.1137/1.9781611973402.65 zbMATHGoogle Scholar
  15. 15.
    Matoušek, J., Sharir, M., Welzl, E.: A subexponential bound for linear programming. Algorithmica 16(4/5), 498–516 (1996).  https://doi.org/10.1007/BF01940877 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Matoušek, J., Szabó, T.: RANDOM EDGE can be exponential on abstract cubes. Adv. Math. 204(1), 262–277 (2006).  https://doi.org/10.1016/j.aim.2005.05.021 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mityagin, A.: On the complexity of finding a local maximum of functions on discrete planar subsets. In: Alt, H., Habib, M. (eds.) Proceedings of STACS 2003, LNCS, vol. 2607, pp. 203–211. Springer, New York (2003)CrossRefGoogle Scholar
  18. 18.
    Schurr, I., Szabó, T.: Finding the sink takes some time: an almost quadratic lower bound for finding the sink of unique sink oriented cubes. Discrete Comput. Geom. 31(4), 627–642 (2004).  https://doi.org/10.1007/s00454-003-0813-8 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Schurr, I., Szabó, T.: Jumping doesn’t help in abstract cubes. In: Jünger, M., Kaibel, V. (eds.) Integer Programming and Combinatorial Optimization, pp. 225–235. Springer, New York (2005).  https://doi.org/10.1007/11496915_17 CrossRefGoogle Scholar
  20. 20.
    Szabó, T., Welzl, E.: Unique sink orientations of cubes. In: Proceedings of FOCS 2001, pp. 547–555. IEEE Computer Society (2001).  https://doi.org/10.1109/SFCS.2001.959931
  21. 21.
    Tschirschnitz, F.: LP-related properties of polytopes with few facets. Ph.D. thesis, ETH Zürich (2003).  https://doi.org/10.3929/ethz-a-004624080

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Computer ScienceETH ZürichZurichSwitzerland
  2. 2.ZurichSwitzerland
  3. 3.CAS Key Lab of Network Data Science and Technology, Institute of Computing TechnologyChinese Academy of Sciences (CAS)BeijingChina
  4. 4.University of Chinese Academy of SciencesBeijingChina

Personalised recommendations